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      "heading": "Preface / 前言 {.unnumbered}",
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    {
      "book": "generic",
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      "heading": "What your materials contribute / 资料如何进入本书",
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      "book": "generic",
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      "heading": "How to study / 如何学习",
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    {
      "book": "generic",
      "line": 31,
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      "book": "generic",
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      "heading": "2.2 Pre-existing nuclei change the problem / 预存气核改变问题性质",
      "page": "course/chapters/G02-nucleation.html",
      "anchor": "s-2-2"
    },
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      "book": "generic",
      "line": 163,
      "heading": "2.3 Static instability is not dynamic activation / 静态失稳不等于动态激活",
      "page": "course/chapters/G02-nucleation.html",
      "anchor": "s-2-3"
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      "book": "generic",
      "line": 173,
      "heading": "3. Deriving the Rayleigh–Plesset equation / 推导Rayleigh–Plesset方程",
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "anchor": "s-3"
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      "book": "generic",
      "line": 175,
      "heading": "3.1 Assumptions before algebra / 先写假设，再做代数",
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "anchor": "s-3-1"
    },
    {
      "book": "generic",
      "line": 181,
      "heading": "3.2 Continuity determines the velocity field / 连续性决定速度场",
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "anchor": "s-3-2"
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      "book": "generic",
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      "page": "course/chapters/G03-rayleigh-plesset.html",
      "anchor": "s-3-3"
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    {
      "book": "generic",
      "line": 208,
      "heading": "3.4 Normal stress supplies capillarity and viscosity / 法向应力给出毛细项和黏性项",
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "anchor": "s-3-4"
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      "book": "generic",
      "line": 227,
      "heading": "3.5 Closing the equation / 方程如何闭合",
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "anchor": "s-3-5"
    },
    {
      "book": "generic",
      "line": 245,
      "heading": "3.6 The same equation from theoretical mechanics / 从理论力学重新得到同一方程",
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "anchor": "s-3-6"
    },
    {
      "book": "generic",
      "line": 271,
      "heading": "4. Gas, vapor, heat, and mass / 气体、蒸气、热量与质量",
      "page": "course/chapters/G04-heat-and-mass.html",
      "anchor": "s-4"
    },
    {
      "book": "generic",
      "line": 273,
      "heading": "4.1 A pressure law is a thermodynamic assumption / 压力定律是一项热力学假设",
      "page": "course/chapters/G04-heat-and-mass.html",
      "anchor": "s-4-1"
    },
    {
      "book": "generic",
      "line": 283,
      "heading": "4.2 Vapor pressure and vapor mass are different closures / 蒸气压与蒸气质量是不同闭合量",
      "page": "course/chapters/G04-heat-and-mass.html",
      "anchor": "s-4-2"
    },
    {
      "book": "generic",
      "line": 293,
      "heading": "4.3 The Stefan condition and interface kinematics / Stefan条件与界面运动学",
      "page": "course/chapters/G04-heat-and-mass.html",
      "anchor": "s-4-3"
    },
    {
      "book": "generic",
      "line": 309,
      "heading": "4.4 Inertia-limited versus heat-limited growth / 惯性控制与热控制的增长",
      "page": "course/chapters/G04-heat-and-mass.html",
      "anchor": "s-4-4"
    },
    {
      "book": "generic",
      "line": 319,
      "heading": "4.5 Dissolved gas and shells / 溶解气体与壳层",
      "page": "course/chapters/G04-heat-and-mass.html",
      "anchor": "s-4-5"
    },
    {
      "book": "generic",
      "line": 330,
      "heading": "5. Inertial collapse, energy focusing, and rebound / 惯性塌缩、能量集中与回弹",
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "anchor": "s-5"
    },
    {
      "book": "generic",
      "line": 332,
      "heading": "5.1 The empty-cavity idealization / 空腔理想化",
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "anchor": "s-5-1"
    },
    {
      "book": "generic",
      "line": 344,
      "heading": "5.2 Integrating once / 第一次积分",
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "anchor": "s-5-2"
    },
    {
      "book": "generic",
      "line": 364,
      "heading": "5.3 The Rayleigh collapse time / Rayleigh塌缩时间",
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "anchor": "s-5-3"
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    {
      "book": "generic",
      "line": 388,
      "heading": "5.4 The near-collapse power law / 接近塌缩时的幂律",
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "anchor": "s-5-4"
    },
    {
      "book": "generic",
      "line": 400,
      "heading": "5.5 Why real bubbles rebound / 真实气泡为何回弹",
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "anchor": "s-5-5"
    },
    {
      "book": "generic",
      "line": 410,
      "heading": "5.6 A conserved mechanical quantity for testing / 用于检验的守恒力学量",
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "anchor": "s-5-6"
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      "book": "generic",
      "line": 423,
      "heading": "6. Bubble oscillations as vibration theory / 用振动理论理解气泡振荡",
      "page": "course/chapters/G06-oscillations.html",
      "anchor": "s-6"
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    {
      "book": "generic",
      "line": 425,
      "heading": "6.1 Linearization around equilibrium / 平衡态附近的线性化",
      "page": "course/chapters/G06-oscillations.html",
      "anchor": "s-6-1"
    },
    {
      "book": "generic",
      "line": 456,
      "heading": "6.2 Harmonic response and phase / 简谐响应与相位",
      "page": "course/chapters/G06-oscillations.html",
      "anchor": "s-6-2"
    },
    {
      "book": "generic",
      "line": 471,
      "heading": "6.3 Three damping mechanisms / 三种阻尼机制",
      "page": "course/chapters/G06-oscillations.html",
      "anchor": "s-6-3"
    },
    {
      "book": "generic",
      "line": 481,
      "heading": "6.4 What becomes nonlinear first / 哪些效应首先变得非线性",
      "page": "course/chapters/G06-oscillations.html",
      "anchor": "s-6-4"
    },
    {
      "book": "generic",
      "line": 491,
      "heading": "7. Compressibility, sound, and model limits / 可压缩性、声辐射与模型极限",
      "page": "course/chapters/G07-compressibility.html",
      "anchor": "s-7"
    },
    {
      "book": "generic",
      "line": 493,
      "heading": "7.1 Two different small parameters / 两个不同的小参数",
      "page": "course/chapters/G07-compressibility.html",
      "anchor": "s-7-1"
    },
    {
      "book": "generic",
      "line": 503,
      "heading": "7.2 Keller–Miksis with constant ambient pressure / 恒定环境压力下的Keller–Miksis方程",
      "page": "course/chapters/G07-compressibility.html",
      "anchor": "s-7-2"
    },
    {
      "book": "generic",
      "line": 522,
      "heading": "7.3 Solving the implicit acceleration correctly / 正确求解隐式加速度",
      "page": "course/chapters/G07-compressibility.html",
      "anchor": "s-7-3"
    },
    {
      "book": "generic",
      "line": 547,
      "heading": "7.4 Gilmore's enthalpy formulation / Gilmore的焓形式",
      "page": "course/chapters/G07-compressibility.html",
      "anchor": "s-7-4"
    },
    {
      "book": "generic",
      "line": 577,
      "heading": "7.5 What the far-field pressure means / 远场压力究竟表示什么",
      "page": "course/chapters/G07-compressibility.html",
      "anchor": "s-7-5"
    },
    {
      "book": "generic",
      "line": 602,
      "heading": "7.6 Stop before you manufacture a result / 在制造伪结果之前停止",
      "page": "course/chapters/G07-compressibility.html",
      "anchor": "s-7-6"
    },
    {
      "book": "generic",
      "line": 961,
      "heading": "13. A reproducible numerical laboratory / 可复现数值实验",
      "page": "course/chapters/G13-numerical-laboratory.html",
      "anchor": "s-13"
    },
    {
      "book": "generic",
      "line": 963,
      "heading": "13.1 What the supplied program actually computes / 配套程序实际计算什么",
      "page": "course/chapters/G13-numerical-laboratory.html",
      "anchor": "s-13-1"
    },
    {
      "book": "generic",
      "line": 973,
      "heading": "13.2 Scaling the equations before integration / 积分前先进行尺度化",
      "page": "course/chapters/G13-numerical-laboratory.html",
      "anchor": "s-13-2"
    },
    {
      "book": "generic",
      "line": 990,
      "heading": "13.3 Running the laboratory / 运行实验",
      "page": "course/chapters/G13-numerical-laboratory.html",
      "anchor": "s-13-3"
    },
    {
      "book": "generic",
      "line": 1023,
      "heading": "13.4 Verification results supplied with this edition / 本版附带的验证结果",
      "page": "course/chapters/G13-numerical-laboratory.html",
      "anchor": "s-13-4"
    },
    {
      "book": "generic",
      "line": 1045,
      "heading": "13.5 What not to do when the solver becomes difficult / 求解困难时不要做什么",
      "page": "course/chapters/G13-numerical-laboratory.html",
      "anchor": "s-13-5"
    },
    {
      "book": "generic",
      "line": 1055,
      "heading": "13.6 Moving from ODEs to field simulation / 从常微分方程走向场仿真",
      "page": "course/chapters/G13-numerical-laboratory.html",
      "anchor": "s-13-6"
    },
    {
      "book": "generic",
      "line": 609,
      "heading": "8. Shape, boundaries, and translation / 形状、边界与平移",
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "anchor": "s-8"
    },
    {
      "book": "generic",
      "line": 611,
      "heading": "8.1 A radius is not a shape / 半径不能代表全部形状",
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "anchor": "s-8-1"
    },
    {
      "book": "generic",
      "line": 633,
      "heading": "8.2 Stand-off distance and image intuition / 离壁距离与镜像直觉",
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "anchor": "s-8-2"
    },
    {
      "book": "generic",
      "line": 647,
      "heading": "8.3 Elastic walls have their own clock / 弹性壁面有自己的时间尺度",
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "anchor": "s-8-3"
    },
    {
      "book": "generic",
      "line": 661,
      "heading": "8.4 Pressure gradients and bubble translation / 压力梯度与气泡平移",
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "anchor": "s-8-4"
    },
    {
      "book": "generic",
      "line": 674,
      "heading": "8.5 What a spherical model can still contribute / 球形模型仍能贡献什么",
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "anchor": "s-8-5"
    },
    {
      "book": "generic",
      "line": 680,
      "heading": "9. Jets, pressure impulse, and liquid transfer / 射流、压力冲量与液体转印",
      "page": "course/chapters/G09-jets-and-transfer.html",
      "anchor": "s-9"
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    {
      "book": "generic",
      "line": 682,
      "heading": "9.1 Two different objects called a jet / 两种不同的“射流”",
      "page": "course/chapters/G09-jets-and-transfer.html",
      "anchor": "s-9-1"
    },
    {
      "book": "generic",
      "line": 692,
      "heading": "9.2 Deriving pressure impulse / 推导压力冲量",
      "page": "course/chapters/G09-jets-and-transfer.html",
      "anchor": "s-9-2"
    },
    {
      "book": "generic",
      "line": 713,
      "heading": "9.3 Impulse and energy answer different questions / 冲量与能量回答不同问题",
      "page": "course/chapters/G09-jets-and-transfer.html",
      "anchor": "s-9-3"
    },
    {
      "book": "generic",
      "line": 723,
      "heading": "9.4 Capillarity and breakup / 毛细作用与断裂",
      "page": "course/chapters/G09-jets-and-transfer.html",
      "anchor": "s-9-4"
    },
    {
      "book": "generic",
      "line": 733,
      "heading": "9.5 Printing windows and failure modes / 打印窗口与失效模式",
      "page": "course/chapters/G09-jets-and-transfer.html",
      "anchor": "s-9-5"
    },
    {
      "book": "generic",
      "line": 743,
      "heading": "10. How lasers create bubbles and loading / 激光如何产生气泡与载荷",
      "page": "course/chapters/G10-laser-source-pathways.html",
      "anchor": "s-10"
    },
    {
      "book": "generic",
      "line": 745,
      "heading": "10.1 Start from the energy pathway / 从能量传递路径开始",
      "page": "course/chapters/G10-laser-source-pathways.html",
      "anchor": "s-10-1"
    },
    {
      "book": "generic",
      "line": 751,
      "heading": "10.2 Optical breakdown / 光学击穿",
      "page": "course/chapters/G10-laser-source-pathways.html",
      "anchor": "s-10-2"
    },
    {
      "book": "generic",
      "line": 761,
      "heading": "10.3 Absorber heating and thermocavitation / 吸收体加热与热空化",
      "page": "course/chapters/G10-laser-source-pathways.html",
      "anchor": "s-10-3"
    },
    {
      "book": "generic",
      "line": 771,
      "heading": "10.4 Thermoelastic sound and confinement / 热弹性声波与约束条件",
      "page": "course/chapters/G10-laser-source-pathways.html",
      "anchor": "s-10-4"
    },
    {
      "book": "generic",
      "line": 784,
      "heading": "10.5 Reading the Lee–Guo experiment correctly / 正确理解Lee–Guo实验",
      "page": "course/chapters/G10-laser-source-pathways.html",
      "anchor": "s-10-5"
    },
    {
      "book": "generic",
      "line": 798,
      "heading": "10.6 Blisters, stress waves, and bubbles / 鼓包、应力波与气泡",
      "page": "course/chapters/G10-laser-source-pathways.html",
      "anchor": "s-10-6"
    },
    {
      "book": "generic",
      "line": 809,
      "heading": "11. Phase-change perfluorocarbon droplets / 相变全氟碳液滴",
      "page": "course/chapters/G11-phase-change-droplets.html",
      "anchor": "s-11"
    },
    {
      "book": "generic",
      "line": 811,
      "heading": "11.1 A droplet is not yet a bubble / 液滴还不是气泡",
      "page": "course/chapters/G11-phase-change-droplets.html",
      "anchor": "s-11-1"
    },
    {
      "book": "generic",
      "line": 821,
      "heading": "11.2 Two interfaces, two pressure jumps / 两个界面与两个压差",
      "page": "course/chapters/G11-phase-change-droplets.html",
      "anchor": "s-11-2"
    },
    {
      "book": "generic",
      "line": 831,
      "heading": "11.3 Acoustic and optical activation / 声激活与光激活",
      "page": "course/chapters/G11-phase-change-droplets.html",
      "anchor": "s-11-3"
    },
    {
      "book": "generic",
      "line": 841,
      "heading": "11.4 A mass-conservation estimate of expansion / 用质量守恒估计膨胀",
      "page": "course/chapters/G11-phase-change-droplets.html",
      "anchor": "s-11-4"
    },
    {
      "book": "generic",
      "line": 859,
      "heading": "11.5 Vaporization, persistence, and inertial cavitation / 汽化、持续存在与惯性空化",
      "page": "course/chapters/G11-phase-change-droplets.html",
      "anchor": "s-11-5"
    },
    {
      "book": "generic",
      "line": 869,
      "heading": "11.6 A terminology trap in your reading list / 阅读清单中的术语陷阱",
      "page": "course/chapters/G11-phase-change-droplets.html",
      "anchor": "s-11-6"
    },
    {
      "book": "generic",
      "line": 875,
      "heading": "12. Bubble-driven films, membranes, and detachment / 气泡驱动薄膜、膜结构与脱离",
      "page": "course/chapters/G12-films-and-detachment.html",
      "anchor": "s-12"
    },
    {
      "book": "generic",
      "line": 877,
      "heading": "12.1 A liquid pressure is only one part of structural loading / 液体压力只是结构载荷的一部分",
      "page": "course/chapters/G12-films-and-detachment.html",
      "anchor": "s-12-1"
    },
    {
      "book": "generic",
      "line": 887,
      "heading": "12.2 Plate and membrane limits / 板与膜的极限",
      "page": "course/chapters/G12-films-and-detachment.html",
      "anchor": "s-12-2"
    },
    {
      "book": "generic",
      "line": 904,
      "heading": "12.3 Modal projection shows why load shape matters / 模态投影解释载荷分布的重要性",
      "page": "course/chapters/G12-films-and-detachment.html",
      "anchor": "s-12-3"
    },
    {
      "book": "generic",
      "line": 930,
      "heading": "12.4 Added mass without double counting / 不重复计算附加质量",
      "page": "course/chapters/G12-films-and-detachment.html",
      "anchor": "s-12-4"
    },
    {
      "book": "generic",
      "line": 940,
      "heading": "12.5 Detachment is a fracture problem / 脱离是断裂问题",
      "page": "course/chapters/G12-films-and-detachment.html",
      "anchor": "s-12-5"
    },
    {
      "book": "generic",
      "line": 954,
      "heading": "12.6 A useful hierarchy of questions / 有效的问题层次",
      "page": "course/chapters/G12-films-and-detachment.html",
      "anchor": "s-12-6"
    },
    {
      "book": "generic",
      "line": 1069,
      "heading": "14. Measurement, inference, and uncertainty / 测量、反演与不确定性",
      "page": "course/chapters/G14-measurement-and-inference.html",
      "anchor": "s-14"
    },
    {
      "book": "generic",
      "line": 1071,
      "heading": "14.1 Define the observable before fitting / 拟合前先定义可观测量",
      "page": "course/chapters/G14-measurement-and-inference.html",
      "anchor": "s-14-1"
    },
    {
      "book": "generic",
      "line": 1081,
      "heading": "14.2 Derivatives amplify image noise / 求导会放大图像噪声",
      "page": "course/chapters/G14-measurement-and-inference.html",
      "anchor": "s-14-2"
    },
    {
      "book": "generic",
      "line": 1091,
      "heading": "14.3 Identifiability: several stories can fit one curve / 可辨识性：一条曲线可能对应多种解释",
      "page": "course/chapters/G14-measurement-and-inference.html",
      "anchor": "s-14-3"
    },
    {
      "book": "generic",
      "line": 1108,
      "heading": "14.4 Pressure measurements are filtered observations / 压力测量是经过滤波的观测",
      "page": "course/chapters/G14-measurement-and-inference.html",
      "anchor": "s-14-4"
    },
    {
      "book": "generic",
      "line": 1118,
      "heading": "14.5 Calibration is not validation / 标定不是验证",
      "page": "course/chapters/G14-measurement-and-inference.html",
      "anchor": "s-14-5"
    },
    {
      "book": "generic",
      "line": 1128,
      "heading": "15. Capstone: a liquid–PVC–film research model / 综合课题：液体—PVC—薄膜研究模型",
      "page": "course/chapters/G15-liquid-pvc-film-capstone.html",
      "anchor": "s-15"
    },
    {
      "book": "generic",
      "line": 1130,
      "heading": "15.1 Define the proposed system, not an imagined reproduction / 定义研究体系，而不是想当然地复现",
      "page": "course/chapters/G15-liquid-pvc-film-capstone.html",
      "anchor": "s-15-1"
    },
    {
      "book": "generic",
      "line": 1140,
      "heading": "15.2 Stage A: the smallest useful baseline / 阶段A：最小但有用的基础模型",
      "page": "course/chapters/G15-liquid-pvc-film-capstone.html",
      "anchor": "s-15-2"
    },
    {
      "book": "generic",
      "line": 1154,
      "heading": "15.3 Stage B: replace the effective source with a physical one / 阶段B：用物理源替代有效源",
      "page": "course/chapters/G15-liquid-pvc-film-capstone.html",
      "anchor": "s-15-3"
    },
    {
      "book": "generic",
      "line": 1164,
      "heading": "15.4 Stage C: introduce release, then capture / 阶段C：先加入释放，再加入捕获",
      "page": "course/chapters/G15-liquid-pvc-film-capstone.html",
      "anchor": "s-15-4"
    },
    {
      "book": "generic",
      "line": 1174,
      "heading": "15.5 Dimensionless organization / 无量纲组织方式",
      "page": "course/chapters/G15-liquid-pvc-film-capstone.html",
      "anchor": "s-15-5"
    },
    {
      "book": "generic",
      "line": 1184,
      "heading": "15.6 A research claim that can actually be tested / 能够真正检验的科研命题",
      "page": "course/chapters/G15-liquid-pvc-film-capstone.html",
      "anchor": "s-15-6"
    },
    {
      "book": "generic",
      "line": 1199,
      "heading": "16. Problems with worked solutions / 习题与详细解答",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16"
    },
    {
      "book": "generic",
      "line": 1205,
      "heading": "16.1 Laplace pressure at three scales / 三个尺度下的Laplace压力",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-1"
    },
    {
      "book": "generic",
      "line": 1215,
      "heading": "16.2 Where the liquid kinetic energy resides / 液体动能分布在哪里",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-2"
    },
    {
      "book": "generic",
      "line": 1225,
      "heading": "16.3 Track a liquid particle / 跟踪液体质点",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-3"
    },
    {
      "book": "generic",
      "line": 1235,
      "heading": "16.4 Diagnose a Bernoulli error / 诊断Bernoulli推导错误",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-4"
    },
    {
      "book": "generic",
      "line": 1245,
      "heading": "16.5 An isothermal Blake threshold / 等温Blake阈值",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-5"
    },
    {
      "book": "generic",
      "line": 1255,
      "heading": "16.6 Probability depends on observation volume / 概率依赖观测体积",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-6"
    },
    {
      "book": "generic",
      "line": 1265,
      "heading": "16.7 Long-time inertial growth speed / 长时间惯性增长速度",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-7"
    },
    {
      "book": "generic",
      "line": 1275,
      "heading": "16.8 Collapse-time scaling / 塌缩时间尺度",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-8"
    },
    {
      "book": "generic",
      "line": 1285,
      "heading": "16.9 Speed at half maximum radius / 半最大半径处的速度",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-9"
    },
    {
      "book": "generic",
      "line": 1295,
      "heading": "16.10 When the incompressible collapse becomes suspect / 不可压缩塌缩何时值得怀疑",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-10"
    },
    {
      "book": "generic",
      "line": 1305,
      "heading": "16.11 Static compliance of a microbubble / 微气泡的静态柔度",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-11"
    },
    {
      "book": "generic",
      "line": 1315,
      "heading": "16.12 Viscosity versus stiffness / 黏度与刚度",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-12"
    },
    {
      "book": "generic",
      "line": 1325,
      "heading": "16.13 The isothermal gas potential / 等温气体势",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-13"
    },
    {
      "book": "generic",
      "line": 1335,
      "heading": "16.14 The hidden acceleration in Keller–Miksis / Keller–Miksis中的隐藏加速度",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-14"
    },
    {
      "book": "generic",
      "line": 1345,
      "heading": "16.15 Radial frequency versus shape frequency / 径向频率与形状频率",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-15"
    },
    {
      "book": "generic",
      "line": 1355,
      "heading": "16.16 A uniform pressure impulse / 均匀压力冲量",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-16"
    },
    {
      "book": "generic",
      "line": 1365,
      "heading": "16.17 Equal impulse, zero residual oscillation / 相同冲量却无残余振动",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-17"
    },
    {
      "book": "generic",
      "line": 1375,
      "heading": "16.18 A necessary adhesion-energy scale / 黏附能的必要尺度",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-18"
    },
    {
      "book": "generic",
      "line": 1385,
      "heading": "16.19 Droplet expansion by mass conservation / 由质量守恒估计液滴膨胀",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-19"
    },
    {
      "book": "generic",
      "line": 1395,
      "heading": "16.20 Geometrical spreading of acoustic energy / 声能的几何扩散",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-20"
    },
    {
      "book": "generic",
      "line": 1405,
      "heading": "16.21 A false temperature prediction / 错误的温度预测",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-21"
    },
    {
      "book": "generic",
      "line": 1415,
      "heading": "16.22 A mesh-convergence calculation / 网格收敛计算",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-22"
    },
    {
      "book": "generic",
      "line": 1425,
      "heading": "16.23 A nonidentifiable gas parameterization / 不可辨识的气体参数化",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-23"
    },
    {
      "book": "generic",
      "line": 1435,
      "heading": "16.24 Why more frames do not automatically fix acceleration / 为什么更多图像帧不自动解决加速度问题",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-24"
    },
    {
      "book": "generic",
      "line": 1445,
      "heading": "16.25 Source-mechanism classification / 源机制分类",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-25"
    },
    {
      "book": "generic",
      "line": 1455,
      "heading": "16.26 A terminology audit / 术语审查",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-26"
    },
    {
      "book": "generic",
      "line": 1465,
      "heading": "16.27 Detect an overconstrained interface / 识别过度约束界面",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-27"
    },
    {
      "book": "generic",
      "line": 1475,
      "heading": "16.28 Avoid double-counting fluid inertia / 避免重复计算流体惯性",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-28"
    },
    {
      "book": "generic",
      "line": 1485,
      "heading": "16.29 Design a discriminating experiment / 设计具有区分能力的实验",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-29"
    },
    {
      "book": "generic",
      "line": 1495,
      "heading": "16.30 Build a falsifiable capstone study / 构建可证伪综合研究",
      "page": "course/chapters/G16-original-worked-problems.html",
      "anchor": "s-16-30"
    },
    {
      "book": "project-theory",
      "line": 9,
      "heading": "Reader’s orientation / 阅读导引",
      "page": "course/chapters/T01-optical-deposition.html",
      "anchor": "reader-s-orientation"
    },
    {
      "book": "project-theory",
      "line": 45,
      "heading": "1. Optical deposition / 光能沉积",
      "page": "course/chapters/T01-optical-deposition.html",
      "anchor": "s-1"
    },
    {
      "book": "project-theory",
      "line": 55,
      "heading": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "page": "course/chapters/T01-optical-deposition.html",
      "anchor": "s-1-1"
    },
    {
      "book": "project-theory",
      "line": 131,
      "heading": "1.2 From pulse energy to a time-dependent source / 从脉冲能量到时变热源",
      "page": "course/chapters/T01-optical-deposition.html",
      "anchor": "s-1-2"
    },
    {
      "book": "project-theory",
      "line": 171,
      "heading": "1.3 From surface intensity to volumetric heating / 从表面光强到体积热源",
      "page": "course/chapters/T01-optical-deposition.html",
      "anchor": "s-1-3"
    },
    {
      "book": "project-theory",
      "line": 240,
      "heading": "1.4 Local dose, peak rate, and temperature rise / 局部剂量、峰值速率与温升",
      "page": "course/chapters/T01-optical-deposition.html",
      "anchor": "s-1-4"
    },
    {
      "book": "project-theory",
      "line": 290,
      "heading": "1.5 Why enough energy does not guarantee a bubble / 为什么能量足够并不保证气泡形成",
      "page": "course/chapters/T01-optical-deposition.html",
      "anchor": "s-1-5"
    },
    {
      "book": "project-theory",
      "line": 322,
      "heading": "2. Activation / 激活",
      "page": "course/chapters/T02-activation-statistics.html",
      "anchor": "s-2"
    },
    {
      "book": "project-theory",
      "line": 332,
      "heading": "2.1 Why use a logistic model? / 为什么使用逻辑概率模型？",
      "page": "course/chapters/T02-activation-statistics.html",
      "anchor": "s-2-1"
    },
    {
      "book": "project-theory",
      "line": 387,
      "heading": "2.2 From one droplet to many droplets / 从单个液滴到液滴群体",
      "page": "course/chapters/T02-activation-statistics.html",
      "anchor": "s-2-2"
    },
    {
      "book": "project-theory",
      "line": 447,
      "heading": "2.3 Adding the detector / 加入探测器模型",
      "page": "course/chapters/T02-activation-statistics.html",
      "anchor": "s-2-3"
    },
    {
      "book": "project-theory",
      "line": 507,
      "heading": "3. Bubble dynamics / 气泡动力学",
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "anchor": "s-3"
    },
    {
      "book": "project-theory",
      "line": 525,
      "heading": "3.1 Continuity determines the liquid velocity / 连续性方程确定液体速度",
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "anchor": "s-3-1"
    },
    {
      "book": "project-theory",
      "line": 567,
      "heading": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "anchor": "s-3-2"
    },
    {
      "book": "project-theory",
      "line": 647,
      "heading": "3.3 Applying the wall condition gives Rayleigh–Plesset / 由壁面条件得到 Rayleigh–Plesset 方程",
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "anchor": "s-3-3"
    },
    {
      "book": "project-theory",
      "line": 709,
      "heading": "Why this equation does not yet predict laser-induced growth / 为什么该方程还不能预测激光诱导增长",
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "anchor": "why-this-equation-does-not-yet-predict-laser-induced-growth"
    },
    {
      "book": "project-theory",
      "line": 725,
      "heading": "3.4 Deriving the Rayleigh collapse time / 推导 Rayleigh 塌缩时间",
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "anchor": "s-3-4"
    },
    {
      "book": "project-theory",
      "line": 793,
      "heading": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "anchor": "s-3-5"
    },
    {
      "book": "project-theory",
      "line": 865,
      "heading": "3.6 Mapping bubble pressure onto a load disk / 将气泡压力映射至载荷圆盘",
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "anchor": "s-3-6"
    },
    {
      "book": "project-theory",
      "line": 913,
      "heading": "3.7 What the signed pressure does—and does not—mean / 带符号压力的含义与局限",
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "anchor": "s-3-7"
    },
    {
      "book": "project-theory",
      "line": 951,
      "heading": "4. Structural motion and interfacial release / 结构运动与界面释放",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "s-4"
    },
    {
      "book": "project-theory",
      "line": 961,
      "heading": "4.1 Reducing the laminate to one generalized coordinate / 将层合结构简化为一个广义坐标",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "s-4-1"
    },
    {
      "book": "project-theory",
      "line": 989,
      "heading": "Modal mass / 模态质量",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "modal-mass"
    },
    {
      "book": "project-theory",
      "line": 1015,
      "heading": "Modal stiffness / 模态刚度",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "modal-stiffness"
    },
    {
      "book": "project-theory",
      "line": 1051,
      "heading": "Generalized load / 广义载荷",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "generalized-load"
    },
    {
      "book": "project-theory",
      "line": 1085,
      "heading": "4.2 Why load timing matters / 为什么载荷时序很重要",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "s-4-2"
    },
    {
      "book": "project-theory",
      "line": 1107,
      "heading": "4.3 Why equal total impulse can also fail spatially / 为什么总冲量相同仍可能产生不同的空间响应",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "s-4-3"
    },
    {
      "book": "project-theory",
      "line": 1145,
      "heading": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "s-4-4"
    },
    {
      "book": "project-theory",
      "line": 1207,
      "heading": "4.5 From transmitted loading to cohesive failure / 从传递载荷到黏聚失效",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "s-4-5"
    },
    {
      "book": "project-theory",
      "line": 1251,
      "heading": "Cohesive-zone length scale / 黏聚区长度尺度",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "cohesive-zone-length-scale"
    },
    {
      "book": "project-theory",
      "line": 1287,
      "heading": "4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值",
      "page": "course/chapters/T04-structural-response-and-release.html",
      "anchor": "s-4-6"
    },
    {
      "book": "project-theory",
      "line": 1329,
      "heading": "5. Sensitivity and the next measurements / 敏感性与下一步测量",
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "anchor": "s-5"
    },
    {
      "book": "project-theory",
      "line": 1339,
      "heading": "5.1 What a Spearman ranking measures / Spearman 排序衡量什么",
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "anchor": "s-5-1"
    },
    {
      "book": "project-theory",
      "line": 1355,
      "heading": "5.2 Optical sensitivity follows directly from the temperature scale / 光学敏感性直接来自温升尺度",
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "anchor": "s-5-2"
    },
    {
      "book": "project-theory",
      "line": 1379,
      "heading": "5.3 Bubble-load sensitivity follows from acceleration scaling / 气泡载荷敏感性来自加速度尺度",
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "anchor": "s-5-3"
    },
    {
      "book": "project-theory",
      "line": 1419,
      "heading": "5.4 Interface toughness enters the availability ratios explicitly / 界面韧度显式进入可用性比值",
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "anchor": "s-5-4"
    },
    {
      "book": "project-theory",
      "line": 1437,
      "heading": "5.5 Why the measurement order matters / 为什么测量顺序很重要",
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "anchor": "s-5-5"
    },
    {
      "book": "project-theory",
      "line": 1463,
      "heading": "The complete picture / 完整图景",
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "anchor": "the-complete-picture"
    },
    {
      "book": "project-theory",
      "line": 1498,
      "heading": "Source notes / 来源说明",
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "anchor": "source-notes"
    },
    {
      "book": "generic",
      "line": 1506,
      "heading": "Appendix A. Symbols, dimensionless form, and model map / 附录A：符号、无量纲形式与模型地图 {.unnumbered}",
      "page": "course/references/notation-and-model-map.html",
      "anchor": "appendix-a-symbols-dimensionless-form-and-model-map"
    },
    {
      "book": "generic",
      "line": 1508,
      "heading": "A.1 Core notation / 核心符号",
      "page": "course/references/notation-and-model-map.html",
      "anchor": "s-a-1"
    },
    {
      "book": "generic",
      "line": 1534,
      "heading": "A.2 Dimensionless Rayleigh–Plesset equation / 无量纲Rayleigh–Plesset方程",
      "page": "course/references/notation-and-model-map.html",
      "anchor": "s-a-2"
    },
    {
      "book": "generic",
      "line": 1554,
      "heading": "A.3 The model map / 模型地图",
      "page": "course/references/notation-and-model-map.html",
      "anchor": "s-a-3"
    },
    {
      "book": "generic",
      "line": 1564,
      "heading": "A.4 Recommended companion textbook / 推荐配套教材",
      "page": "course/references/notation-and-model-map.html",
      "anchor": "s-a-4"
    },
    {
      "book": "generic",
      "line": 1575,
      "heading": "Appendix B. Annotated atlas of your 35 papers / 附录B：所提供35篇论文的注释地图 {.unnumbered}",
      "page": "course/references/paper-atlas.html",
      "anchor": "appendix-b-annotated-atlas-of-your-35-papers"
    },
    {
      "book": "generic",
      "line": 1585,
      "heading": "S72. Laser-driven noncontact bubble transfer printing via a hydrogel composite stamp {#src-s72 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s72"
    },
    {
      "book": "generic",
      "line": 1599,
      "heading": "S250. Laser-induced droplet deformation: curvature inversion explained from instantaneous pressure impulse {#src-s250 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s250"
    },
    {
      "book": "generic",
      "line": 1613,
      "heading": "S251. Nozzle-Free Printing of CNT Electronics Using Laser-Generated Focused Ultrasound {#src-s251 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s251"
    },
    {
      "book": "generic",
      "line": 1627,
      "heading": "S252. Super-Resolution Ultrasound Imaging in Vivo with Transient Laser-Activated Nanodroplets {#src-s252 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s252"
    },
    {
      "book": "generic",
      "line": 1641,
      "heading": "S339. Dynamics of laser-induced cavitation bubbles near an elastic boundary {#src-s339 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s339"
    },
    {
      "book": "generic",
      "line": 1655,
      "heading": "S340. Shock-wave-induced Thin-film Delamination (SWIFD): A Non-thermal Structuring Method of Functional Layers {#src-s340 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s340"
    },
    {
      "book": "generic",
      "line": 1669,
      "heading": "S343. Cavitation bubble evidence in BA-LIFT processes {#src-s343 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s343"
    },
    {
      "book": "generic",
      "line": 1683,
      "heading": "S345. Destructive mechanisms in laser induced forward transfer {#src-s345 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s345"
    },
    {
      "book": "generic",
      "line": 1697,
      "heading": "S346. Time-resolved imaging of the laser forward transfer of liquids {#src-s346 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s346"
    },
    {
      "book": "generic",
      "line": 1711,
      "heading": "S348. Nanodroplet vaporization with pulsed-laser excitation repeatedly amplifies photoacoustic signals at low vaporization thresholds {#src-s348 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s348"
    },
    {
      "book": "generic",
      "line": 1725,
      "heading": "S349. Factors Influencing the Repeated Transient Optical Droplet Vaporization Threshold and Lifetimes of Phase Change, Perfluorocarbon Nanodroplets {#src-s349 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s349"
    },
    {
      "book": "generic",
      "line": 1739,
      "heading": "S350. Biomedical photoacoustics beyond thermal expansion using triggered nanodroplet vaporization for contrast-enhanced imaging {#src-s350 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s350"
    },
    {
      "book": "generic",
      "line": 1753,
      "heading": "S351. Vaporization of perfluorocarbon droplets using optical irradiation {#src-s351 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s351"
    },
    {
      "book": "generic",
      "line": 1767,
      "heading": "S355. Dynamics of pulsed laser-induced cavities on a liquid–gas interface: from a conical splash to a ‘bullet’ jet {#src-s355 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s355"
    },
    {
      "book": "generic",
      "line": 1781,
      "heading": "S362. Cavitation bubble induced wall shear stress on an elastic boundary {#src-s362 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s362"
    },
    {
      "book": "generic",
      "line": 1795,
      "heading": "S371. Spontaneous Nucleation of Stable Perfluorocarbon Emulsions for Ultrasound Contrast Agents {#src-s371 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s371"
    },
    {
      "book": "generic",
      "line": 1809,
      "heading": "S377. Effect of laser-induced cavitation bubble on a thin elastic membrane {#src-s377 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s377"
    },
    {
      "book": "generic",
      "line": 1823,
      "heading": "S381. Laser-induced cavitation in nanoemulsion with gold nanospheres for blood clot disruption: in vitro results {#src-s381 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s381"
    },
    {
      "book": "generic",
      "line": 1837,
      "heading": "S384. Photoacoustic and ultrasound imaging using dual contrast perfluorocarbon nanodroplets triggered by laser pulses at 1064 nm {#src-s384 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s384"
    },
    {
      "book": "generic",
      "line": 1851,
      "heading": "S387. Perfluorocarbon nanodroplet size, acoustic vaporization, and inertial cavitation affected by lipid shell composition <i>in vitro</i> {#src-s387 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s387"
    },
    {
      "book": "generic",
      "line": 1865,
      "heading": "S390. Nozzle-Free Liquid Microjetting via Homogeneous Bubble Nucleation {#src-s390 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s390"
    },
    {
      "book": "generic",
      "line": 1879,
      "heading": "S405. Acoustic Droplet Vaporization of Perfluorohexane Emulsions Induced by Heterogeneous Nucleation at an Ultrasonic Frequency of 1.1 MHz {#src-s405 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s405"
    },
    {
      "book": "generic",
      "line": 1893,
      "heading": "S406. Intrathecal Delivery of Macromolecules to the Spinal Cord Enabled by Laser-Activated Perfluorocarbon Nanodroplets {#src-s406 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s406"
    },
    {
      "book": "generic",
      "line": 1907,
      "heading": "S407. Microfluidics control the ballistic energy of thermocavitation liquid jets for needle-free injections {#src-s407 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s407"
    },
    {
      "book": "generic",
      "line": 1921,
      "heading": "S408. Near threshold nucleation and growth of cavitation bubbles generated with a picosecond laser {#src-s408 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s408"
    },
    {
      "book": "generic",
      "line": 1935,
      "heading": "S409. Comprehensive analysis of spherical bubble oscillations and shock wave emission in laser-induced cavitation {#src-s409 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s409"
    },
    {
      "book": "generic",
      "line": 1949,
      "heading": "S410. A Quantitative Printability Framework for Programmable Assembly of Pre-Vascular Patterns via Laser-Induced Forward Transfer {#src-s410 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s410"
    },
    {
      "book": "generic",
      "line": 1963,
      "heading": "S411. Growth and collapse of cavitation bubbles near a curved rigid boundary {#src-s411 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s411"
    },
    {
      "book": "generic",
      "line": 1977,
      "heading": "S412. Laser-generated liquid microjets: correlation between bubble dynamics and liquid ejection {#src-s412 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s412"
    },
    {
      "book": "generic",
      "line": 1991,
      "heading": "S413. Bubble oscillations of large amplitude {#src-s413 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s413"
    },
    {
      "book": "generic",
      "line": 2005,
      "heading": "S414. Giant and explosive plasmonic bubbles by delayed nucleation {#src-s414 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s414"
    },
    {
      "book": "generic",
      "line": 2019,
      "heading": "S415. Nucleation dynamics around single microabsorbers in water heated by nanosecond laser irradiation {#src-s415 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s415"
    },
    {
      "book": "generic",
      "line": 2033,
      "heading": "S416. Phase-Transition Temperature of Gold-Nanorod-Coated Nanodroplets to Microbubbles by Pulsed Laser {#src-s416 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s416"
    },
    {
      "book": "generic",
      "line": 2047,
      "heading": "S418. Numerical investigation of laser-induced cavitation bubble dynamics near a rigid surface based on three-dimensional fully compressible model {#src-s418 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s418"
    },
    {
      "book": "generic",
      "line": 2061,
      "heading": "S419. The Dynamics of Cavitation Bubbles {#src-s419 .unnumbered .unlisted}",
      "page": "course/references/paper-atlas.html",
      "anchor": "src-s419"
    },
    {
      "book": "project-theory",
      "line": 1516,
      "heading": "Appendix A. Original theory pages / 附录 A：原始理论页面",
      "page": "course/references/theory-original-pages-note.html",
      "anchor": "appendix-a-original-theory-pages"
    }
  ],
  "equations": [
    {
      "page": "course/chapters/G01-mechanical-system.html",
      "tex": "p_B-p_\\infty=\\frac{2\\sigma}{R}. \\tag{1.1}",
      "context": "1.1 The object you are modeling / 你究竟在研究什么",
      "source": {
        "book": "generic",
        "line": 53,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-d0a8cd2fb39e76d4.svg",
      "variables": [
        {
          "tex": "p_B",
          "symbol": "p_B",
          "role": "total interior bubble pressure"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/G01-mechanical-system.html",
      "tex": "U_* = \\sqrt{\\frac{\\Delta p}{\\rho}},\\qquad\n t_I=\\frac{R_*}{U_*},\\qquad t_a=\\frac{R_*}{c},\\qquad\n t_\\sigma=\\sqrt{\\frac{\\rho R_*^3}{\\sigma}}, \\tag{1.2}",
      "context": "1.4 The first scale analysis / 第一次尺度分析",
      "source": {
        "book": "generic",
        "line": 85,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-30426bf0c3ea7dee.svg",
      "variables": [
        {
          "tex": "U",
          "symbol": "U",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "t_I",
          "symbol": "t_I",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "t_a",
          "symbol": "t_a",
          "role": "acoustic crossing time"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "t_\\sigma",
          "symbol": "t_σ",
          "role": "physical time"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        }
      ]
    },
    {
      "page": "course/chapters/G01-mechanical-system.html",
      "tex": "t_\\nu=\\frac{R_*^2}{\\nu},\\qquad\n t_\\alpha=\\frac{R_*^2}{\\alpha},\\qquad \\nu=\\frac{\\mu}{\\rho}. \\tag{1.3}",
      "context": "1.4 The first scale analysis / 第一次尺度分析",
      "source": {
        "book": "generic",
        "line": 89,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-305858af5b4cd520.svg",
      "variables": [
        {
          "tex": "t_\\nu",
          "symbol": "t_ν",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\nu",
          "symbol": "ν",
          "role": "Poisson ratio"
        },
        {
          "tex": "t_\\alpha",
          "symbol": "t_α",
          "role": "physical time"
        },
        {
          "tex": "\\alpha",
          "symbol": "α",
          "role": "thermal diffusivity"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G01-mechanical-system.html",
      "tex": "p_\\sigma=2\\sigma/R,\\qquad U_* = \\sqrt{\\Delta p/\\rho}",
      "context": "A bubble as a mechanical system · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G01",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-183013ec06c95739.svg",
      "variables": [
        {
          "tex": "p_\\sigma",
          "symbol": "p_σ",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "U",
          "symbol": "U",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G01-mechanical-system.html",
      "tex": "t_I=R\\sqrt{\\rho/\\Delta p},\\qquad t_a=R/c,\\qquad M_R=4\\pi\\rho R^3",
      "context": "A bubble as a mechanical system · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G01",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-ce43c25f7bdb3e3a.svg",
      "variables": [
        {
          "tex": "t_I",
          "symbol": "t_I",
          "role": "physical time"
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        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t_a",
          "symbol": "t_a",
          "role": "acoustic crossing time"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "M_R",
          "symbol": "M_R",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        }
      ]
    },
    {
      "page": "course/chapters/G01-mechanical-system.html",
      "tex": "\\ddot R=\\big[(p_B-p_\\infty-2\\sigma/R-4\\mu\\dot R/R)/\\rho-\\tfrac32\\dot R^2\\big]/R",
      "context": "A bubble as a mechanical system · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G01",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-64ecb634e9123ec5.svg",
      "variables": [
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "p_B",
          "symbol": "p_B",
          "role": "total interior bubble pressure"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G02-nucleation.html",
      "tex": "\\Delta G(r)=4\\pi\\sigma r^2-\\frac{4\\pi}{3}\\Delta p\\,r^3. \\tag{2.1}",
      "context": "2.1 Why a favorable pressure does not guarantee a bubble / 为什么有利压差并不保证成泡",
      "source": {
        "book": "generic",
        "line": 110,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-84af3c29b13e316c.svg",
      "variables": [
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "G",
          "symbol": "G",
          "role": "formation / fracture energy"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/G02-nucleation.html",
      "tex": "r_* = \\frac{2\\sigma}{\\Delta p},\\qquad\n\\Delta G_* = \\frac{16\\pi\\sigma^3}{3\\Delta p^2}. \\tag{2.2}",
      "context": "2.1 Why a favorable pressure does not guarantee a bubble / 为什么有利压差并不保证成泡",
      "source": {
        "book": "generic",
        "line": 116,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-dc40d411688b9018.svg",
      "variables": [
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "G",
          "symbol": "G",
          "role": "formation / fracture energy"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        }
      ]
    },
    {
      "page": "course/chapters/G02-nucleation.html",
      "tex": "J\\sim J_0\\exp\\!\\left[-\\frac{\\Delta G_*}{k_BT}\\right],\\qquad\nP=1-\\exp\\!\\left[-\\int\\!\\!\\int_{V_{\\mathrm{obs}}}J(\\mathbf x,t)\\,dV\\,dt\\right]. \\tag{2.3}",
      "context": "2.1 Why a favorable pressure does not guarantee a bubble / 为什么有利压差并不保证成泡",
      "source": {
        "book": "generic",
        "line": 127,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-5b76d24390829925.svg",
      "variables": [
        {
          "tex": "J",
          "symbol": "J",
          "role": "nucleation rate / areal impulse"
        },
        {
          "tex": "J_0",
          "symbol": "J_0",
          "role": "nucleation rate / areal impulse"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "G",
          "symbol": "G",
          "role": "formation / fracture energy"
        },
        {
          "tex": "k_B",
          "symbol": "k_B",
          "role": "wave number / Boltzmann constant"
        },
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        }
      ]
    },
    {
      "page": "course/chapters/G02-nucleation.html",
      "tex": "p_{\\mathrm{eq}}(R)=p_v+\\frac{C}{R^{3\\kappa}}-\\frac{2\\sigma}{R}=p_\\infty. \\tag{2.4}",
      "context": "2.2 Pre-existing nuclei change the problem / 预存气核改变问题性质",
      "source": {
        "book": "generic",
        "line": 144,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-bef7df08ad3fe34d.svg",
      "variables": [
        {
          "tex": "p_{\\mathrm{eq}}",
          "symbol": "p_eq",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "C",
          "symbol": "C",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        }
      ]
    },
    {
      "page": "course/chapters/G02-nucleation.html",
      "tex": "R_B=\\left(\\frac{3\\kappa C}{2\\sigma}\\right)^{1/(3\\kappa-1)},\\qquad\np_B^{\\mathrm{crit}}=p_v-\\frac{2\\sigma}{R_B}\\left(1-\\frac{1}{3\\kappa}\\right). \\tag{2.5}",
      "context": "2.2 Pre-existing nuclei change the problem / 预存气核改变问题性质",
      "source": {
        "book": "generic",
        "line": 154,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-d57f90d971dfba94.svg",
      "variables": [
        {
          "tex": "R_B",
          "symbol": "R_B",
          "role": "bubble wall radius"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "C",
          "symbol": "C",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        }
      ]
    },
    {
      "page": "course/chapters/G02-nucleation.html",
      "tex": "r_*=2\\sigma/\\Delta p,\\qquad \\Delta G_*=16\\pi\\sigma^3/(3\\Delta p^2)",
      "context": "Nucleation and static stability · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G02",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-1ed59c68c866c670.svg",
      "variables": [
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "G",
          "symbol": "G",
          "role": "formation / fracture energy"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        }
      ]
    },
    {
      "page": "course/chapters/G02-nucleation.html",
      "tex": "P=1-e^{-JV\\tau}",
      "context": "Nucleation and static stability · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G02",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-0d1f96032533e367.svg",
      "variables": [
        {
          "tex": "P",
          "symbol": "P",
          "role": "event probability"
        },
        {
          "tex": "J",
          "symbol": "J",
          "role": "nucleation rate / areal impulse"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\tau",
          "symbol": "τ",
          "role": "duration / dimensionless time / dummy time"
        }
      ]
    },
    {
      "page": "course/chapters/G02-nucleation.html",
      "tex": "p_{g0}=p_0-p_v+2\\sigma/R_0,\\quad R_B=\\sqrt{3p_{g0}R_0^3/(2\\sigma)},\\quad p_{\\mathrm{Blake}}=p_v-4\\sigma/(3R_B)",
      "context": "Nucleation and static stability · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G02",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-cad83b8b44df83fc.svg",
      "variables": [
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "R_B",
          "symbol": "R_B",
          "role": "bubble wall radius"
        },
        {
          "tex": "p_{\\mathrm{Blake}}",
          "symbol": "p_Blake",
          "role": "critical ambient pressure"
        }
      ]
    },
    {
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "tex": "u(r,t)=\\frac{R^2\\dot R}{r^2},\\qquad\n\\phi(r,t)=-\\frac{R^2\\dot R}{r},\\qquad u=\\partial_r\\phi. \\tag{3.1}",
      "context": "3.2 Continuity determines the velocity field / 连续性决定速度场",
      "source": {
        "book": "generic",
        "line": 187,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-6816404a33b418c2.svg",
      "variables": [
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "velocity potential / mode shape"
        }
      ]
    },
    {
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "tex": "\\left.\\partial_t\\phi\\right|_{r=R}\n=-(2\\dot R^2+R\\ddot R),\\qquad\n\\frac{p_L-p_\\infty}{\\rho}=R\\ddot R+\\frac32\\dot R^2. \\tag{3.2}",
      "context": "3.3 Bernoulli produces the inertia terms / Bernoulli方程给出惯性项",
      "source": {
        "book": "generic",
        "line": 200,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-d6da46a520e61977.svg",
      "variables": [
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "velocity potential / mode shape"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "p_L",
          "symbol": "p_L",
          "role": "liquid pressure next to wall"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "tex": "p_L=p_B-\\frac{2\\sigma}{R}-\\frac{4\\mu\\dot R}{R}. \\tag{3.3}",
      "context": "3.4 Normal stress supplies capillarity and viscosity / 法向应力给出毛细项和黏性项",
      "source": {
        "book": "generic",
        "line": 214,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-75802c652fe3e973.svg",
      "variables": [
        {
          "tex": "p_L",
          "symbol": "p_L",
          "role": "liquid pressure next to wall"
        },
        {
          "tex": "p_B",
          "symbol": "p_B",
          "role": "total interior bubble pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "tex": "\\boxed{\\rho\\left(R\\ddot R+\\frac32\\dot R^2\\right)\n=p_B(t)-p_\\infty(t)-\\frac{2\\sigma}{R}-\\frac{4\\mu\\dot R}{R}.} \\tag{3.4}",
      "context": "3.4 Normal stress supplies capillarity and viscosity / 法向应力给出毛细项和黏性项",
      "source": {
        "book": "generic",
        "line": 220,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f0f22db1b3e69e58.svg",
      "variables": [
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "p_B",
          "symbol": "p_B",
          "role": "total interior bubble pressure"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "tex": "p_B=p_g+p_v,\\qquad\np_g=p_{g0}\\left(\\frac{R_0}{R}\\right)^{3\\kappa},\\qquad\np_{g0}=p_0-p_v+\\frac{2\\sigma}{R_0}. \\tag{3.5}",
      "context": "3.5 Closing the equation / 方程如何闭合",
      "source": {
        "book": "generic",
        "line": 233,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-dc4f6a2f6455978b.svg",
      "variables": [
        {
          "tex": "p_B",
          "symbol": "p_B",
          "role": "total interior bubble pressure"
        },
        {
          "tex": "p_g",
          "symbol": "p_g",
          "role": "noncondensable gas pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        }
      ]
    },
    {
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "tex": "T=\\frac12\\rho\\int_R^\\infty u^2\\,4\\pi r^2dr\n=2\\pi\\rho R^3\\dot R^2\n=\\frac12 M_R(R)\\dot R^2,\\qquad M_R=4\\pi\\rho R^3. \\tag{3.6}",
      "context": "3.6 The same equation from theoretical mechanics / 从理论力学重新得到同一方程",
      "source": {
        "book": "generic",
        "line": 251,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-d2b9ffb856ef636e.svg",
      "variables": [
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "M_R",
          "symbol": "M_R",
          "role": "generalized mass / molar mass"
        }
      ]
    },
    {
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "tex": "\\frac{dT}{dt}=(p_B-p_\\infty)\\frac{dV}{dt}\n-\\frac{d}{dt}(4\\pi\\sigma R^2)-16\\pi\\mu R\\dot R^2,\n\\qquad V=\\frac{4\\pi R^3}{3}. \\tag{3.7}",
      "context": "3.6 The same equation from theoretical mechanics / 从理论力学重新得到同一方程",
      "source": {
        "book": "generic",
        "line": 263,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-eaced8dcf0d2c757.svg",
      "variables": [
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "p_B",
          "symbol": "p_B",
          "role": "total interior bubble pressure"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "tex": "\\rho(R\\ddot R+\\tfrac32\\dot R^2)=p_g+p_v-p_\\infty-2\\sigma/R-4\\mu\\dot R/R",
      "context": "Deriving Rayleigh–Plesset · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G03",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-782e087d688769ed.svg",
      "variables": [
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "p_g",
          "symbol": "p_g",
          "role": "noncondensable gas pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "tex": "u=R^2\\dot R/r^2,\\quad T=2\\pi\\rho R^3\\dot R^2,\\quad T_{r>aR}/T=1/a,\\quad \\frac{d\\phi(R(t),t)}{dt}=\\partial_t\\phi+\\dot R\\partial_r\\phi",
      "context": "Deriving Rayleigh–Plesset · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G03",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-f101ef508b22dfd9.svg",
      "variables": [
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "T_{r>aR}",
          "symbol": "T_r>aR",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "velocity potential / mode shape"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/G03-rayleigh-plesset.html",
      "tex": "\\ddot R>0\\ \\Longleftrightarrow\\ p_g+p_v-p_\\infty-2\\sigma/R-4\\mu\\dot R/R>\\tfrac32\\rho\\dot R^2",
      "context": "Deriving Rayleigh–Plesset · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G03",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-28248b37201b7f44.svg",
      "variables": [
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "p_g",
          "symbol": "p_g",
          "role": "noncondensable gas pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G04-heat-and-mass.html",
      "tex": "jL_v=q_{L\\to i}+q_{V\\to i}. \\tag{4.1}",
      "context": "4.3 The Stefan condition and interface kinematics / Stefan条件与界面运动学",
      "source": {
        "book": "generic",
        "line": 303,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-391a03c538b6a0ff.svg",
      "variables": [
        {
          "tex": "j",
          "symbol": "j",
          "role": "evaporative mass flux"
        },
        {
          "tex": "L_v",
          "symbol": "L_v",
          "role": "length / latent heat / load disk radius"
        },
        {
          "tex": "q_{L\\to i}",
          "symbol": "q_Lto i",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "q_{V\\to i}",
          "symbol": "q_Vto i",
          "role": "heat flux / mode displacement"
        }
      ]
    },
    {
      "page": "course/chapters/G04-heat-and-mass.html",
      "tex": "p_g/p_{g0}=(R_0/R)^{3\\kappa},\\qquad T_g/T_{g0}=(R_0/R)^{3(\\kappa-1)}",
      "context": "Gas, vapor, heat and mass · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G04",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-73774f94b9fcc8f9.svg",
      "variables": [
        {
          "tex": "p_g",
          "symbol": "p_g",
          "role": "noncondensable gas pressure"
        },
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "T_g",
          "symbol": "T_g",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "T_{g0}",
          "symbol": "T_g0",
          "role": "temperature / kinetic energy as defined"
        }
      ]
    },
    {
      "page": "course/chapters/G04-heat-and-mass.html",
      "tex": "jL_v=q_{L\\to i}+q_{V\\to i},\\qquad u_L=\\dot R-j/\\rho_L",
      "context": "Gas, vapor, heat and mass · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G04",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-7a5ffbaa29f1f4ee.svg",
      "variables": [
        {
          "tex": "j",
          "symbol": "j",
          "role": "evaporative mass flux"
        },
        {
          "tex": "L_v",
          "symbol": "L_v",
          "role": "length / latent heat / load disk radius"
        },
        {
          "tex": "q_{L\\to i}",
          "symbol": "q_Lto i",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "q_{V\\to i}",
          "symbol": "q_Vto i",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "u_L",
          "symbol": "u_L",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\rho_L",
          "symbol": "ρ_L",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G04-heat-and-mass.html",
      "tex": "Ja=c_{p,L}(T_\\infty-T_{\\mathrm{sat}})/L_v,\\qquad R_{\\mathrm{thermal}}\\sim\\sqrt{\\alpha t}",
      "context": "Gas, vapor, heat and mass · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G04",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-073d5b2a6d4ac692.svg",
      "variables": [
        {
          "tex": "Ja",
          "symbol": "Ja",
          "role": "sensible-to-latent heat ratio"
        },
        {
          "tex": "c_{p,L}",
          "symbol": "c_p,L",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "T_\\infty",
          "symbol": "T∞",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "T_{\\mathrm{sat}}",
          "symbol": "T_sat",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "L_v",
          "symbol": "L_v",
          "role": "length / latent heat / load disk radius"
        },
        {
          "tex": "R_{\\mathrm{thermal}}",
          "symbol": "R_thermal",
          "role": "bubble wall radius"
        },
        {
          "tex": "\\alpha",
          "symbol": "α",
          "role": "thermal diffusivity"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "tex": "R\\ddot R+\\frac32\\dot R^2=-\\frac{\\Delta p}{\\rho}. \\tag{5.1}",
      "context": "5.1 The empty-cavity idealization / 空腔理想化",
      "source": {
        "book": "generic",
        "line": 338,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f23489f0c0d1cf8c.svg",
      "variables": [
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "tex": "\\dot R^2=\\frac{2\\Delta p}{3\\rho}\n\\left[\\left(\\frac{R_{\\max}}{R}\\right)^3-1\\right]. \\tag{5.2}",
      "context": "5.2 Integrating once / 第一次积分",
      "source": {
        "book": "generic",
        "line": 350,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-2fa4e9e565187312.svg",
      "variables": [
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "tex": "2\\pi\\rho R^3\\dot R^2\n=\\frac{4\\pi}{3}\\Delta p\\,(R_{\\max}^3-R^3). \\tag{5.3}",
      "context": "5.2 Integrating once / 第一次积分",
      "source": {
        "book": "generic",
        "line": 357,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-9898ed4e4b11cf34.svg",
      "variables": [
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        }
      ]
    },
    {
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "tex": "t_c=R_{\\max}\\sqrt{\\frac{3\\rho}{2\\Delta p}}\n\\int_0^1\\frac{x^{3/2}}{\\sqrt{1-x^3}}\\,dx\n=0.914681\\ldots\\,R_{\\max}\\sqrt{\\frac{\\rho}{\\Delta p}}. \\tag{5.4}",
      "context": "5.3 The Rayleigh collapse time / Rayleigh塌缩时间",
      "source": {
        "book": "generic",
        "line": 370,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-9a8038f15d0fb570.svg",
      "variables": [
        {
          "tex": "t_c",
          "symbol": "t_c",
          "role": "collapse duration"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        }
      ]
    },
    {
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "tex": "R\\simeq\\left[\\frac52K(t_c-t)\\right]^{2/5}. \\tag{5.5}",
      "context": "5.4 The near-collapse power law / 接近塌缩时的幂律",
      "source": {
        "book": "generic",
        "line": 394,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-35ac931dc7533cf6.svg",
      "variables": [
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "K",
          "symbol": "K",
          "role": "stiffness / collapse constant"
        },
        {
          "tex": "t_c",
          "symbol": "t_c",
          "role": "collapse duration"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "tex": "\\mathcal E=2\\pi\\rho R^3\\dot R^2+(p_0-p_v)V\n+4\\pi\\sigma R^2+\\frac{p_gV}{\\kappa-1}. \\tag{5.6}",
      "context": "5.6 A conserved mechanical quantity for testing / 用于检验的守恒力学量",
      "source": {
        "book": "generic",
        "line": 416,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f8a766d0ef705b22.svg",
      "variables": [
        {
          "tex": "E",
          "symbol": "E",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "p_g",
          "symbol": "p_g",
          "role": "noncondensable gas pressure"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        }
      ]
    },
    {
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "tex": "R\\ddot R+\\tfrac32\\dot R^2=-\\Delta p/\\rho,\\quad \\dot R^2=\\frac{2\\Delta p}{3\\rho}\\left[(R_{\\max}/R)^3-1\\right],\\quad t_c=R_{\\max}\\sqrt{\\rho/\\Delta p}\\sqrt{\\tfrac32}\\int_0^1\\frac{x^{3/2}}{\\sqrt{1-x^3}}dx",
      "context": "Inertial collapse, energy focusing and rebound · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G05",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-6cdd8fde334d12b3.svg",
      "variables": [
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "t_c",
          "symbol": "t_c",
          "role": "collapse duration"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        }
      ]
    },
    {
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "tex": "\\dot R=-\\sqrt{\\frac{2\\Delta p}{3\\rho}[(R_{\\max}/R)^3-1]},\\qquad t_c\\propto\\Delta p^{-1/2}",
      "context": "Inertial collapse, energy focusing and rebound · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G05",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-ced3c7ddcd7909ac.svg",
      "variables": [
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "t_c",
          "symbol": "t_c",
          "role": "collapse duration"
        }
      ]
    },
    {
      "page": "course/chapters/G05-collapse-and-rebound.html",
      "tex": "E=2\\pi\\rho R^3\\dot R^2+(p_0-p_v)V+4\\pi\\sigma R^2+W_g,\\quad \\dot E=-16\\pi\\mu R\\dot R^2,\\quad W_{g,\\kappa=1}=-p_{g0}V_0\\ln(V/V_0)",
      "context": "Inertial collapse, energy focusing and rebound · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G05",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-e83411456be37444.svg",
      "variables": [
        {
          "tex": "E",
          "symbol": "E",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "W_g",
          "symbol": "W_g",
          "role": "work / potential energy"
        },
        {
          "tex": "\\dot E",
          "symbol": "Ė",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "W_{g,\\kappa=1}",
          "symbol": "W_g,κ=1",
          "role": "work / potential energy"
        },
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "V_0",
          "symbol": "V_0",
          "role": "bubble or observation volume"
        }
      ]
    },
    {
      "page": "course/chapters/G06-oscillations.html",
      "tex": "\\ddot x+2\\beta_\\mu\\dot x+\\omega_0^2x\n=-\\frac{p_a(t)}{\\rho R_0^2}, \\tag{6.1}",
      "context": "6.1 Linearization around equilibrium / 平衡态附近的线性化",
      "source": {
        "book": "generic",
        "line": 431,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-258a4c52baa9f6a7.svg",
      "variables": [
        {
          "tex": "\\ddot x",
          "symbol": "ẍ",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "\\beta_\\mu",
          "symbol": "β_μ",
          "role": "damping / logistic coefficient"
        },
        {
          "tex": "\\dot x",
          "symbol": "ẋ",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "\\omega_0",
          "symbol": "ω_0",
          "role": "linear radial angular frequency"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "p_a",
          "symbol": "p_a",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        }
      ]
    },
    {
      "page": "course/chapters/G06-oscillations.html",
      "tex": "\\omega_0^2=\\frac{3\\kappa p_{g0}-2\\sigma/R_0}{\\rho R_0^2}\n=\\frac{3\\kappa(p_0-p_v)+(3\\kappa-1)2\\sigma/R_0}{\\rho R_0^2},\n\\qquad \\beta_\\mu=\\frac{2\\mu}{\\rho R_0^2}. \\tag{6.2}",
      "context": "6.1 Linearization around equilibrium / 平衡态附近的线性化",
      "source": {
        "book": "generic",
        "line": 434,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-9579c37ead519c55.svg",
      "variables": [
        {
          "tex": "\\omega_0",
          "symbol": "ω_0",
          "role": "linear radial angular frequency"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "\\beta_\\mu",
          "symbol": "β_μ",
          "role": "damping / logistic coefficient"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/G06-oscillations.html",
      "tex": "f_0\\simeq\\frac{1}{2\\pi R_0}\\sqrt{\\frac{3\\kappa p_0}{\\rho}}. \\tag{6.3}",
      "context": "6.1 Linearization around equilibrium / 平衡态附近的线性化",
      "source": {
        "book": "generic",
        "line": 446,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-8354f4421f207929.svg",
      "variables": [
        {
          "tex": "f_0",
          "symbol": "f_0",
          "role": "frequency / predictor / false detections"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G06-oscillations.html",
      "tex": "|\\hat x|=\\frac{P_a/(\\rho R_0^2)}\n{\\sqrt{(\\omega_0^2-\\omega^2)^2+(2\\beta\\omega)^2}}. \\tag{6.4}",
      "context": "6.2 Harmonic response and phase / 简谐响应与相位",
      "source": {
        "book": "generic",
        "line": 462,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-406d8f63f9980f8b.svg",
      "variables": [
        {
          "tex": "\\hat x",
          "symbol": "x^",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "P_a",
          "symbol": "P_a",
          "role": "event probability"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\omega_0",
          "symbol": "ω_0",
          "role": "linear radial angular frequency"
        },
        {
          "tex": "\\omega",
          "symbol": "ω",
          "role": "angular frequency"
        },
        {
          "tex": "\\beta",
          "symbol": "β",
          "role": "damping / logistic coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/G06-oscillations.html",
      "tex": "p_{g0}=p_0-p_v+2\\sigma/R_0,\\quad \\omega_0^2=(3\\kappa p_{g0}-2\\sigma/R_0)/(\\rho R_0^2),\\quad x_{\\mathrm{static}}=-\\delta p/(3\\kappa p_{g0}-2\\sigma/R_0)",
      "context": "Bubble oscillations as vibration theory · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G06",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-1abf8315fca3dc0e.svg",
      "variables": [
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\omega_0",
          "symbol": "ω_0",
          "role": "linear radial angular frequency"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "x_{\\mathrm{static}}",
          "symbol": "x_static",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "\\delta",
          "symbol": "δ",
          "role": "separation / perturbation / penetration depth"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/G06-oscillations.html",
      "tex": "|x|=\\frac{p_a/(\\rho R_0^2)}{\\sqrt{(\\omega_0^2-\\omega^2)^2+(2\\beta\\omega)^2}},\\qquad \\beta_\\mu=2\\mu/(\\rho R_0^2)",
      "context": "Bubble oscillations as vibration theory · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G06",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-a70bd33587ba0372.svg",
      "variables": [
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "p_a",
          "symbol": "p_a",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\omega_0",
          "symbol": "ω_0",
          "role": "linear radial angular frequency"
        },
        {
          "tex": "\\omega",
          "symbol": "ω",
          "role": "angular frequency"
        },
        {
          "tex": "\\beta",
          "symbol": "β",
          "role": "damping / logistic coefficient"
        },
        {
          "tex": "\\beta_\\mu",
          "symbol": "β_μ",
          "role": "damping / logistic coefficient"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/G06-oscillations.html",
      "tex": "\\beta=\\beta_\\mu+\\beta_{\\mathrm{thermal}}+\\beta_{\\mathrm{radiation}}",
      "context": "Bubble oscillations as vibration theory · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G06",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-b2709cbf290ba285.svg",
      "variables": [
        {
          "tex": "\\beta",
          "symbol": "β",
          "role": "damping / logistic coefficient"
        },
        {
          "tex": "\\beta_\\mu",
          "symbol": "β_μ",
          "role": "damping / logistic coefficient"
        },
        {
          "tex": "\\beta_{\\mathrm{thermal}}",
          "symbol": "β_thermal",
          "role": "damping / logistic coefficient"
        },
        {
          "tex": "\\beta_{\\mathrm{radiation}}",
          "symbol": "β_radiation",
          "role": "damping / logistic coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "\\left(1-\\frac{\\dot R}{c}\\right)R\\ddot R\n+\\frac32\\left(1-\\frac{\\dot R}{3c}\\right)\\dot R^2\n=\\left(1+\\frac{\\dot R}{c}\\right)\\frac{p_L-p_0}{\\rho}\n+\\frac{R}{\\rho c}\\frac{dp_L}{dt}. \\tag{7.1}",
      "context": "7.2 Keller–Miksis with constant ambient pressure / 恒定环境压力下的Keller–Miksis方程",
      "source": {
        "book": "generic",
        "line": 509,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f2757db6f4b73406.svg",
      "variables": [
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "p_L",
          "symbol": "p_L",
          "role": "liquid pressure next to wall"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "\\frac{dp_L}{dt}=-\\frac{3\\kappa p_gU}{R}\n+\\frac{2\\sigma U}{R^2}+\\frac{4\\mu U^2}{R^2}\n-\\frac{4\\mu}{R}\\dot U. \\tag{7.2}",
      "context": "7.3 Solving the implicit acceleration correctly / 正确求解隐式加速度",
      "source": {
        "book": "generic",
        "line": 528,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-fb1c6797cadcd134.svg",
      "variables": [
        {
          "tex": "p_L",
          "symbol": "p_L",
          "role": "liquid pressure next to wall"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "p_g",
          "symbol": "p_g",
          "role": "noncondensable gas pressure"
        },
        {
          "tex": "U",
          "symbol": "U",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\dot U",
          "symbol": "U̇",
          "role": "velocity / velocity scale"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "\\dot U=\\frac{\n(1+U/c)(p_L-p_0)/\\rho+RD_p/(\\rho c)\n-\\tfrac32(1-U/(3c))U^2}\n{R(1-U/c)+4\\mu/(\\rho c)}. \\tag{7.3}",
      "context": "7.3 Solving the implicit acceleration correctly / 正确求解隐式加速度",
      "source": {
        "book": "generic",
        "line": 536,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-4bc8b1626d90c787.svg",
      "variables": [
        {
          "tex": "\\dot U",
          "symbol": "U̇",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "U",
          "symbol": "U",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "p_L",
          "symbol": "p_L",
          "role": "liquid pressure next to wall"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "D_p",
          "symbol": "D_p",
          "role": "bending rigidity / dose / residual derivative"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "H=\\int_{p_\\infty}^{p_L}\\frac{dp}{\\rho(p)},\\qquad\nC^2=\\left.\\frac{dp}{d\\rho}\\right|_{p_L}. \\tag{7.4}",
      "context": "7.4 Gilmore's enthalpy formulation / Gilmore的焓形式",
      "source": {
        "book": "generic",
        "line": 553,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-92b60e3a39ac34d2.svg",
      "variables": [
        {
          "tex": "H",
          "symbol": "H",
          "role": "enthalpy / absorbed dose / liquid depth"
        },
        {
          "tex": "p_L",
          "symbol": "p_L",
          "role": "liquid pressure next to wall"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "\\left(1-\\frac{U}{C}\\right)R\\dot U\n+\\frac32\\left(1-\\frac{U}{3C}\\right)U^2\n=\\left(1+\\frac{U}{C}\\right)H\n+\\left(1-\\frac{U}{C}\\right)\\frac{R}{C}\\dot H. \\tag{7.5}",
      "context": "7.4 Gilmore's enthalpy formulation / Gilmore的焓形式",
      "source": {
        "book": "generic",
        "line": 560,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-5dbaea01364fa09d.svg",
      "variables": [
        {
          "tex": "U",
          "symbol": "U",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "C",
          "symbol": "C",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot U",
          "symbol": "U̇",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "H",
          "symbol": "H",
          "role": "enthalpy / absorbed dose / liquid depth"
        },
        {
          "tex": "\\dot H",
          "symbol": "Ḣ",
          "role": "enthalpy / absorbed dose / liquid depth"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "H=\\frac{n(p_\\infty+B)}{(n-1)\\rho_\\infty}\n\\left[\\left(\\frac{p_L+B}{p_\\infty+B}\\right)^{(n-1)/n}-1\\right],\n\\qquad C^2=c_\\infty^2+(n-1)H. \\tag{7.6}",
      "context": "7.4 Gilmore's enthalpy formulation / Gilmore的焓形式",
      "source": {
        "book": "generic",
        "line": 569,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-7f8fe991271d3f1a.svg",
      "variables": [
        {
          "tex": "H",
          "symbol": "H",
          "role": "enthalpy / absorbed dose / liquid depth"
        },
        {
          "tex": "n",
          "symbol": "n",
          "role": "mode / bubble number density"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "B",
          "symbol": "B",
          "role": "kinetic pressure coefficient"
        },
        {
          "tex": "\\rho_\\infty",
          "symbol": "ρ∞",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "p_L",
          "symbol": "p_L",
          "role": "liquid pressure next to wall"
        },
        {
          "tex": "C",
          "symbol": "C",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "c_\\infty",
          "symbol": "c∞",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "p'(r,t)\\simeq\\frac{\\rho}{4\\pi r}\\ddot V(t-r/c)\n=\\frac{\\rho}{r}\\left[R^2\\ddot R+2R\\dot R^2\\right]_{t-r/c}. \\tag{7.7}",
      "context": "7.5 What the far-field pressure means / 远场压力究竟表示什么",
      "source": {
        "book": "generic",
        "line": 583,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-3515d8158a40f3db.svg",
      "variables": [
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\ddot V",
          "symbol": "V̈",
          "role": "bubble or observation volume"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "E_{\\mathrm{ac}}\\simeq\\frac{4\\pi r^2}{\\rho c}\\int p'(r,t)^2\\,dt. \\tag{7.8}",
      "context": "7.5 What the far-field pressure means / 远场压力究竟表示什么",
      "source": {
        "book": "generic",
        "line": 594,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1fd2ed6b731e3e67.svg",
      "variables": [
        {
          "tex": "E_{\\mathrm{ac}}",
          "symbol": "E_ac",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "M_w=|\\dot R|/c,\\qquad kR=\\omega R/c",
      "context": "Compressibility, sound and model limits · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G07",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-cd593fbe81b1a917.svg",
      "variables": [
        {
          "tex": "M_w",
          "symbol": "M_w",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "k",
          "symbol": "k",
          "role": "wave number / Boltzmann constant"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\omega",
          "symbol": "ω",
          "role": "angular frequency"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "\\dot p_L=D_p-4\\mu\\ddot R/R,\\quad D_p=-3\\kappa p_g\\dot R/R+2\\sigma\\dot R/R^2+4\\mu\\dot R^2/R^2,\\quad \\mathcal D=R(1-\\dot R/c)+4\\mu/(\\rho c)",
      "context": "Compressibility, sound and model limits · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G07",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-79e4f09400861fc9.svg",
      "variables": [
        {
          "tex": "\\dot p_L",
          "symbol": "ṗ_L",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "D_p",
          "symbol": "D_p",
          "role": "bending rigidity / dose / residual derivative"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "p_g",
          "symbol": "p_g",
          "role": "noncondensable gas pressure"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "D",
          "symbol": "D",
          "role": "bending rigidity / dose / residual derivative"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G07-compressibility.html",
      "tex": "p'(r,t)=\\rho\\ddot V(t-r/c)/(4\\pi r)",
      "context": "Compressibility, sound and model limits · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G07",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-15db0d47d0691d80.svg",
      "variables": [
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\ddot V",
          "symbol": "V̈",
          "role": "bubble or observation volume"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        }
      ]
    },
    {
      "page": "course/chapters/G13-numerical-laboratory.html",
      "tex": "\\frac{dx}{d\\tau}=v,\\qquad\n\\frac{dv}{d\\tau}=\\frac{R_0}{U_*^2}\\,a(R_0x,U_*v), \\tag{13.1}",
      "context": "13.2 Scaling the equations before integration / 积分前先进行尺度化",
      "source": {
        "book": "generic",
        "line": 979,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-2fe61f4062aeb59c.svg",
      "variables": [
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "\\tau",
          "symbol": "τ",
          "role": "duration / dimensionless time / dummy time"
        },
        {
          "tex": "v",
          "symbol": "v",
          "role": "velocity / squared wall speed as defined"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "U",
          "symbol": "U",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        }
      ]
    },
    {
      "page": "course/chapters/G13-numerical-laboratory.html",
      "tex": "U_*=\\sqrt{(p_0-p_v)/\\rho},\\quad t_*=R_0/U_*,\\quad x=R/R_0,\\quad v=\\dot R/U_*",
      "context": "A reproducible numerical laboratory · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G13",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-160ff82232614766.svg",
      "variables": [
        {
          "tex": "U",
          "symbol": "U",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "v",
          "symbol": "v",
          "role": "velocity / squared wall speed as defined"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/G13-numerical-laboratory.html",
      "tex": "\\epsilon_E=\\max_t|E(t)-E(0)+\\int_0^t16\\pi\\mu R\\dot R^2dt|/|E(0)|",
      "context": "A reproducible numerical laboratory · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G13",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-c43c8f059e938c5e.svg",
      "variables": [
        {
          "tex": "\\epsilon_E",
          "symbol": "ε_E",
          "role": "small perturbation / error"
        },
        {
          "tex": "E",
          "symbol": "E",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/G13-numerical-laboratory.html",
      "tex": "M_w=|\\dot R|/c,\\qquad t_{\\mathrm{cut}}=t_*\\sqrt{3/2}\\int_{x_{\\mathrm{cut}}}^1\\frac{x^{3/2}}{\\sqrt{1-x^3}}dx",
      "context": "A reproducible numerical laboratory · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G13",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-9235afe619c72b3a.svg",
      "variables": [
        {
          "tex": "M_w",
          "symbol": "M_w",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "t_{\\mathrm{cut}}",
          "symbol": "t_cut",
          "role": "physical time"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        }
      ]
    },
    {
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "tex": "\\ddot a_\\ell+3\\frac{\\dot R}{R}\\dot a_\\ell+\n\\left[\\frac{(\\ell-1)(\\ell+1)(\\ell+2)\\sigma}{\\rho R^3}\n-(\\ell-1)\\frac{\\ddot R}{R}\\right]a_\\ell=0. \\tag{8.1}",
      "context": "8.1 A radius is not a shape / 半径不能代表全部形状",
      "source": {
        "book": "generic",
        "line": 621,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-33f650d0417ade06.svg",
      "variables": [
        {
          "tex": "\\ddot a_\\ell",
          "symbol": "ä_ℓ",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot a_\\ell",
          "symbol": "ȧ_ℓ",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\ell",
          "symbol": "ℓ",
          "role": "absorption / cohesive-zone length"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "a_\\ell",
          "symbol": "a_ℓ",
          "role": "droplet radius / film span / acceleration"
        }
      ]
    },
    {
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "tex": "\\mathbf F_p\\simeq -V(t)\\nabla p_{\\mathrm{ext}},\\qquad\n\\langle\\mathbf F_p\\rangle\\simeq-\\langle V\\nabla p_{\\mathrm{ext}}\\rangle. \\tag{8.2}",
      "context": "8.4 Pressure gradients and bubble translation / 压力梯度与气泡平移",
      "source": {
        "book": "generic",
        "line": 667,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c48cd41e762c22c5.svg",
      "variables": [
        {
          "tex": "F_p",
          "symbol": "F_p",
          "role": "force / fluence"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "p_{\\mathrm{ext}}",
          "symbol": "p_ext",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "tex": "\\gamma=h/R_{\\max},\\qquad g_0=h-R_{\\max}",
      "context": "Shape, boundaries and translation · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G08",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-3fcb5b0011adc85a.svg",
      "variables": [
        {
          "tex": "\\gamma",
          "symbol": "γ",
          "role": "stand-off / heat-capacity ratio"
        },
        {
          "tex": "h",
          "symbol": "h",
          "role": "stand-off / layer thickness"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "g_0",
          "symbol": "g_0",
          "role": "temporal pulse profile / wall gap"
        }
      ]
    },
    {
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "tex": "\\omega_l^2=(l-1)(l+1)(l+2)\\sigma/(\\rho R^3),\\quad \\ddot a_l+3\\dot R\\dot a_l/R+[\\omega_l^2-(l-1)\\ddot R/R]a_l=0",
      "context": "Shape, boundaries and translation · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G08",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-1e8ebb5ae08afcb9.svg",
      "variables": [
        {
          "tex": "\\omega_l",
          "symbol": "ω_l",
          "role": "angular frequency"
        },
        {
          "tex": "l",
          "symbol": "l",
          "role": "shape harmonic order"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot a_l",
          "symbol": "ä_l",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\dot a_l",
          "symbol": "ȧ_l",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "a_l",
          "symbol": "a_l",
          "role": "droplet radius / film span / acceleration"
        }
      ]
    },
    {
      "page": "course/chapters/G08-shape-and-boundaries.html",
      "tex": "\\mathbf F=-V\\nabla p,\\qquad V=4\\pi R^3/3,\\qquad \\langle\\mathbf F\\rangle=-\\langle V\\nabla p\\rangle",
      "context": "Shape, boundaries and translation · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G08",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-29a30b9184d55901.svg",
      "variables": [
        {
          "tex": "F",
          "symbol": "F",
          "role": "force / fluence"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/G09-jets-and-transfer.html",
      "tex": "\\Pi(\\mathbf x)=\\int_{t_-}^{t_+}[p(\\mathbf x,t)-p_{\\mathrm{ref}}(\\mathbf x,t)]\\,dt,\n\\qquad \\Delta\\mathbf u\\simeq-\\frac{1}{\\rho}\\nabla\\Pi. \\tag{9.1}",
      "context": "9.2 Deriving pressure impulse / 推导压力冲量",
      "source": {
        "book": "generic",
        "line": 698,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-db93f789dd728416.svg",
      "variables": [
        {
          "tex": "\\Pi",
          "symbol": "Π",
          "role": "pressure impulse / shell traction / availability"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_{\\mathrm{ref}}",
          "symbol": "p_ref",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G09-jets-and-transfer.html",
      "tex": "I_p=\\int\\Delta p\\,dt\\approx\\Delta p\\tau,\\quad \\Delta U_{\\mathrm{slug}}=I_p/(\\rho L),\\quad t_a=L/c",
      "context": "Jets, pressure impulse and liquid transfer · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G09",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-0a8f7851e7495991.svg",
      "variables": [
        {
          "tex": "I_p",
          "symbol": "I_p",
          "role": "pressure impulse per unit area"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\tau",
          "symbol": "τ",
          "role": "duration / dimensionless time / dummy time"
        },
        {
          "tex": "U_{\\mathrm{slug}}",
          "symbol": "U_slug",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "L",
          "symbol": "L",
          "role": "length / latent heat / load disk radius"
        },
        {
          "tex": "t_a",
          "symbol": "t_a",
          "role": "acoustic crossing time"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/G09-jets-and-transfer.html",
      "tex": "We_j=\\rho U_j^2d_j/\\sigma,\\quad Oh_j=\\mu/\\sqrt{\\rho\\sigma d_j},\\quad q_j=\\rho U_j^2/2,\\quad F_{\\mathrm{steady}}=\\rho(\\pi d_j^2/4)U_j^2",
      "context": "Jets, pressure impulse and liquid transfer · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G09",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-d6c312e6eff046b1.svg",
      "variables": [
        {
          "tex": "We_j",
          "symbol": "We_j",
          "role": "inertia-to-capillary ratio"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "U_j",
          "symbol": "U_j",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "Oh_j",
          "symbol": "Oh_j",
          "role": "viscous-to-capillary ratio"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "q_j",
          "symbol": "q_j",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "F_{\\mathrm{steady}}",
          "symbol": "F_steady",
          "role": "force / fluence"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        }
      ]
    },
    {
      "page": "course/chapters/G09-jets-and-transfer.html",
      "tex": "\\Delta\\mathbf u=-\\nabla\\Pi/\\rho,\\qquad \\Pi(\\mathbf x)=\\int(p-p_{\\mathrm{ref}})dt",
      "context": "Jets, pressure impulse and liquid transfer · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G09",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-e7b5bb7715912201.svg",
      "variables": [
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "\\Pi",
          "symbol": "Π",
          "role": "pressure impulse / shell traction / availability"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_{\\mathrm{ref}}",
          "symbol": "p_ref",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/G10-laser-source-pathways.html",
      "tex": "p_{\\mathrm{init}}\\sim\\Gamma H_{\\mathrm{abs}},\\qquad\n\\Gamma=\\frac{\\beta_T c^2}{c_p}, \\tag{10.1}",
      "context": "10.4 Thermoelastic sound and confinement / 热弹性声波与约束条件",
      "source": {
        "book": "generic",
        "line": 777,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-909df61a724e33cc.svg",
      "variables": [
        {
          "tex": "p_{\\mathrm{init}}",
          "symbol": "p_init",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\Gamma",
          "symbol": "Γ",
          "role": "Grüneisen parameter"
        },
        {
          "tex": "H_{\\mathrm{abs}}",
          "symbol": "H_abs",
          "role": "enthalpy / absorbed dose / liquid depth"
        },
        {
          "tex": "\\beta_T",
          "symbol": "β_T",
          "role": "damping / logistic coefficient"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "c_p",
          "symbol": "c_p",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/G10-laser-source-pathways.html",
      "tex": "p_{\\mathrm{init}}=\\Gamma H_{\\mathrm{abs}},\\qquad t_{\\mathrm{stress}}=\\ell/c,\\qquad t_{\\mathrm{thermal}}=\\ell^2/\\alpha",
      "context": "How lasers create bubbles and loading · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G10",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-00d162eaf7a66c97.svg",
      "variables": [
        {
          "tex": "p_{\\mathrm{init}}",
          "symbol": "p_init",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\Gamma",
          "symbol": "Γ",
          "role": "Grüneisen parameter"
        },
        {
          "tex": "H_{\\mathrm{abs}}",
          "symbol": "H_abs",
          "role": "enthalpy / absorbed dose / liquid depth"
        },
        {
          "tex": "t_{\\mathrm{stress}}",
          "symbol": "t_stress",
          "role": "physical time"
        },
        {
          "tex": "\\ell",
          "symbol": "ℓ",
          "role": "absorption / cohesive-zone length"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "t_{\\mathrm{thermal}}",
          "symbol": "t_thermal",
          "role": "physical time"
        },
        {
          "tex": "\\alpha",
          "symbol": "α",
          "role": "thermal diffusivity"
        }
      ]
    },
    {
      "page": "course/chapters/G10-laser-source-pathways.html",
      "tex": "\\eta_BE_{\\mathrm{abs}}=\\Delta p\\,4\\pi R_{\\max}^3/3",
      "context": "How lasers create bubbles and loading · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G10",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-acdde12028b1ce40.svg",
      "variables": [
        {
          "tex": "\\eta_B",
          "symbol": "η_B",
          "role": "bubble-work conversion fraction"
        },
        {
          "tex": "E_{\\mathrm{abs}}",
          "symbol": "E_abs",
          "role": "total absorbed energy"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        }
      ]
    },
    {
      "page": "course/chapters/G10-laser-source-pathways.html",
      "tex": "E_{\\mathrm{laser}}\\longrightarrow E_{\\mathrm{absorber}}\\longrightarrow p_{\\mathrm{acoustic}}(t)\\longrightarrow R(t)",
      "context": "How lasers create bubbles and loading · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G10",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-f7ed20cebdcd1113.svg",
      "variables": [
        {
          "tex": "E_{\\mathrm{laser}}",
          "symbol": "E_laser",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "E_{\\mathrm{absorber}}",
          "symbol": "E_absorber",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "p_{\\mathrm{acoustic}}",
          "symbol": "p_acoustic",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/G11-phase-change-droplets.html",
      "tex": "\\rho_{d}\\frac{4\\pi a_0^3}{3}\n=\\frac{M p_{\\mathrm{PF}}}{R_uT_b}\\frac{4\\pi R_b^3}{3},\\qquad\n\\frac{R_b}{a_0}=\\left(\\frac{\\rho_dR_uT_b}{Mp_{\\mathrm{PF}}}\\right)^{1/3}. \\tag{11.1}",
      "context": "11.4 A mass-conservation estimate of expansion / 用质量守恒估计膨胀",
      "source": {
        "book": "generic",
        "line": 847,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-3160b3f77d77f052.svg",
      "variables": [
        {
          "tex": "\\rho_{d}",
          "symbol": "ρ_d",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "a_0",
          "symbol": "a_0",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "M",
          "symbol": "M",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "p_{\\mathrm{PF}}",
          "symbol": "p_PF",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "R_u",
          "symbol": "R_u",
          "role": "bubble wall radius"
        },
        {
          "tex": "T_b",
          "symbol": "T_b",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "R_b",
          "symbol": "R_b",
          "role": "bubble wall radius"
        },
        {
          "tex": "\\rho_d",
          "symbol": "ρ_d",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G11-phase-change-droplets.html",
      "tex": "p_d-p_w=2\\sigma_{dw}/a+\\Pi_{\\mathrm{shell}},\\qquad p_{v,\\mathrm{emb}}-p_d=2\\sigma_{vd}/r",
      "context": "Phase-change perfluorocarbon droplets · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G11",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-afa813c1d2f3ae58.svg",
      "variables": [
        {
          "tex": "p_d",
          "symbol": "p_d",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_w",
          "symbol": "p_w",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\sigma_{dw}",
          "symbol": "σ_dw",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\Pi_{\\mathrm{shell}}",
          "symbol": "Π_shell",
          "role": "pressure impulse / shell traction / availability"
        },
        {
          "tex": "p_{v,\\mathrm{emb}}",
          "symbol": "p_v,emb",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\sigma_{vd}",
          "symbol": "σ_vd",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        }
      ]
    },
    {
      "page": "course/chapters/G11-phase-change-droplets.html",
      "tex": "R_b/a_0=[\\rho_dR_uT/(Mp_{PF})]^{1/3}",
      "context": "Phase-change perfluorocarbon droplets · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G11",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-5f9c6c054aba6cfe.svg",
      "variables": [
        {
          "tex": "R_b",
          "symbol": "R_b",
          "role": "bubble wall radius"
        },
        {
          "tex": "a_0",
          "symbol": "a_0",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\rho_d",
          "symbol": "ρ_d",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_u",
          "symbol": "R_u",
          "role": "bubble wall radius"
        },
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "M",
          "symbol": "M",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "p_{PF}",
          "symbol": "p_PF",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/G11-phase-change-droplets.html",
      "tex": "m_{PF}=\\rho_d(4\\pi a_0^3/3),\\qquad p_{PF}V_b=m_{PF}R_uT/M",
      "context": "Phase-change perfluorocarbon droplets · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G11",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-c6f06d9fbfdf81ec.svg",
      "variables": [
        {
          "tex": "m_{PF}",
          "symbol": "m_PF",
          "role": "mass or areal mass"
        },
        {
          "tex": "\\rho_d",
          "symbol": "ρ_d",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "a_0",
          "symbol": "a_0",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "p_{PF}",
          "symbol": "p_PF",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "V_b",
          "symbol": "V_b",
          "role": "bubble or observation volume"
        },
        {
          "tex": "R_u",
          "symbol": "R_u",
          "role": "bubble wall radius"
        },
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "M",
          "symbol": "M",
          "role": "generalized mass / molar mass"
        }
      ]
    },
    {
      "page": "course/chapters/G12-films-and-detachment.html",
      "tex": "m_A w_{tt}+c_Aw_t+D\\nabla^4w-N_0\\nabla^2w=\\Delta p(\\mathbf x,t),\n\\qquad D=\\frac{Eh^3}{12(1-\\nu_s^2)}. \\tag{12.1}",
      "context": "12.2 Plate and membrane limits / 板与膜的极限",
      "source": {
        "book": "generic",
        "line": 893,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-2283c1937941b73c.svg",
      "variables": [
        {
          "tex": "m_A",
          "symbol": "m_A",
          "role": "moving mass per unit area"
        },
        {
          "tex": "w_{tt}",
          "symbol": "w_tt",
          "role": "transverse film displacement"
        },
        {
          "tex": "c_A",
          "symbol": "c_A",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "w_t",
          "symbol": "w_t",
          "role": "transverse film displacement"
        },
        {
          "tex": "D",
          "symbol": "D",
          "role": "plate bending rigidity"
        },
        {
          "tex": "w",
          "symbol": "w",
          "role": "transverse film displacement"
        },
        {
          "tex": "N_0",
          "symbol": "N_0",
          "role": "event count / membrane tension"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\nu_s",
          "symbol": "ν_s",
          "role": "Poisson ratio"
        }
      ]
    },
    {
      "page": "course/chapters/G12-films-and-detachment.html",
      "tex": "M_n\\ddot q_n+C_n\\dot q_n+K_nq_n=F_n(t),\\qquad\nF_n(t)=\\int_A\\Delta p(\\mathbf x,t)\\phi_n(\\mathbf x)\\,dA. \\tag{12.2}",
      "context": "12.3 Modal projection shows why load shape matters / 模态投影解释载荷分布的重要性",
      "source": {
        "book": "generic",
        "line": 910,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1c08afd7ffb46017.svg",
      "variables": [
        {
          "tex": "M_n",
          "symbol": "M_n",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "\\ddot q_n",
          "symbol": "q̈_n",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "C_n",
          "symbol": "C_n",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "\\dot q_n",
          "symbol": "q̇_n",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "K_n",
          "symbol": "K_n",
          "role": "stiffness / collapse constant"
        },
        {
          "tex": "q_n",
          "symbol": "q_n",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "F_n",
          "symbol": "F_n",
          "role": "force / fluence"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "\\phi_n",
          "symbol": "φ_n",
          "role": "velocity potential / mode shape"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/G12-films-and-detachment.html",
      "tex": "A_n=\\frac{|I_n|}{M_n\\omega_n}\n\\left|\\frac{2\\sin(\\omega_n\\tau_p/2)}{\\omega_n\\tau_p}\\right|. \\tag{12.3}",
      "context": "12.3 Modal projection shows why load shape matters / 模态投影解释载荷分布的重要性",
      "source": {
        "book": "generic",
        "line": 921,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-e15f838c92178927.svg",
      "variables": [
        {
          "tex": "A_n",
          "symbol": "A_n",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "I_n",
          "symbol": "I_n",
          "role": "pressure or modal impulse"
        },
        {
          "tex": "M_n",
          "symbol": "M_n",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "\\omega_n",
          "symbol": "ω_n",
          "role": "structural modal angular frequency"
        },
        {
          "tex": "\\tau_p",
          "symbol": "τ_p",
          "role": "duration / dimensionless time / dummy time"
        }
      ]
    },
    {
      "page": "course/chapters/G12-films-and-detachment.html",
      "tex": "D=Eh^3/[12(1-\\nu^2)],\\qquad \\Pi_b=\\Delta p\\,a^4/(Dh)",
      "context": "Bubble-driven films, membranes and detachment · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G12",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-94addbbeb06f3e00.svg",
      "variables": [
        {
          "tex": "D",
          "symbol": "D",
          "role": "bending rigidity / dose / residual derivative"
        },
        {
          "tex": "\\nu",
          "symbol": "ν",
          "role": "Poisson ratio"
        },
        {
          "tex": "\\Pi_b",
          "symbol": "Π_b",
          "role": "pressure impulse / shell traction / availability"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        }
      ]
    },
    {
      "page": "course/chapters/G12-films-and-detachment.html",
      "tex": "q_{\\mathrm{free}}=\\frac{|I_n|}{M_n\\omega_n}\\left|\\frac{2\\sin(\\omega_n\\tau_p/2)}{\\omega_n\\tau_p}\\right|",
      "context": "Bubble-driven films, membranes and detachment · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G12",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-3157c3f62ee69800.svg",
      "variables": [
        {
          "tex": "q_{\\mathrm{free}}",
          "symbol": "q_free",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "I_n",
          "symbol": "I_n",
          "role": "pressure or modal impulse"
        },
        {
          "tex": "M_n",
          "symbol": "M_n",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "\\omega_n",
          "symbol": "ω_n",
          "role": "structural modal angular frequency"
        },
        {
          "tex": "\\tau_p",
          "symbol": "τ_p",
          "role": "duration / dimensionless time / dummy time"
        }
      ]
    },
    {
      "page": "course/chapters/G12-films-and-detachment.html",
      "tex": "\\tfrac12m_Av^2=G_c,\\qquad I_{p,\\mathrm{nom}}=\\sqrt{2m_AG_c}",
      "context": "Bubble-driven films, membranes and detachment · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G12",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-96c1376e81617d85.svg",
      "variables": [
        {
          "tex": "m_A",
          "symbol": "m_A",
          "role": "moving mass per unit area"
        },
        {
          "tex": "v",
          "symbol": "v",
          "role": "velocity / squared wall speed as defined"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        },
        {
          "tex": "I_{p,\\mathrm{nom}}",
          "symbol": "I_p,nom",
          "role": "pressure or modal impulse"
        }
      ]
    },
    {
      "page": "course/chapters/G14-measurement-and-inference.html",
      "tex": "\\chi^2(\\boldsymbol\\theta)=\\sum_i\n\\frac{[R_{\\mathrm{model}}(t_i;\\boldsymbol\\theta)-R_i]^2}{s_i^2}. \\tag{14.1}",
      "context": "14.3 Identifiability: several stories can fit one curve / 可辨识性：一条曲线可能对应多种解释",
      "source": {
        "book": "generic",
        "line": 1097,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1816aeed9751a604.svg",
      "variables": [
        {
          "tex": "\\chi",
          "symbol": "χ",
          "role": "confinement / susceptibility"
        },
        {
          "tex": "\\theta",
          "symbol": "θ",
          "role": "polar coordinate"
        },
        {
          "tex": "R_{\\mathrm{model}}",
          "symbol": "R_model",
          "role": "bubble wall radius"
        },
        {
          "tex": "t_i",
          "symbol": "t_i",
          "role": "physical time"
        },
        {
          "tex": "R_i",
          "symbol": "R_i",
          "role": "bubble wall radius"
        },
        {
          "tex": "s_i",
          "symbol": "s_i",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/G14-measurement-and-inference.html",
      "tex": "\\ddot R_i\\approx(R_{i+1}-2R_i+R_{i-1})/\\Delta t^2,\\qquad \\sigma_{\\ddot R}=\\sqrt6\\,\\sigma_R/\\Delta t^2",
      "context": "Measurement, inference and uncertainty · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G14",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-90920ac440e50551.svg",
      "variables": [
        {
          "tex": "\\ddot R_i",
          "symbol": "R̈_i",
          "role": "bubble wall radius"
        },
        {
          "tex": "R_{i+1}",
          "symbol": "R_i+1",
          "role": "bubble wall radius"
        },
        {
          "tex": "R_i",
          "symbol": "R_i",
          "role": "bubble wall radius"
        },
        {
          "tex": "R_{i-1}",
          "symbol": "R_i-1",
          "role": "bubble wall radius"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\sigma_{\\ddot R}",
          "symbol": "σ_R̈",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\sigma_R",
          "symbol": "σ_R",
          "role": "interface tension / solid stress"
        }
      ]
    },
    {
      "page": "course/chapters/G14-measurement-and-inference.html",
      "tex": "C=p_{g0}R_0^{3\\kappa},\\qquad p_{\\mathrm{meas}}=h_{\\mathrm{det}}*p_{\\mathrm{true}}",
      "context": "Measurement, inference and uncertainty · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G14",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-7efbad68e8ac7034.svg",
      "variables": [
        {
          "tex": "C",
          "symbol": "C",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "p_{\\mathrm{meas}}",
          "symbol": "p_meas",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "h_{\\mathrm{det}}",
          "symbol": "h_det",
          "role": "stand-off / layer thickness"
        },
        {
          "tex": "p_{\\mathrm{true}}",
          "symbol": "p_true",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/G14-measurement-and-inference.html",
      "tex": "p=\\log_2[(Q_h-Q_{h/2})/(Q_{h/2}-Q_{h/4})],\\qquad Q_*=Q_{h/4}+(Q_{h/4}-Q_{h/2})/(2^p-1)",
      "context": "Measurement, inference and uncertainty · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G14",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-f8d2267765513a21.svg",
      "variables": [
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "Q_h",
          "symbol": "Q_h",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "Q_{h/2}",
          "symbol": "Q_h/2",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "Q_{h/4}",
          "symbol": "Q_h/4",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "Q",
          "symbol": "Q",
          "role": "heat or generalized load as defined"
        }
      ]
    },
    {
      "page": "course/chapters/G15-liquid-pvc-film-capstone.html",
      "tex": "V_b=V_{\\mathrm{base}}+\\Delta V\\sin^4(\\pi t/T_b),\\qquad R=(3V_b/4\\pi)^{1/3}",
      "context": "Capstone: a liquid–PVC–film research model · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G15",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-03074ee97ce19eb6.svg",
      "variables": [
        {
          "tex": "V_b",
          "symbol": "V_b",
          "role": "bubble or observation volume"
        },
        {
          "tex": "V_{\\mathrm{base}}",
          "symbol": "V_base",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "T_b",
          "symbol": "T_b",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/G15-liquid-pvc-film-capstone.html",
      "tex": "\\gamma=h/R_{\\max},\\quad \\chi=R_{\\max}/H_L,\\quad \\lambda_f=R_{\\max}/a_f,\\quad \\Theta=\\omega_nT_b",
      "context": "Capstone: a liquid–PVC–film research model · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G15",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-7e73bf0f2a1928d4.svg",
      "variables": [
        {
          "tex": "\\gamma",
          "symbol": "γ",
          "role": "stand-off / heat-capacity ratio"
        },
        {
          "tex": "h",
          "symbol": "h",
          "role": "stand-off / layer thickness"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "\\chi",
          "symbol": "χ",
          "role": "confinement / susceptibility"
        },
        {
          "tex": "H_L",
          "symbol": "H_L",
          "role": "enthalpy / absorbed dose / liquid depth"
        },
        {
          "tex": "\\lambda_f",
          "symbol": "λ_f",
          "role": "bubble-to-film size ratio"
        },
        {
          "tex": "a_f",
          "symbol": "a_f",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\Theta",
          "symbol": "Θ",
          "role": "loading-to-mode timing ratio"
        },
        {
          "tex": "\\omega_n",
          "symbol": "ω_n",
          "role": "structural modal angular frequency"
        },
        {
          "tex": "T_b",
          "symbol": "T_b",
          "role": "temperature / kinetic energy as defined"
        }
      ]
    },
    {
      "page": "course/chapters/G15-liquid-pvc-film-capstone.html",
      "tex": "Q_n(t)=\\int_A\\Delta p(\\mathbf x,t)\\phi_n\\,dA,\\qquad M_n\\ddot q_n+C_n\\dot q_n+K_nq_n=Q_n(t)",
      "context": "Capstone: a liquid–PVC–film research model · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G15",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-89f79872a9de33be.svg",
      "variables": [
        {
          "tex": "Q_n",
          "symbol": "Q_n",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "\\phi_n",
          "symbol": "φ_n",
          "role": "assumed spatial mode shape"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "M_n",
          "symbol": "M_n",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "\\ddot q_n",
          "symbol": "q̈_n",
          "role": "generalized mode displacement"
        },
        {
          "tex": "C_n",
          "symbol": "C_n",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "\\dot q_n",
          "symbol": "q̇_n",
          "role": "generalized mode displacement"
        },
        {
          "tex": "K_n",
          "symbol": "K_n",
          "role": "stiffness / collapse constant"
        },
        {
          "tex": "q_n",
          "symbol": "q_n",
          "role": "generalized mode displacement"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "p_\\sigma=2\\sigma/R",
      "context": "16.1 Laplace pressure at three scales / 三个尺度下的Laplace压力",
      "source": {
        "type": "original-problem-law",
        "number": 1
      },
      "figure": "assets/curriculum-figures/formula-e0df73df907cc3ec.svg",
      "variables": [
        {
          "tex": "p_\\sigma",
          "symbol": "p_σ",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "T=\\tfrac12\\rho\\int_R^\\infty u^2 4\\pi r^2dr,\\quad u=R^2\\dot R/r^2",
      "context": "16.2 Where the liquid kinetic energy resides / 液体动能分布在哪里",
      "source": {
        "type": "original-problem-law",
        "number": 2
      },
      "figure": "assets/curriculum-figures/formula-544b200bd511c654.svg",
      "variables": [
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "\\dot r=u(r,t)=R^2\\dot R/r^2",
      "context": "16.3 Track a liquid particle / 跟踪液体质点",
      "source": {
        "type": "original-problem-law",
        "number": 3
      },
      "figure": "assets/curriculum-figures/formula-e10972baaabc73e8.svg",
      "variables": [
        {
          "tex": "\\dot r",
          "symbol": "ṙ",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "\\phi=-R^2\\dot R/r,\\quad d\\phi(R(t),t)/dt=\\partial_t\\phi+\\dot R\\partial_r\\phi",
      "context": "16.4 Diagnose a Bernoulli error / 诊断Bernoulli推导错误",
      "source": {
        "type": "original-problem-law",
        "number": 4
      },
      "figure": "assets/curriculum-figures/formula-3aa4d8e8d149fa92.svg",
      "variables": [
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "velocity potential / mode shape"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "p_{g0}=p_0-p_v+2\\sigma/R_0,\\quad R_B=\\sqrt{3p_{g0}R_0^3/(2\\sigma)},\\quad p_{\\mathrm{Blake}}=p_v-4\\sigma/(3R_B)",
      "context": "16.5 An isothermal Blake threshold / 等温Blake阈值",
      "source": {
        "type": "original-problem-law",
        "number": 5
      },
      "figure": "assets/curriculum-figures/formula-9cba76bc4a9ed787.svg",
      "variables": [
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "p_0",
          "symbol": "p_0",
          "role": "reference ambient pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "R_B",
          "symbol": "R_B",
          "role": "bubble wall radius"
        },
        {
          "tex": "p_{\\mathrm{Blake}}",
          "symbol": "p_Blake",
          "role": "critical ambient pressure"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "P=1-e^{-JV\\tau}",
      "context": "16.6 Probability depends on observation volume / 概率依赖观测体积",
      "source": {
        "type": "original-problem-law",
        "number": 6
      },
      "figure": "assets/curriculum-figures/formula-46027d27d0b7c0d0.svg",
      "variables": [
        {
          "tex": "P",
          "symbol": "P",
          "role": "event probability"
        },
        {
          "tex": "J",
          "symbol": "J",
          "role": "nucleation rate / areal impulse"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\tau",
          "symbol": "τ",
          "role": "duration / dimensionless time / dummy time"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "R\\ddot R+\\tfrac32\\dot R^2=\\Delta p/\\rho",
      "context": "16.7 Long-time inertial growth speed / 长时间惯性增长速度",
      "source": {
        "type": "original-problem-law",
        "number": 7
      },
      "figure": "assets/curriculum-figures/formula-17dd878103d140a7.svg",
      "variables": [
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "t_c=0.914681R_{\\max}\\sqrt{\\rho/\\Delta p}",
      "context": "16.8 Collapse-time scaling / 塌缩时间尺度",
      "source": {
        "type": "original-problem-law",
        "number": 8
      },
      "figure": "assets/curriculum-figures/formula-8211bc1d80fbc322.svg",
      "variables": [
        {
          "tex": "t_c",
          "symbol": "t_c",
          "role": "collapse duration"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "\\dot R^2=\\frac{2\\Delta p}{3\\rho}[(R_{\\max}/R)^3-1]",
      "context": "16.9 Speed at half maximum radius / 半最大半径处的速度",
      "source": {
        "type": "original-problem-law",
        "number": 9
      },
      "figure": "assets/curriculum-figures/formula-84f71ecb407e8775.svg",
      "variables": [
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "M_w=|\\dot R|/c,\\quad R/R_{\\max}=[1+3\\rho c^2M_w^2/(2\\Delta p)]^{-1/3}",
      "context": "16.10 When the incompressible collapse becomes suspect / 不可压缩塌缩何时值得怀疑",
      "source": {
        "type": "original-problem-law",
        "number": 10
      },
      "figure": "assets/curriculum-figures/formula-7e4ffcdc98d80c96.svg",
      "variables": [
        {
          "tex": "M_w",
          "symbol": "M_w",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "\\omega_0^2=(3\\kappa p_{g0}-2\\sigma/R_0)/(\\rho R_0^2),\\quad x=-\\delta p/(\\rho R_0^2\\omega_0^2)",
      "context": "16.11 Static compliance of a microbubble / 微气泡的静态柔度",
      "source": {
        "type": "original-problem-law",
        "number": 11
      },
      "figure": "assets/curriculum-figures/formula-57c9320186701203.svg",
      "variables": [
        {
          "tex": "\\omega_0",
          "symbol": "ω_0",
          "role": "linear radial angular frequency"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "\\delta",
          "symbol": "δ",
          "role": "separation / perturbation / penetration depth"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "\\beta_\\mu=2\\mu/(\\rho R_0^2)",
      "context": "16.12 Viscosity versus stiffness / 黏度与刚度",
      "source": {
        "type": "original-problem-law",
        "number": 12
      },
      "figure": "assets/curriculum-figures/formula-db111e0c2675ac20.svg",
      "variables": [
        {
          "tex": "\\beta_\\mu",
          "symbol": "β_μ",
          "role": "damping / logistic coefficient"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "W_g=\\int -p_g\\,dV,\\quad p_g=p_{g0}V_0/V",
      "context": "16.13 The isothermal gas potential / 等温气体势",
      "source": {
        "type": "original-problem-law",
        "number": 13
      },
      "figure": "assets/curriculum-figures/formula-fc578272ceb5b3b7.svg",
      "variables": [
        {
          "tex": "W_g",
          "symbol": "W_g",
          "role": "work / potential energy"
        },
        {
          "tex": "p_g",
          "symbol": "p_g",
          "role": "noncondensable gas pressure"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "V_0",
          "symbol": "V_0",
          "role": "bubble or observation volume"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "\\dot p_L=D_p-4\\mu\\ddot R/R",
      "context": "16.14 The hidden acceleration in Keller–Miksis / Keller–Miksis中的隐藏加速度",
      "source": {
        "type": "original-problem-law",
        "number": 14
      },
      "figure": "assets/curriculum-figures/formula-6be0daea6f4a0315.svg",
      "variables": [
        {
          "tex": "\\dot p_L",
          "symbol": "ṗ_L",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "D_p",
          "symbol": "D_p",
          "role": "bending rigidity / dose / residual derivative"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "\\omega_l^2=(l-1)(l+1)(l+2)\\sigma/(\\rho R_0^3)",
      "context": "16.15 Radial frequency versus shape frequency / 径向频率与形状频率",
      "source": {
        "type": "original-problem-law",
        "number": 15
      },
      "figure": "assets/curriculum-figures/formula-775df456a1106547.svg",
      "variables": [
        {
          "tex": "\\omega_l",
          "symbol": "ω_l",
          "role": "angular frequency"
        },
        {
          "tex": "l",
          "symbol": "l",
          "role": "shape harmonic order"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "I_p=\\Delta p\\tau,\\quad \\Delta U=I_p/(\\rho L)",
      "context": "16.16 A uniform pressure impulse / 均匀压力冲量",
      "source": {
        "type": "original-problem-law",
        "number": 16
      },
      "figure": "assets/curriculum-figures/formula-503ace99cf84f3b6.svg",
      "variables": [
        {
          "tex": "I_p",
          "symbol": "I_p",
          "role": "pressure impulse per unit area"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\tau",
          "symbol": "τ",
          "role": "duration / dimensionless time / dummy time"
        },
        {
          "tex": "U",
          "symbol": "U",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "L",
          "symbol": "L",
          "role": "length / latent heat / load disk radius"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "q_{\\mathrm{free}}=\\frac{|I_n|}{M_n\\omega_n}\\left|\\frac{2\\sin(\\omega_n\\tau_p/2)}{\\omega_n\\tau_p}\\right|",
      "context": "16.17 Equal impulse, zero residual oscillation / 相同冲量却无残余振动",
      "source": {
        "type": "original-problem-law",
        "number": 17
      },
      "figure": "assets/curriculum-figures/formula-7a665f5681fe072f.svg",
      "variables": [
        {
          "tex": "q_{\\mathrm{free}}",
          "symbol": "q_free",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "I_n",
          "symbol": "I_n",
          "role": "pressure or modal impulse"
        },
        {
          "tex": "M_n",
          "symbol": "M_n",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "\\omega_n",
          "symbol": "ω_n",
          "role": "structural modal angular frequency"
        },
        {
          "tex": "\\tau_p",
          "symbol": "τ_p",
          "role": "duration / dimensionless time / dummy time"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "\\tfrac12m_Av^2=G_c,\\quad I_p=m_Av",
      "context": "16.18 A necessary adhesion-energy scale / 黏附能的必要尺度",
      "source": {
        "type": "original-problem-law",
        "number": 18
      },
      "figure": "assets/curriculum-figures/formula-880b70e5f2610bb8.svg",
      "variables": [
        {
          "tex": "m_A",
          "symbol": "m_A",
          "role": "moving mass per unit area"
        },
        {
          "tex": "v",
          "symbol": "v",
          "role": "velocity / squared wall speed as defined"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        },
        {
          "tex": "I_p",
          "symbol": "I_p",
          "role": "pressure impulse per unit area"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "R_b/a_0=[\\rho_dR_uT/(Mp_{PF})]^{1/3}",
      "context": "16.19 Droplet expansion by mass conservation / 由质量守恒估计液滴膨胀",
      "source": {
        "type": "original-problem-law",
        "number": 19
      },
      "figure": "assets/curriculum-figures/formula-b5fac0217eb89565.svg",
      "variables": [
        {
          "tex": "R_b",
          "symbol": "R_b",
          "role": "bubble wall radius"
        },
        {
          "tex": "a_0",
          "symbol": "a_0",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\rho_d",
          "symbol": "ρ_d",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_u",
          "symbol": "R_u",
          "role": "bubble wall radius"
        },
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "M",
          "symbol": "M",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "p_{PF}",
          "symbol": "p_PF",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "E_a=\\frac{4\\pi r^2}{\\rho c}\\int p'^2dt",
      "context": "16.20 Geometrical spreading of acoustic energy / 声能的几何扩散",
      "source": {
        "type": "original-problem-law",
        "number": 20
      },
      "figure": "assets/curriculum-figures/formula-c7ba9358060826c8.svg",
      "variables": [
        {
          "tex": "E_a",
          "symbol": "E_a",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "T_g/T_{g0}=(R_0/R)^{3(\\kappa-1)}",
      "context": "16.21 A false temperature prediction / 错误的温度预测",
      "source": {
        "type": "original-problem-law",
        "number": 21
      },
      "figure": "assets/curriculum-figures/formula-dbe87d5648644723.svg",
      "variables": [
        {
          "tex": "T_g",
          "symbol": "T_g",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "T_{g0}",
          "symbol": "T_g0",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "p=\\log_2[(Q_h-Q_{h/2})/(Q_{h/2}-Q_{h/4})],\\quad Q_*=Q_{h/4}+(Q_{h/4}-Q_{h/2})/(2^p-1)",
      "context": "16.22 A mesh-convergence calculation / 网格收敛计算",
      "source": {
        "type": "original-problem-law",
        "number": 22
      },
      "figure": "assets/curriculum-figures/formula-72d2c50da4c98973.svg",
      "variables": [
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "Q_h",
          "symbol": "Q_h",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "Q_{h/2}",
          "symbol": "Q_h/2",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "Q_{h/4}",
          "symbol": "Q_h/4",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "Q",
          "symbol": "Q",
          "role": "heat or generalized load as defined"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "p_g=C/R^{3\\kappa},\\quad C=p_{g0}R_0^{3\\kappa}",
      "context": "16.23 A nonidentifiable gas parameterization / 不可辨识的气体参数化",
      "source": {
        "type": "original-problem-law",
        "number": 23
      },
      "figure": "assets/curriculum-figures/formula-777b54c961c9c006.svg",
      "variables": [
        {
          "tex": "p_g",
          "symbol": "p_g",
          "role": "noncondensable gas pressure"
        },
        {
          "tex": "C",
          "symbol": "C",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "p_{g0}",
          "symbol": "p_g0",
          "role": "reference gas pressure"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "\\ddot R_i\\approx(R_{i+1}-2R_i+R_{i-1})/\\Delta t^2",
      "context": "16.24 Why more frames do not automatically fix acceleration / 为什么更多图像帧不自动解决加速度问题",
      "source": {
        "type": "original-problem-law",
        "number": 24
      },
      "figure": "assets/curriculum-figures/formula-d8fda3ae30a37676.svg",
      "variables": [
        {
          "tex": "\\ddot R_i",
          "symbol": "R̈_i",
          "role": "bubble wall radius"
        },
        {
          "tex": "R_{i+1}",
          "symbol": "R_i+1",
          "role": "bubble wall radius"
        },
        {
          "tex": "R_i",
          "symbol": "R_i",
          "role": "bubble wall radius"
        },
        {
          "tex": "R_{i-1}",
          "symbol": "R_i-1",
          "role": "bubble wall radius"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "E_{\\mathrm{laser}}\\longrightarrow p_{\\mathrm{acoustic}}(t)\\longrightarrow R(t)",
      "context": "16.25 Source-mechanism classification / 源机制分类",
      "source": {
        "type": "original-problem-law",
        "number": 25
      },
      "figure": "assets/curriculum-figures/formula-57249b23d8250228.svg",
      "variables": [
        {
          "tex": "E_{\\mathrm{laser}}",
          "symbol": "E_laser",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "p_{\\mathrm{acoustic}}",
          "symbol": "p_acoustic",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "p_{PF}V_b=m_{PF}R_uT/M",
      "context": "16.26 A terminology audit / 术语审查",
      "source": {
        "type": "original-problem-law",
        "number": 26
      },
      "figure": "assets/curriculum-figures/formula-29e0b5b12a14c80e.svg",
      "variables": [
        {
          "tex": "p_{PF}",
          "symbol": "p_PF",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "V_b",
          "symbol": "V_b",
          "role": "bubble or observation volume"
        },
        {
          "tex": "m_{PF}",
          "symbol": "m_PF",
          "role": "mass or areal mass"
        },
        {
          "tex": "R_u",
          "symbol": "R_u",
          "role": "bubble wall radius"
        },
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "M",
          "symbol": "M",
          "role": "generalized mass / molar mass"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "p_{B,\\mathrm{required}}=p_\\infty+\\rho(R\\ddot R+\\tfrac32\\dot R^2)+2\\sigma/R+4\\mu\\dot R/R",
      "context": "16.27 Detect an overconstrained interface / 识别过度约束界面",
      "source": {
        "type": "original-problem-law",
        "number": 27
      },
      "figure": "assets/curriculum-figures/formula-f1c02f5fb34cf9ea.svg",
      "variables": [
        {
          "tex": "p_{B,\\mathrm{required}}",
          "symbol": "p_B,required",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "(M_s+M_a)\\ddot q+C\\dot q+Kq=Q(t)",
      "context": "16.28 Avoid double-counting fluid inertia / 避免重复计算流体惯性",
      "source": {
        "type": "original-problem-law",
        "number": 28
      },
      "figure": "assets/curriculum-figures/formula-ad7280829f816669.svg",
      "variables": [
        {
          "tex": "M_s",
          "symbol": "M_s",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "M_a",
          "symbol": "M_a",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "\\ddot q",
          "symbol": "q̈",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "C",
          "symbol": "C",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "\\dot q",
          "symbol": "q̇",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "K",
          "symbol": "K",
          "role": "stiffness / collapse constant"
        },
        {
          "tex": "q",
          "symbol": "q",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "Q",
          "symbol": "Q",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "t_a=\\ell/c,\\quad t_{\\mathrm{th}}=\\ell^2/\\alpha",
      "context": "16.29 Design a discriminating experiment / 设计具有区分能力的实验",
      "source": {
        "type": "original-problem-law",
        "number": 29
      },
      "figure": "assets/curriculum-figures/formula-ef20d2b130628b9e.svg",
      "variables": [
        {
          "tex": "t_a",
          "symbol": "t_a",
          "role": "acoustic crossing time"
        },
        {
          "tex": "\\ell",
          "symbol": "ℓ",
          "role": "absorption / cohesive-zone length"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "t_{\\mathrm{th}}",
          "symbol": "t_th",
          "role": "physical time"
        },
        {
          "tex": "\\alpha",
          "symbol": "α",
          "role": "thermal diffusivity"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "\\Theta=\\omega_nt_b,\\quad Q_n=\\int_A\\Delta p\\phi_n dA",
      "context": "16.30 Build a falsifiable capstone study / 构建可证伪综合研究",
      "source": {
        "type": "original-problem-law",
        "number": 30
      },
      "figure": "assets/curriculum-figures/formula-263db7b92eb41d1e.svg",
      "variables": [
        {
          "tex": "\\Theta",
          "symbol": "Θ",
          "role": "loading-to-mode timing ratio"
        },
        {
          "tex": "\\omega_n",
          "symbol": "ω_n",
          "role": "structural modal angular frequency"
        },
        {
          "tex": "t_b",
          "symbol": "t_b",
          "role": "physical time"
        },
        {
          "tex": "Q_n",
          "symbol": "Q_n",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\phi_n",
          "symbol": "φ_n",
          "role": "velocity potential / mode shape"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "t_c=0.914681R_{\\max}\\sqrt{\\rho/\\Delta p},\\quad \\dot R=-\\sqrt{2\\Delta p[(R_{\\max}/R)^3-1]/(3\\rho)},\\quad M_w=|\\dot R|/c",
      "context": "Thirty original worked problems · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G16",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-604bf61b2b947fd3.svg",
      "variables": [
        {
          "tex": "t_c",
          "symbol": "t_c",
          "role": "collapse duration"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "M_w",
          "symbol": "M_w",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/G16-original-worked-problems.html",
      "tex": "I_{p,\\mathrm{nom}}=\\sqrt{2m_AG_c},\\qquad Q_n=\\int_A\\Delta p\\phi_n\\,dA",
      "context": "Thirty original worked problems · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "G16",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-3017746d726b9f62.svg",
      "variables": [
        {
          "tex": "I_{p,\\mathrm{nom}}",
          "symbol": "I_p,nom",
          "role": "pressure or modal impulse"
        },
        {
          "tex": "m_A",
          "symbol": "m_A",
          "role": "moving mass per unit area"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        },
        {
          "tex": "Q_n",
          "symbol": "Q_n",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\phi_n",
          "symbol": "φ_n",
          "role": "velocity potential / mode shape"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\n\\begin{gathered}\n\\text{Optical energy deposition / 光能沉积}\\\\\n\\downarrow\\\\\n\\text{Activation probability / 激活概率}\\\\\n\\downarrow\\\\\n\\text{Bubble motion and pressure / 气泡运动与压力}\\\\\n\\downarrow\\\\\n\\text{Structural response and release / 结构响应与释放}\\\\\n\\downarrow\\\\\n\\text{Sensitivity and measurements / 敏感性与测量}\n\\end{gathered}\n",
      "context": "Reader’s orientation / 阅读导引",
      "source": {
        "book": "project-theory",
        "line": 27,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-a91d6964ed9fba33.svg",
      "variables": []
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "[F]=\\mathrm{J\\,m^{-2}}.",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 61,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-ce1391dee48e0e1f.svg",
      "variables": [
        {
          "tex": "F",
          "symbol": "F",
          "role": "local beam fluence"
        },
        {
          "tex": "J",
          "symbol": "J",
          "role": "nucleation rate / areal impulse"
        },
        {
          "tex": "m",
          "symbol": "m",
          "role": "mass or areal mass"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "[I]=\\mathrm{W\\,m^{-2}}.",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 67,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-743b2a9e60ce2293.svg",
      "variables": [
        {
          "tex": "I",
          "symbol": "I",
          "role": "pressure or modal impulse"
        },
        {
          "tex": "W",
          "symbol": "W",
          "role": "work / potential energy"
        },
        {
          "tex": "m",
          "symbol": "m",
          "role": "mass or areal mass"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "F(r)=F_0\\exp\\left(-\\frac{2r^2}{w_0^2}\\right),",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 73,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-6d9d8c57868efabd.svg",
      "variables": [
        {
          "tex": "F",
          "symbol": "F",
          "role": "local beam fluence"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "distance from beam axis"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "w_0",
          "symbol": "w_0",
          "role": "Gaussian beam waist radius"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "E_p=2\\pi F_0\\int_0^\\infty r\\exp\\left(-\\frac{2r^2}{w_0^2}\\right)\\,dr.",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 79,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-444941e112a42707.svg",
      "variables": [
        {
          "tex": "E_p",
          "symbol": "E_p",
          "role": "incident laser-pulse energy"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "distance from beam axis"
        },
        {
          "tex": "w_0",
          "symbol": "w_0",
          "role": "Gaussian beam waist radius"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "u=\\frac{2r^2}{w_0^2},\\qquad r\\,dr=\\frac{w_0^2}{4}\\,du.",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 85,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-8b4ca96c4fa034ad.svg",
      "variables": [
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "distance from beam axis"
        },
        {
          "tex": "w_0",
          "symbol": "w_0",
          "role": "Gaussian beam waist radius"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "E_p=\\frac{\\pi w_0^2F_0}{2}\\int_0^\\infty e^{-u}\\,du,",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 91,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-d653637a332abefc.svg",
      "variables": [
        {
          "tex": "E_p",
          "symbol": "E_p",
          "role": "incident laser-pulse energy"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "w_0",
          "symbol": "w_0",
          "role": "Gaussian beam waist radius"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{E_p=\\frac{\\pi w_0^2F_0}{2}}.",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 97,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-a6af8d3445008443.svg",
      "variables": [
        {
          "tex": "E_p",
          "symbol": "E_p",
          "role": "incident laser-pulse energy"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "w_0",
          "symbol": "w_0",
          "role": "Gaussian beam waist radius"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "w_0=25\\times10^{-6}\\ \\mathrm m,\\qquad F_0=0.10\\ \\mathrm{J\\,cm^{-2}}=1000\\ \\mathrm{J\\,m^{-2}},",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 107,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-4861d404ac9b80d5.svg",
      "variables": [
        {
          "tex": "w_0",
          "symbol": "w_0",
          "role": "Gaussian beam waist radius"
        },
        {
          "tex": "m",
          "symbol": "m",
          "role": "mass or areal mass"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "J",
          "symbol": "J",
          "role": "nucleation rate / areal impulse"
        },
        {
          "tex": "c",
          "symbol": "c",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "E_p=\\frac{\\pi(25\\times10^{-6})^2(1000)}{2}=9.8175\\times10^{-7}\\ \\mathrm J.",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 113,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-ccfb2d6fbfec4acc.svg",
      "variables": [
        {
          "tex": "E_p",
          "symbol": "E_p",
          "role": "incident laser-pulse energy"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "J",
          "symbol": "J",
          "role": "nucleation rate / areal impulse"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{E_p=0.98175\\ \\mu\\mathrm J}.",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 119,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-47d101b22f42c6c0.svg",
      "variables": [
        {
          "tex": "E_p",
          "symbol": "E_p",
          "role": "incident laser-pulse energy"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "J",
          "symbol": "J",
          "role": "nucleation rate / areal impulse"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{E_{\\mathrm{abs}}=\\eta_{\\mathrm{abs}}E_p=0.29452\\ \\mu\\mathrm J}.",
      "context": "1.1 From beam fluence to total pulse energy / 从光束能量面密度到脉冲总能量",
      "source": {
        "book": "project-theory",
        "line": 125,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-951d1d69a02631e2.svg",
      "variables": [
        {
          "tex": "E_{\\mathrm{abs}}",
          "symbol": "E_abs",
          "role": "total absorbed energy"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "E_p",
          "symbol": "E_p",
          "role": "incident laser-pulse energy"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "J",
          "symbol": "J",
          "role": "nucleation rate / areal impulse"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "I(r,t)=F(r)g(t),",
      "context": "1.2 From pulse energy to a time-dependent source / 从脉冲能量到时变热源",
      "source": {
        "book": "project-theory",
        "line": 137,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-ee085563f040fd37.svg",
      "variables": [
        {
          "tex": "I",
          "symbol": "I",
          "role": "pressure or modal impulse"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "distance from beam axis"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "F",
          "symbol": "F",
          "role": "local beam fluence"
        },
        {
          "tex": "g",
          "symbol": "g",
          "role": "unit-integral pulse profile"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\int_{-\\infty}^{\\infty}g(t)\\,dt=1.",
      "context": "1.2 From pulse energy to a time-dependent source / 从脉冲能量到时变热源",
      "source": {
        "book": "project-theory",
        "line": 143,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-b5d7df6b75c9e9f6.svg",
      "variables": [
        {
          "tex": "g",
          "symbol": "g",
          "role": "temporal pulse profile / wall gap"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\int_{-\\infty}^{\\infty}I(r,t)\\,dt=F(r).",
      "context": "1.2 From pulse energy to a time-dependent source / 从脉冲能量到时变热源",
      "source": {
        "book": "project-theory",
        "line": 149,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1960cc4c3f2dfec5.svg",
      "variables": [
        {
          "tex": "I",
          "symbol": "I",
          "role": "pressure or modal impulse"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "F",
          "symbol": "F",
          "role": "force / fluence"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{g(t)=\\frac{1}{\\tau_L}\\sqrt{\\frac{4\\ln2}{\\pi}}\\exp\\left[-4\\ln2\\left(\\frac{t-t_c}{\\tau_L}\\right)^2\\right]}.",
      "context": "1.2 From pulse energy to a time-dependent source / 从脉冲能量到时变热源",
      "source": {
        "book": "project-theory",
        "line": 155,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-bca25e4a80ba951f.svg",
      "variables": [
        {
          "tex": "g",
          "symbol": "g",
          "role": "unit-integral pulse profile"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\tau_L",
          "symbol": "τ_L",
          "role": "laser pulse FWHM"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "t_c",
          "symbol": "t_c",
          "role": "collapse duration"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "[g]=\\mathrm{s^{-1}}.",
      "context": "1.2 From pulse energy to a time-dependent source / 从脉冲能量到时变热源",
      "source": {
        "book": "project-theory",
        "line": 165,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-47109c9f66fb0a45.svg",
      "variables": [
        {
          "tex": "g",
          "symbol": "g",
          "role": "temporal pulse profile / wall gap"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\frac{\\partial I}{\\partial z}=-\\mu_a I,",
      "context": "1.3 From surface intensity to volumetric heating / 从表面光强到体积热源",
      "source": {
        "book": "project-theory",
        "line": 177,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-29f26b3ce8db3e26.svg",
      "variables": [
        {
          "tex": "I",
          "symbol": "I",
          "role": "pressure or modal impulse"
        },
        {
          "tex": "z",
          "symbol": "z",
          "role": "depth / formulation predictor"
        },
        {
          "tex": "\\mu_a",
          "symbol": "μ_a",
          "role": "optical absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "I(z)=I(0)e^{-\\mu_a z}.",
      "context": "1.3 From surface intensity to volumetric heating / 从表面光强到体积热源",
      "source": {
        "book": "project-theory",
        "line": 183,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-2c6cfc0c07bbf23f.svg",
      "variables": [
        {
          "tex": "I",
          "symbol": "I",
          "role": "pressure or modal impulse"
        },
        {
          "tex": "z",
          "symbol": "z",
          "role": "depth / formulation predictor"
        },
        {
          "tex": "\\mu_a",
          "symbol": "μ_a",
          "role": "optical absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "-\\frac{\\partial I}{\\partial z}=\\mu_a I(0)e^{-\\mu_a z}.",
      "context": "1.3 From surface intensity to volumetric heating / 从表面光强到体积热源",
      "source": {
        "book": "project-theory",
        "line": 189,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-8bfd47b7cdccc26a.svg",
      "variables": [
        {
          "tex": "I",
          "symbol": "I",
          "role": "pressure or modal impulse"
        },
        {
          "tex": "z",
          "symbol": "z",
          "role": "depth / formulation predictor"
        },
        {
          "tex": "\\mu_a",
          "symbol": "μ_a",
          "role": "optical absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{Q'''(r,z,t)=\\eta_{\\mathrm{abs}}F(r)g(t)\\mu_a e^{-\\mu_a z}},\\qquad \\mu_a=\\frac{1}{\\delta_{\\mathrm{abs}}}.",
      "context": "1.3 From surface intensity to volumetric heating / 从表面光强到体积热源",
      "source": {
        "book": "project-theory",
        "line": 195,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f94b36aea7c16d98.svg",
      "variables": [
        {
          "tex": "Q'''",
          "symbol": "Q'''",
          "role": "volumetric heat deposition rate"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "distance from beam axis"
        },
        {
          "tex": "z",
          "symbol": "z",
          "role": "depth into absorber"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "F",
          "symbol": "F",
          "role": "local beam fluence"
        },
        {
          "tex": "g",
          "symbol": "g",
          "role": "unit-integral pulse profile"
        },
        {
          "tex": "\\mu_a",
          "symbol": "μ_a",
          "role": "optical absorption coefficient"
        },
        {
          "tex": "\\delta_{\\mathrm{abs}}",
          "symbol": "δ_abs",
          "role": "optical absorption depth"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "[Q''']=\\underbrace{\\mathrm{J\\,m^{-2}}}_{F}\\underbrace{\\mathrm{s^{-1}}}_{g}\\underbrace{\\mathrm{m^{-1}}}_{\\mu_a}=\\mathrm{W\\,m^{-3}}.",
      "context": "1.3 From surface intensity to volumetric heating / 从表面光强到体积热源",
      "source": {
        "book": "project-theory",
        "line": 201,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1e44171ff51130d3.svg",
      "variables": [
        {
          "tex": "Q'''",
          "symbol": "Q'''",
          "role": "volumetric heat deposition rate"
        },
        {
          "tex": "J",
          "symbol": "J",
          "role": "nucleation rate / areal impulse"
        },
        {
          "tex": "m",
          "symbol": "m",
          "role": "mass or areal mass"
        },
        {
          "tex": "F",
          "symbol": "F",
          "role": "local beam fluence"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "g",
          "symbol": "g",
          "role": "unit-integral pulse profile"
        },
        {
          "tex": "\\mu_a",
          "symbol": "μ_a",
          "role": "optical absorption coefficient"
        },
        {
          "tex": "W",
          "symbol": "W",
          "role": "work / potential energy"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\n\\begin{aligned}\n\\int_{-\\infty}^{\\infty}\\int_V Q'''\\,dV\\,dt\n&=\\eta_{\\mathrm{abs}}\\left(\\int_A F\\,dA\\right)\\left(\\int g\\,dt\\right)\\\\\n&\\quad\\times\\left(\\int_0^\\infty\\mu_a e^{-\\mu_a z}\\,dz\\right)\\\\\n&=\\eta_{\\mathrm{abs}}E_p.\n\\end{aligned}\n",
      "context": "1.3 From surface intensity to volumetric heating / 从表面光强到体积热源",
      "source": {
        "book": "project-theory",
        "line": 207,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-0d24287bb7ade347.svg",
      "variables": [
        {
          "tex": "Q'''",
          "symbol": "Q'''",
          "role": "volumetric heat deposition rate"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "F",
          "symbol": "F",
          "role": "local beam fluence"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "g",
          "symbol": "g",
          "role": "unit-integral pulse profile"
        },
        {
          "tex": "\\mu_a",
          "symbol": "μ_a",
          "role": "optical absorption coefficient"
        },
        {
          "tex": "z",
          "symbol": "z",
          "role": "depth into absorber"
        },
        {
          "tex": "E_p",
          "symbol": "E_p",
          "role": "incident laser-pulse energy"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "1-e^{-\\mu_a h}.",
      "context": "1.3 From surface intensity to volumetric heating / 从表面光强到体积热源",
      "source": {
        "book": "project-theory",
        "line": 224,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-d1eccd6d18d4f8e5.svg",
      "variables": [
        {
          "tex": "\\mu_a",
          "symbol": "μ_a",
          "role": "optical absorption coefficient"
        },
        {
          "tex": "h",
          "symbol": "h",
          "role": "stand-off / layer thickness"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{\\epsilon_E=\\frac{\\left|\\displaystyle\\int\\!\\!\\int_V Q'''\\,dV\\,dt-E_{\\mathrm{abs,target}}\\right|}{E_{\\mathrm{abs,target}}}<0.01}.",
      "context": "1.3 From surface intensity to volumetric heating / 从表面光强到体积热源",
      "source": {
        "book": "project-theory",
        "line": 234,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-38a8bce61f9a7451.svg",
      "variables": [
        {
          "tex": "\\epsilon_E",
          "symbol": "ε_E",
          "role": "small perturbation / error"
        },
        {
          "tex": "Q'''",
          "symbol": "Q'''",
          "role": "volumetric heat deposition rate"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "E_{\\mathrm{abs,target}}",
          "symbol": "E_abs,target",
          "role": "energy / elastic modulus as defined"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "D_{\\mathrm{abs}}(\\mathbf x)=\\int Q'''(\\mathbf x,t)\\,dt.",
      "context": "1.4 Local dose, peak rate, and temperature rise / 局部剂量、峰值速率与温升",
      "source": {
        "book": "project-theory",
        "line": 246,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c3bd012fb8496d77.svg",
      "variables": [
        {
          "tex": "D_{\\mathrm{abs}}",
          "symbol": "D_abs",
          "role": "absorbed dose per unit volume"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "Q'''",
          "symbol": "Q'''",
          "role": "volumetric heat deposition rate"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{D_{\\mathrm{abs}}(r,z)=\\eta_{\\mathrm{abs}}F(r)\\mu_a e^{-\\mu_a z}},\\qquad [D_{\\mathrm{abs}}]=\\mathrm{J\\,m^{-3}}.",
      "context": "1.4 Local dose, peak rate, and temperature rise / 局部剂量、峰值速率与温升",
      "source": {
        "book": "project-theory",
        "line": 252,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-18fef646e2b15535.svg",
      "variables": [
        {
          "tex": "D_{\\mathrm{abs}}",
          "symbol": "D_abs",
          "role": "absorbed dose per unit volume"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "distance from beam axis"
        },
        {
          "tex": "z",
          "symbol": "z",
          "role": "depth into absorber"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "F",
          "symbol": "F",
          "role": "local beam fluence"
        },
        {
          "tex": "\\mu_a",
          "symbol": "μ_a",
          "role": "optical absorption coefficient"
        },
        {
          "tex": "J",
          "symbol": "J",
          "role": "nucleation rate / areal impulse"
        },
        {
          "tex": "m",
          "symbol": "m",
          "role": "mass or areal mass"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{\\dot q'''_{\\max}(\\mathbf x)=D_{\\mathrm{abs}}(\\mathbf x)\\frac{1}{\\tau_L}\\sqrt{\\frac{4\\ln2}{\\pi}}}.",
      "context": "1.4 Local dose, peak rate, and temperature rise / 局部剂量、峰值速率与温升",
      "source": {
        "book": "project-theory",
        "line": 258,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-6a2495a5e393d91f.svg",
      "variables": [
        {
          "tex": "\\dot q'''_{\\max}",
          "symbol": "q̇'''_max",
          "role": "peak volumetric deposition rate"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "D_{\\mathrm{abs}}",
          "symbol": "D_abs",
          "role": "absorbed dose per unit volume"
        },
        {
          "tex": "\\tau_L",
          "symbol": "τ_L",
          "role": "laser pulse FWHM"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\rho_{\\mathrm{abs}}c_p\\frac{\\partial T}{\\partial t}=Q'''.",
      "context": "1.4 Local dose, peak rate, and temperature rise / 局部剂量、峰值速率与温升",
      "source": {
        "book": "project-theory",
        "line": 268,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-79479ef1e8dec19c.svg",
      "variables": [
        {
          "tex": "\\rho_{\\mathrm{abs}}",
          "symbol": "ρ_abs",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "c_p",
          "symbol": "c_p",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "Q'''",
          "symbol": "Q'''",
          "role": "volumetric heat deposition rate"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{\\Delta T_{\\mathrm{ad}}=\\frac{D_{\\mathrm{abs}}}{\\rho_{\\mathrm{abs}}c_p}}.",
      "context": "1.4 Local dose, peak rate, and temperature rise / 局部剂量、峰值速率与温升",
      "source": {
        "book": "project-theory",
        "line": 274,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-0534c2fae77f63df.svg",
      "variables": [
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "T_{\\mathrm{ad}}",
          "symbol": "T_ad",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "D_{\\mathrm{abs}}",
          "symbol": "D_abs",
          "role": "absorbed dose per unit volume"
        },
        {
          "tex": "\\rho_{\\mathrm{abs}}",
          "symbol": "ρ_abs",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "c_p",
          "symbol": "c_p",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{\\Delta T_{\\mathrm{ad},0}=\\frac{\\eta_{\\mathrm{abs}}F_0}{\\rho_{\\mathrm{abs}}c_p\\delta_{\\mathrm{abs}}}}.",
      "context": "1.4 Local dose, peak rate, and temperature rise / 局部剂量、峰值速率与温升",
      "source": {
        "book": "project-theory",
        "line": 280,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-33c4b2209cd60b67.svg",
      "variables": [
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "T_{\\mathrm{ad},0}",
          "symbol": "T_ad,0",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "\\rho_{\\mathrm{abs}}",
          "symbol": "ρ_abs",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "c_p",
          "symbol": "c_p",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "\\delta_{\\mathrm{abs}}",
          "symbol": "δ_abs",
          "role": "optical absorption depth"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "W_{PV}=\\int_{V_0}^{V_{\\max}}\\Delta p\\,dV=\\frac{4\\pi}{3}\\Delta p\\left(R_{\\max}^3-R_0^3\\right).",
      "context": "1.5 Why enough energy does not guarantee a bubble / 为什么能量足够并不保证气泡形成",
      "source": {
        "book": "project-theory",
        "line": 300,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1fb60639373f6c2c.svg",
      "variables": [
        {
          "tex": "W_{PV}",
          "symbol": "W_PV",
          "role": "work / potential energy"
        },
        {
          "tex": "V_{\\max}",
          "symbol": "V_max",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "\\boxed{R_{\\mathrm{ceiling}}=\\left[R_0^3+\\frac{3\\eta_BE_{\\mathrm{abs}}}{4\\pi\\Delta p}\\right]^{1/3}}.",
      "context": "1.5 Why enough energy does not guarantee a bubble / 为什么能量足够并不保证气泡形成",
      "source": {
        "book": "project-theory",
        "line": 306,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-52319cb0d1ca12aa.svg",
      "variables": [
        {
          "tex": "R_{\\mathrm{ceiling}}",
          "symbol": "R_ceiling",
          "role": "bubble wall radius"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\eta_B",
          "symbol": "η_B",
          "role": "bubble-work conversion fraction"
        },
        {
          "tex": "E_{\\mathrm{abs}}",
          "symbol": "E_abs",
          "role": "total absorbed energy"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "R_{\\mathrm{ceiling}}\\sim\\left(\\frac{\\eta_BE_{\\mathrm{abs}}}{\\Delta p}\\right)^{1/3}.",
      "context": "1.5 Why enough energy does not guarantee a bubble / 为什么能量足够并不保证气泡形成",
      "source": {
        "book": "project-theory",
        "line": 312,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-448dbfba8f643040.svg",
      "variables": [
        {
          "tex": "R_{\\mathrm{ceiling}}",
          "symbol": "R_ceiling",
          "role": "bubble wall radius"
        },
        {
          "tex": "\\eta_B",
          "symbol": "η_B",
          "role": "bubble-work conversion fraction"
        },
        {
          "tex": "E_{\\mathrm{abs}}",
          "symbol": "E_abs",
          "role": "total absorbed energy"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "E_p=\\pi w_0^2F_0/2,\\quad E_{\\mathrm{abs}}=\\eta_{\\mathrm{abs}}E_p,\\quad E_{0<h}=E_{\\mathrm{abs}}(1-e^{-\\mu_ah})",
      "context": "Optical deposition · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T01",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-ff0a62da3867fdd7.svg",
      "variables": [
        {
          "tex": "E_p",
          "symbol": "E_p",
          "role": "incident laser-pulse energy"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "w_0",
          "symbol": "w_0",
          "role": "Gaussian beam waist radius"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "E_{\\mathrm{abs}}",
          "symbol": "E_abs",
          "role": "total absorbed energy"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "E_{0<h}",
          "symbol": "E_0<h",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "\\mu_a",
          "symbol": "μ_a",
          "role": "optical absorption coefficient"
        },
        {
          "tex": "h",
          "symbol": "h",
          "role": "stand-off / layer thickness"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "D_{\\mathrm{abs},0}=\\eta_{\\mathrm{abs}}F_0/\\delta_{\\mathrm{abs}},\\quad \\dot q'''_{\\max}=D_{\\mathrm{abs},0}\\sqrt{4\\ln2/\\pi}/\\tau_L,\\quad \\Delta T_{\\mathrm{ad}}=D_{\\mathrm{abs},0}/(\\rho_{\\mathrm{abs}}c_p)",
      "context": "Optical deposition · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T01",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-143eb436367d461c.svg",
      "variables": [
        {
          "tex": "D_{\\mathrm{abs},0}",
          "symbol": "D_abs,0",
          "role": "absorbed energy density"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "\\delta_{\\mathrm{abs}}",
          "symbol": "δ_abs",
          "role": "optical absorption depth"
        },
        {
          "tex": "\\dot q'''_{\\max}",
          "symbol": "q̇'''_max",
          "role": "peak volumetric deposition rate"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\tau_L",
          "symbol": "τ_L",
          "role": "laser pulse FWHM"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "T_{\\mathrm{ad}}",
          "symbol": "T_ad",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "\\rho_{\\mathrm{abs}}",
          "symbol": "ρ_abs",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "c_p",
          "symbol": "c_p",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/T01-optical-deposition.html",
      "tex": "R_{\\mathrm{ceiling}}=[3\\eta_BE_{\\mathrm{abs}}/(4\\pi\\Delta p)]^{1/3}",
      "context": "Optical deposition · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T01",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-5c7157dbbc379696.svg",
      "variables": [
        {
          "tex": "R_{\\mathrm{ceiling}}",
          "symbol": "R_ceiling",
          "role": "bubble wall radius"
        },
        {
          "tex": "\\eta_B",
          "symbol": "η_B",
          "role": "bubble-work conversion fraction"
        },
        {
          "tex": "E_{\\mathrm{abs}}",
          "symbol": "E_abs",
          "role": "total absorbed energy"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "0\\le p_1\\le1.",
      "context": "2.1 Why use a logistic model? / 为什么使用逻辑概率模型？",
      "source": {
        "book": "project-theory",
        "line": 338,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-cdc9c43b9e05f65d.svg",
      "variables": [
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "\\operatorname{logit}(p_1)=\\ln\\left(\\frac{p_1}{1-p_1}\\right)=\\ell.",
      "context": "2.1 Why use a logistic model? / 为什么使用逻辑概率模型？",
      "source": {
        "book": "project-theory",
        "line": 344,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-b0ca2018a239f08e.svg",
      "variables": [
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "\\ell",
          "symbol": "ℓ",
          "role": "logit / specified length"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "\\frac{p_1}{1-p_1}=e^\\ell\\quad\\Longrightarrow\\quad\\boxed{p_1=\\frac{1}{1+e^{-\\ell}}}.",
      "context": "2.1 Why use a logistic model? / 为什么使用逻辑概率模型？",
      "source": {
        "book": "project-theory",
        "line": 350,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f5d3e13ef17d47da.svg",
      "variables": [
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "\\ell",
          "symbol": "ℓ",
          "role": "absorption / cohesive-zone length"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "\n\\begin{aligned}\n\\ell={}&\\beta_0+b_{\\mathrm{batch}}\n+\\beta_D\\ln\\frac{D_{\\mathrm{abs}}}{D_{\\mathrm{ref}}}\n+\\beta_Q\\ln\\frac{\\dot q'''_{\\max}}{\\dot q'''_{\\mathrm{ref}}}\\\\\n&+\\beta_\\tau\\ln\\frac{\\tau_L}{\\tau_{\\mathrm{ref}}}\n+f_a(a_d)\n+\\mathbf z_{\\mathrm{form}}^T\\boldsymbol\\beta_{\\mathrm{form}}\\\\\n&+\\mathbf z_{\\mathrm{det}}^T\\boldsymbol\\beta_{\\mathrm{det}}\n+\\mathbf z_{\\mathrm{hist}}^T\\boldsymbol\\beta_{\\mathrm{hist}}.\n\\end{aligned}\n",
      "context": "2.1 Why use a logistic model? / 为什么使用逻辑概率模型？",
      "source": {
        "book": "project-theory",
        "line": 356,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-084089d8c5625482.svg",
      "variables": [
        {
          "tex": "\\ell",
          "symbol": "ℓ",
          "role": "logit / specified length"
        },
        {
          "tex": "\\beta_0",
          "symbol": "β_0",
          "role": "activation regression coefficient"
        },
        {
          "tex": "b_{\\mathrm{batch}}",
          "symbol": "b_batch",
          "role": "spatial profile / batch effect"
        },
        {
          "tex": "\\beta_D",
          "symbol": "β_D",
          "role": "activation regression coefficient"
        },
        {
          "tex": "D_{\\mathrm{abs}}",
          "symbol": "D_abs",
          "role": "absorbed dose per unit volume"
        },
        {
          "tex": "D_{\\mathrm{ref}}",
          "symbol": "D_ref",
          "role": "reference absorbed dose"
        },
        {
          "tex": "\\beta_Q",
          "symbol": "β_Q",
          "role": "activation regression coefficient"
        },
        {
          "tex": "\\dot q'''_{\\max}",
          "symbol": "q̇'''_max",
          "role": "peak volumetric deposition rate"
        },
        {
          "tex": "\\dot q'''_{\\mathrm{ref}}",
          "symbol": "q̇'''_ref",
          "role": "peak volumetric deposition rate"
        },
        {
          "tex": "\\beta_\\tau",
          "symbol": "β_τ",
          "role": "activation regression coefficient"
        },
        {
          "tex": "\\tau_L",
          "symbol": "τ_L",
          "role": "laser pulse FWHM"
        },
        {
          "tex": "\\tau_{\\mathrm{ref}}",
          "symbol": "τ_ref",
          "role": "duration / dimensionless time / dummy time"
        },
        {
          "tex": "f_a",
          "symbol": "f_a",
          "role": "frequency / predictor / false detections"
        },
        {
          "tex": "a_d",
          "symbol": "a_d",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "z_{\\mathrm{form}}",
          "symbol": "z_form",
          "role": "depth / formulation predictor"
        },
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "\\beta_{\\mathrm{form}}",
          "symbol": "β_form",
          "role": "activation regression coefficient"
        },
        {
          "tex": "z_{\\mathrm{det}}",
          "symbol": "z_det",
          "role": "depth / formulation predictor"
        },
        {
          "tex": "\\beta_{\\mathrm{det}}",
          "symbol": "β_det",
          "role": "activation regression coefficient"
        },
        {
          "tex": "z_{\\mathrm{hist}}",
          "symbol": "z_hist",
          "role": "depth / formulation predictor"
        },
        {
          "tex": "\\beta_{\\mathrm{hist}}",
          "symbol": "β_hist",
          "role": "activation regression coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "\\ln\\dot q'''_{\\max}=\\ln D_{\\mathrm{abs}}-\\ln\\tau_L+\\text{constant}.",
      "context": "2.1 Why use a logistic model? / 为什么使用逻辑概率模型？",
      "source": {
        "book": "project-theory",
        "line": 377,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-4a1695a39c22c479.svg",
      "variables": [
        {
          "tex": "\\dot q'''_{\\max}",
          "symbol": "q̇'''_max",
          "role": "peak volumetric deposition rate"
        },
        {
          "tex": "D_{\\mathrm{abs}}",
          "symbol": "D_abs",
          "role": "absorbed dose per unit volume"
        },
        {
          "tex": "\\tau_L",
          "symbol": "τ_L",
          "role": "laser pulse FWHM"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "P(\\text{no event})=1-p_1.",
      "context": "2.2 From one droplet to many droplets / 从单个液滴到液滴群体",
      "source": {
        "book": "project-theory",
        "line": 397,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-5b790f32f51a92b7.svg",
      "variables": [
        {
          "tex": "P",
          "symbol": "P",
          "role": "event probability"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "P(\\text{no events among all droplets})=(1-p_1)^{N_{\\mathrm{active}}}.",
      "context": "2.2 From one droplet to many droplets / 从单个液滴到液滴群体",
      "source": {
        "book": "project-theory",
        "line": 403,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c1ad0e30189fba03.svg",
      "variables": [
        {
          "tex": "P",
          "symbol": "P",
          "role": "event probability"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "N_{\\mathrm{active}}",
          "symbol": "N_active",
          "role": "event count / membrane tension"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "\\boxed{P_{\\ge1}^{(N)}=1-(1-p_1)^{N_{\\mathrm{active}}}}.",
      "context": "2.2 From one droplet to many droplets / 从单个液滴到液滴群体",
      "source": {
        "book": "project-theory",
        "line": 409,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-ff887d0960984836.svg",
      "variables": [
        {
          "tex": "P_{\\ge1}",
          "symbol": "P_ge1",
          "role": "at least one true event"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "N_{\\mathrm{active}}",
          "symbol": "N_active",
          "role": "event count / membrane tension"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "P_{\\ge1}^{(1)}=0.01,",
      "context": "2.2 From one droplet to many droplets / 从单个液滴到液滴群体",
      "source": {
        "book": "project-theory",
        "line": 419,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-75f8599fcf87d550.svg",
      "variables": [
        {
          "tex": "P_{\\ge1}",
          "symbol": "P_ge1",
          "role": "at least one true event"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "P_{\\ge1}^{(100)}=1-0.99^{100}\\approx0.634.",
      "context": "2.2 From one droplet to many droplets / 从单个液滴到液滴群体",
      "source": {
        "book": "project-theory",
        "line": 425,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-ec5d5923415f520a.svg",
      "variables": [
        {
          "tex": "P_{\\ge1}",
          "symbol": "P_ge1",
          "role": "at least one true event"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "1-P_{\\ge1}^{(N)}=(1-p_1)^N,",
      "context": "2.2 From one droplet to many droplets / 从单个液滴到液滴群体",
      "source": {
        "book": "project-theory",
        "line": 435,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-229f47582439600e.svg",
      "variables": [
        {
          "tex": "P_{\\ge1}",
          "symbol": "P_ge1",
          "role": "at least one true event"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "\\boxed{p_1=1-\\left(1-P_{\\ge1}^{(N)}\\right)^{1/N}}.",
      "context": "2.2 From one droplet to many droplets / 从单个液滴到液滴群体",
      "source": {
        "book": "project-theory",
        "line": 441,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-7cffcda8733ad682.svg",
      "variables": [
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "P_{\\ge1}",
          "symbol": "P_ge1",
          "role": "at least one true event"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "P(\\text{detected real event})=q_dp_1.",
      "context": "2.3 Adding the detector / 加入探测器模型",
      "source": {
        "book": "project-theory",
        "line": 457,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-615abe92beb12405.svg",
      "variables": [
        {
          "tex": "P",
          "symbol": "P",
          "role": "event probability"
        },
        {
          "tex": "q_d",
          "symbol": "q_d",
          "role": "single-event detection efficiency"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "(1-q_dp_1)^N.",
      "context": "2.3 Adding the detector / 加入探测器模型",
      "source": {
        "book": "project-theory",
        "line": 463,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-954e816b58a7dd6a.svg",
      "variables": [
        {
          "tex": "q_d",
          "symbol": "q_d",
          "role": "single-event detection efficiency"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "P(\\text{no recorded event})=(1-f_d)(1-q_dp_1)^N.",
      "context": "2.3 Adding the detector / 加入探测器模型",
      "source": {
        "book": "project-theory",
        "line": 469,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-6acb96cce408521d.svg",
      "variables": [
        {
          "tex": "P",
          "symbol": "P",
          "role": "event probability"
        },
        {
          "tex": "f_d",
          "symbol": "f_d",
          "role": "false-detection probability"
        },
        {
          "tex": "q_d",
          "symbol": "q_d",
          "role": "single-event detection efficiency"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "\\boxed{P_{\\mathrm{det}}=1-(1-f_d)(1-q_dp_1)^N},",
      "context": "2.3 Adding the detector / 加入探测器模型",
      "source": {
        "book": "project-theory",
        "line": 475,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-e845bcb1d1dbb650.svg",
      "variables": [
        {
          "tex": "P_{\\mathrm{det}}",
          "symbol": "P_det",
          "role": "detected-event probability"
        },
        {
          "tex": "f_d",
          "symbol": "f_d",
          "role": "false-detection probability"
        },
        {
          "tex": "q_d",
          "symbol": "q_d",
          "role": "single-event detection efficiency"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "\\boxed{P_{\\mathrm{det}}=f_d+(1-f_d)\\left[1-(1-q_dp_1)^N\\right]}.",
      "context": "2.3 Adding the detector / 加入探测器模型",
      "source": {
        "book": "project-theory",
        "line": 481,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-96f5754e9c28f6d2.svg",
      "variables": [
        {
          "tex": "P_{\\mathrm{det}}",
          "symbol": "P_det",
          "role": "detected-event probability"
        },
        {
          "tex": "f_d",
          "symbol": "f_d",
          "role": "false-detection probability"
        },
        {
          "tex": "q_d",
          "symbol": "q_d",
          "role": "single-event detection efficiency"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "\\boxed{p_1=\\frac{1-\\left(\\dfrac{1-P_{\\mathrm{det}}}{1-f_d}\\right)^{1/N}}{q_d}}.",
      "context": "2.3 Adding the detector / 加入探测器模型",
      "source": {
        "book": "project-theory",
        "line": 491,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-95ddb1b063d46efe.svg",
      "variables": [
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "P_{\\mathrm{det}}",
          "symbol": "P_det",
          "role": "detected-event probability"
        },
        {
          "tex": "f_d",
          "symbol": "f_d",
          "role": "false-detection probability"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        },
        {
          "tex": "q_d",
          "symbol": "q_d",
          "role": "single-event detection efficiency"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "f_d\\le P_{\\mathrm{det}}\\le f_d+(1-f_d)\\left[1-(1-q_d)^N\\right].",
      "context": "2.3 Adding the detector / 加入探测器模型",
      "source": {
        "book": "project-theory",
        "line": 497,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-b2f5db8055dc80ae.svg",
      "variables": [
        {
          "tex": "f_d",
          "symbol": "f_d",
          "role": "false-detection probability"
        },
        {
          "tex": "P_{\\mathrm{det}}",
          "symbol": "P_det",
          "role": "detected-event probability"
        },
        {
          "tex": "q_d",
          "symbol": "q_d",
          "role": "single-event detection efficiency"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "P_{\\ge1}=1-(1-p_1)^N,\\qquad P_{\\mathrm{det}}=1-(1-f_d)(1-q_dp_1)^N",
      "context": "Activation · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T02",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-c5e46af017ca78c7.svg",
      "variables": [
        {
          "tex": "P_{\\ge1}",
          "symbol": "P_ge1",
          "role": "at least one true event"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        },
        {
          "tex": "P_{\\mathrm{det}}",
          "symbol": "P_det",
          "role": "detected-event probability"
        },
        {
          "tex": "f_d",
          "symbol": "f_d",
          "role": "false-detection probability"
        },
        {
          "tex": "q_d",
          "symbol": "q_d",
          "role": "single-event detection efficiency"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "p_1=\\frac{1-[(1-P_{\\mathrm{det}})/(1-f_d)]^{1/N}}{q_d},\\quad f_d\\le P_{\\mathrm{det}}\\le1-(1-f_d)(1-q_d)^N",
      "context": "Activation · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T02",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-07f952efc72538d7.svg",
      "variables": [
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "P_{\\mathrm{det}}",
          "symbol": "P_det",
          "role": "detected-event probability"
        },
        {
          "tex": "f_d",
          "symbol": "f_d",
          "role": "false-detection probability"
        },
        {
          "tex": "N",
          "symbol": "N",
          "role": "event count / membrane tension"
        },
        {
          "tex": "q_d",
          "symbol": "q_d",
          "role": "single-event detection efficiency"
        }
      ]
    },
    {
      "page": "course/chapters/T02-activation-statistics.html",
      "tex": "\\ln\\dot q'''_{\\max}=\\ln D_{\\mathrm{abs}}-\\ln\\tau_L+\\ln\\sqrt{4\\ln2/\\pi}",
      "context": "Activation · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T02",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-71bee352e404c53b.svg",
      "variables": [
        {
          "tex": "\\dot q'''_{\\max}",
          "symbol": "q̇'''_max",
          "role": "peak volumetric deposition rate"
        },
        {
          "tex": "D_{\\mathrm{abs}}",
          "symbol": "D_abs",
          "role": "absorbed dose per unit volume"
        },
        {
          "tex": "\\tau_L",
          "symbol": "τ_L",
          "role": "laser pulse FWHM"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\nabla\\cdot\\mathbf u=\\frac{1}{r^2}\\frac{\\partial}{\\partial r}(r^2u)=0.",
      "context": "3.1 Continuity determines the liquid velocity / 连续性方程确定液体速度",
      "source": {
        "book": "project-theory",
        "line": 531,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f380952ac8f9d904.svg",
      "variables": [
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "r^2u=C(t).",
      "context": "3.1 Continuity determines the liquid velocity / 连续性方程确定液体速度",
      "source": {
        "book": "project-theory",
        "line": 537,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-febc280b3de5eb42.svg",
      "variables": [
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "C",
          "symbol": "C",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "u(R,t)=\\dot R.",
      "context": "3.1 Continuity determines the liquid velocity / 连续性方程确定液体速度",
      "source": {
        "book": "project-theory",
        "line": 543,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-34ae402818e60bc8.svg",
      "variables": [
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "C(t)=R^2\\dot R,",
      "context": "3.1 Continuity determines the liquid velocity / 连续性方程确定液体速度",
      "source": {
        "book": "project-theory",
        "line": 549,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c0508a8cef3c81f9.svg",
      "variables": [
        {
          "tex": "C",
          "symbol": "C",
          "role": "gas-content constant / damping"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{u(r,t)=\\frac{R^2\\dot R}{r^2}}.",
      "context": "3.1 Continuity determines the liquid velocity / 连续性方程确定液体速度",
      "source": {
        "book": "project-theory",
        "line": 555,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-949f0cab680b386d.svg",
      "variables": [
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "4\\pi r^2u=4\\pi R^2\\dot R.",
      "context": "3.1 Continuity determines the liquid velocity / 连续性方程确定液体速度",
      "source": {
        "book": "project-theory",
        "line": 561,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-b4b52dd55f8bac4f.svg",
      "variables": [
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "u=\\frac{\\partial\\Phi}{\\partial r}.",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 573,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-eefd9f87e751154d.svg",
      "variables": [
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "\\Phi",
          "symbol": "Φ",
          "role": "velocity potential"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{\\Phi(r,t)=-\\frac{R^2\\dot R}{r}}.",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 579,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-898b44d2cca84970.svg",
      "variables": [
        {
          "tex": "\\Phi",
          "symbol": "Φ",
          "role": "velocity potential"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\frac{\\partial\\Phi}{\\partial t}+\\frac12u^2+\\frac{p-p_\\infty}{\\rho_l}=0.",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 585,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-736c7760152e12f9.svg",
      "variables": [
        {
          "tex": "\\Phi",
          "symbol": "Φ",
          "role": "velocity potential"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\frac{\\partial\\Phi}{\\partial t}=-\\frac{R^2\\ddot R+2R\\dot R^2}{r}.",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 591,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-99159a2693fd5100.svg",
      "variables": [
        {
          "tex": "\\Phi",
          "symbol": "Φ",
          "role": "velocity potential"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\frac12u^2=\\frac{R^4\\dot R^2}{2r^4}.",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 597,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-2f819172b684eec8.svg",
      "variables": [
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{p'(r,t)\\equiv p(r,t)-p_\\infty(t)=\\rho_l\\left[\\frac{R^2\\ddot R+2R\\dot R^2}{r}-\\frac{R^4\\dot R^2}{2r^4}\\right]}.",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 603,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c04eb2c3e7542c4f.svg",
      "variables": [
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "A(t)=R^2\\ddot R+2R\\dot R^2,\\qquad B(t)=R^4\\dot R^2.",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 613,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-0a209a7a9b7e1db0.svg",
      "variables": [
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "B",
          "symbol": "B",
          "role": "kinetic pressure coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "p'=\\rho_l\\left(\\frac{A}{r}-\\frac{B}{2r^4}\\right).",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 619,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-96ae3346653776a1.svg",
      "variables": [
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "B",
          "symbol": "B",
          "role": "kinetic pressure coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "V=\\frac{4\\pi}{3}R^3,",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 625,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-03977f419a64d0e2.svg",
      "variables": [
        {
          "tex": "V",
          "symbol": "V",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\ddot V=4\\pi\\left(R^2\\ddot R+2R\\dot R^2\\right)=4\\pi A.",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 631,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-8d9ca928b10dfbeb.svg",
      "variables": [
        {
          "tex": "\\ddot V",
          "symbol": "V̈",
          "role": "bubble or observation volume"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\frac{\\rho_lA}{r}=\\frac{\\rho_l}{4\\pi r}\\ddot V.",
      "context": "3.2 The velocity potential gives the pressure field / 由速度势求得压力场",
      "source": {
        "book": "project-theory",
        "line": 637,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-5cb8fb4c55db933d.svg",
      "variables": [
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "\\ddot V",
          "symbol": "V̈",
          "role": "bubble or observation volume"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "p_l(R,t)-p_\\infty=\\rho_l\\left[R\\ddot R+2\\dot R^2-\\frac12\\dot R^2\\right].",
      "context": "3.3 Applying the wall condition gives Rayleigh–Plesset / 由壁面条件得到 Rayleigh–Plesset 方程",
      "source": {
        "book": "project-theory",
        "line": 653,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-a53f86c53fa077eb.svg",
      "variables": [
        {
          "tex": "p_l",
          "symbol": "p_l",
          "role": "liquid-side pressure"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{p_l(R,t)-p_\\infty=\\rho_l\\left(R\\ddot R+\\frac32\\dot R^2\\right)}.",
      "context": "3.3 Applying the wall condition gives Rayleigh–Plesset / 由壁面条件得到 Rayleigh–Plesset 方程",
      "source": {
        "book": "project-theory",
        "line": 659,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-8e67c4dc4f8216b9.svg",
      "variables": [
        {
          "tex": "p_l",
          "symbol": "p_l",
          "role": "liquid-side pressure"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "p_B-p_l(R)=\\frac{2\\sigma}{R}+\\frac{4\\mu_l\\dot R}{R}.",
      "context": "3.3 Applying the wall condition gives Rayleigh–Plesset / 由壁面条件得到 Rayleigh–Plesset 方程",
      "source": {
        "book": "project-theory",
        "line": 669,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-6401ac5ba3714e05.svg",
      "variables": [
        {
          "tex": "p_B",
          "symbol": "p_B",
          "role": "total interior bubble pressure"
        },
        {
          "tex": "p_l",
          "symbol": "p_l",
          "role": "liquid-side pressure"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\mu_l",
          "symbol": "μ_l",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\left.\\frac{\\partial u}{\\partial r}\\right|_{r=R}=-\\frac{2\\dot R}{R}",
      "context": "3.3 Applying the wall condition gives Rayleigh–Plesset / 由壁面条件得到 Rayleigh–Plesset 方程",
      "source": {
        "book": "project-theory",
        "line": 675,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-7fa50c3c721dae43.svg",
      "variables": [
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{\\rho_l\\left(R\\ddot R+\\frac32\\dot R^2\\right)=p_B-p_\\infty-\\frac{2\\sigma}{R}-\\frac{4\\mu_l\\dot R}{R}}.",
      "context": "3.3 Applying the wall condition gives Rayleigh–Plesset / 由壁面条件得到 Rayleigh–Plesset 方程",
      "source": {
        "book": "project-theory",
        "line": 685,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-9acc398aba93482d.svg",
      "variables": [
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "p_B",
          "symbol": "p_B",
          "role": "total interior bubble pressure"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\mu_l",
          "symbol": "μ_l",
          "role": "dynamic viscosity / absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\n\\underbrace{\\rho_l(R\\ddot R+\\tfrac32\\dot R^2)}_{\\text{liquid inertia / 液体惯性}}\n=\n\\underbrace{p_B-p_\\infty}_{\\text{pressure driving / 压力驱动}}\n-\n\\underbrace{2\\sigma/R}_{\\text{surface tension / 表面张力}}\n-\n\\underbrace{4\\mu_l\\dot R/R}_{\\text{viscous resistance / 黏性阻力}}.\n",
      "context": "3.3 Applying the wall condition gives Rayleigh–Plesset / 由壁面条件得到 Rayleigh–Plesset 方程",
      "source": {
        "book": "project-theory",
        "line": 695,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1708707f95a306d3.svg",
      "variables": [
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "p_B",
          "symbol": "p_B",
          "role": "total interior bubble pressure"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R}_{",
          "symbol": "R_",
          "role": "bubble wall radius"
        },
        {
          "tex": "\\mu_l",
          "symbol": "μ_l",
          "role": "dynamic viscosity / absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{p_{B,\\mathrm{required}}(t)=p_\\infty+\\rho_l\\left(R\\ddot R+\\frac32\\dot R^2\\right)+\\frac{2\\sigma}{R}+\\frac{4\\mu_l\\dot R}{R}}.",
      "context": "Why this equation does not yet predict laser-induced growth / 为什么该方程还不能预测激光诱导增长",
      "source": {
        "book": "project-theory",
        "line": 719,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-d6d7e24c05d22851.svg",
      "variables": [
        {
          "tex": "p_{B,\\mathrm{required}}",
          "symbol": "p_B,required",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\mu_l",
          "symbol": "μ_l",
          "role": "dynamic viscosity / absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\Delta p=p_\\infty-p_v>0.",
      "context": "3.4 Deriving the Rayleigh collapse time / 推导 Rayleigh 塌缩时间",
      "source": {
        "book": "project-theory",
        "line": 735,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-a7508b063a46fd10.svg",
      "variables": [
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "p_v",
          "symbol": "p_v",
          "role": "vapor partial pressure"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "R\\ddot R+\\frac32\\dot R^2=-\\frac{\\Delta p}{\\rho_l}.",
      "context": "3.4 Deriving the Rayleigh collapse time / 推导 Rayleigh 塌缩时间",
      "source": {
        "book": "project-theory",
        "line": 741,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-cc49e2ad654b7de0.svg",
      "variables": [
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\ddot R=\\frac12\\frac{dv}{dR},",
      "context": "3.4 Deriving the Rayleigh collapse time / 推导 Rayleigh 塌缩时间",
      "source": {
        "book": "project-theory",
        "line": 747,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-4f92fd56a077641e.svg",
      "variables": [
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "v",
          "symbol": "v",
          "role": "velocity / squared wall speed as defined"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "R\\frac{dv}{dR}+3v=-\\frac{2\\Delta p}{\\rho_l}.",
      "context": "3.4 Deriving the Rayleigh collapse time / 推导 Rayleigh 塌缩时间",
      "source": {
        "book": "project-theory",
        "line": 753,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-0ad5d5c793d7b656.svg",
      "variables": [
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "v",
          "symbol": "v",
          "role": "velocity / squared wall speed as defined"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\frac{d}{dR}(R^3v)=-\\frac{2\\Delta p}{\\rho_l}R^2.",
      "context": "3.4 Deriving the Rayleigh collapse time / 推导 Rayleigh 塌缩时间",
      "source": {
        "book": "project-theory",
        "line": 759,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-95f114f2652924fa.svg",
      "variables": [
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "v",
          "symbol": "v",
          "role": "velocity / squared wall speed as defined"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{\\dot R^2=\\frac{2\\Delta p}{3\\rho_l}\\left[\\left(\\frac{R_{\\max}}{R}\\right)^3-1\\right]}.",
      "context": "3.4 Deriving the Rayleigh collapse time / 推导 Rayleigh 塌缩时间",
      "source": {
        "book": "project-theory",
        "line": 765,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-2879e784131ef803.svg",
      "variables": [
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "t_R=\\int_0^{R_{\\max}}\\frac{dR}{\\sqrt{\\dfrac{2\\Delta p}{3\\rho_l}\\left[\\left(\\dfrac{R_{\\max}}{R}\\right)^3-1\\right]}}.",
      "context": "3.4 Deriving the Rayleigh collapse time / 推导 Rayleigh 塌缩时间",
      "source": {
        "book": "project-theory",
        "line": 771,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-537174be35ea21a6.svg",
      "variables": [
        {
          "tex": "t_R",
          "symbol": "t_R",
          "role": "physical time"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "t_R=R_{\\max}\\sqrt{\\frac{3\\rho_l}{2\\Delta p}}\\int_0^1\\frac{x^{3/2}}{\\sqrt{1-x^3}}\\,dx.",
      "context": "3.4 Deriving the Rayleigh collapse time / 推导 Rayleigh 塌缩时间",
      "source": {
        "book": "project-theory",
        "line": 777,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f24b354c5c5547b4.svg",
      "variables": [
        {
          "tex": "t_R",
          "symbol": "t_R",
          "role": "physical time"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{t_R\\approx0.915R_{\\max}\\sqrt{\\frac{\\rho_l}{\\Delta p}}}.",
      "context": "3.4 Deriving the Rayleigh collapse time / 推导 Rayleigh 塌缩时间",
      "source": {
        "book": "project-theory",
        "line": 783,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1a5430ac88496b46.svg",
      "variables": [
        {
          "tex": "t_R",
          "symbol": "t_R",
          "role": "physical time"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "S(0)=0,\\qquad S(1)=1,",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 803,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-0d341f52f520d85d.svg",
      "variables": [
        {
          "tex": "S",
          "symbol": "S",
          "role": "smoothstep / capillary scale"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "S'(0)=S'(1)=0,\\qquad S''(0)=S''(1)=0.",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 809,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-15ba428b49c24725.svg",
      "variables": [
        {
          "tex": "S'",
          "symbol": "S'",
          "role": "smoothstep / capillary scale"
        },
        {
          "tex": "S''",
          "symbol": "S''",
          "role": "smoothstep / capillary scale"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{S(\\xi)=10\\xi^3-15\\xi^4+6\\xi^5}.",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 815,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-7f3fbcff0d2149af.svg",
      "variables": [
        {
          "tex": "S",
          "symbol": "S",
          "role": "smoothstep / capillary scale"
        },
        {
          "tex": "\\xi",
          "symbol": "ξ",
          "role": "normalized branch time"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "S'(\\xi)=30\\xi^2(1-\\xi)^2,",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 821,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-5276445f2167f62c.svg",
      "variables": [
        {
          "tex": "S'",
          "symbol": "S'",
          "role": "smoothstep / capillary scale"
        },
        {
          "tex": "\\xi",
          "symbol": "ξ",
          "role": "normalized branch time"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "S''(\\xi)=60\\xi(1-\\xi)(1-2\\xi).",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 823,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-4ad7a8cb2007ccdb.svg",
      "variables": [
        {
          "tex": "S''",
          "symbol": "S''",
          "role": "smoothstep / capillary scale"
        },
        {
          "tex": "\\xi",
          "symbol": "ξ",
          "role": "normalized branch time"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{R(t)=\\begin{cases}R_0+\\Delta R\\,S(t/t_g),&0\\le t\\le t_g,\\\\[4pt]R_{\\max}-\\Delta R\\,S((t-t_g)/t_c),&t_g<t\\le t_g+t_c.\\end{cases}}",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 829,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-e4a2c3641dfd0f1c.svg",
      "variables": [
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "S",
          "symbol": "S",
          "role": "smoothstep / capillary scale"
        },
        {
          "tex": "t_g",
          "symbol": "t_g",
          "role": "growth duration"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "t_c",
          "symbol": "t_c",
          "role": "collapse duration"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "S'_{\\max}=\\frac{30}{16}=1.875.",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 839,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-6ae56fea27fb7fe3.svg",
      "variables": [
        {
          "tex": "S'_{\\max}",
          "symbol": "S'_max",
          "role": "smoothstep / capillary scale"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\dot R_{\\max,\\mathrm{growth}}=1.875\\frac{\\Delta R}{t_g},\\qquad |\\dot R|_{\\max,\\mathrm{collapse}}=1.875\\frac{\\Delta R}{t_c}.",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 845,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-11575aa22f4c8625.svg",
      "variables": [
        {
          "tex": "\\dot R_{\\max,\\mathrm{growth}}",
          "symbol": "Ṙ_max,growth",
          "role": "bubble wall radius"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "t_g",
          "symbol": "t_g",
          "role": "growth duration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "t_c",
          "symbol": "t_c",
          "role": "collapse duration"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "R_0=1\\ \\mu\\mathrm m,\\quad R_{\\max}=8\\ \\mu\\mathrm m,\\quad t_g=0.7\\ \\mu\\mathrm s,\\quad t_c=0.6\\ \\mu\\mathrm s,",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 851,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-cd7d29ba5af3282a.svg",
      "variables": [
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "m",
          "symbol": "m",
          "role": "mass or areal mass"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        },
        {
          "tex": "t_g",
          "symbol": "t_g",
          "role": "growth duration"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "t_c",
          "symbol": "t_c",
          "role": "collapse duration"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\dot R_{\\max,\\mathrm{growth}}=18.75\\ \\mathrm{m\\,s^{-1}},",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 857,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-8bbc0b2d8b3ffbe4.svg",
      "variables": [
        {
          "tex": "\\dot R_{\\max,\\mathrm{growth}}",
          "symbol": "Ṙ_max,growth",
          "role": "bubble wall radius"
        },
        {
          "tex": "m",
          "symbol": "m",
          "role": "mass or areal mass"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{|\\dot R|_{\\max,\\mathrm{collapse}}=21.875\\ \\mathrm{m\\,s^{-1}}}.",
      "context": "3.5 Deriving the prescribed smooth trajectory / 推导规定的平滑轨迹",
      "source": {
        "book": "project-theory",
        "line": 859,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-a7f5f2569bc82049.svg",
      "variables": [
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "m",
          "symbol": "m",
          "role": "mass or areal mass"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "r=\\sqrt{d^2+s^2}.",
      "context": "3.6 Mapping bubble pressure onto a load disk / 将气泡压力映射至载荷圆盘",
      "source": {
        "book": "project-theory",
        "line": 871,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1558b599bc0170b5.svg",
      "variables": [
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "p'(s,t)=\\rho_l\\left[\\frac{A(t)}{\\sqrt{d^2+s^2}}-\\frac{B(t)}{2(d^2+s^2)^2}\\right].",
      "context": "3.6 Mapping bubble pressure onto a load disk / 将气泡压力映射至载荷圆盘",
      "source": {
        "book": "project-theory",
        "line": 877,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-d9cdadd6e79895ad.svg",
      "variables": [
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "B",
          "symbol": "B",
          "role": "kinetic pressure coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\bar p(t)=\\frac{1}{\\pi L^2}\\int_0^L p'(s,t)\\,2\\pi s\\,ds.",
      "context": "3.6 Mapping bubble pressure onto a load disk / 将气泡压力映射至载荷圆盘",
      "source": {
        "book": "project-theory",
        "line": 883,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-63d41734cfa9df71.svg",
      "variables": [
        {
          "tex": "\\bar p",
          "symbol": "p̄",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "L",
          "symbol": "L",
          "role": "length / latent heat / load disk radius"
        },
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\int_0^L\\frac{s}{\\sqrt{d^2+s^2}}\\,ds=\\sqrt{d^2+L^2}-d,",
      "context": "3.6 Mapping bubble pressure onto a load disk / 将气泡压力映射至载荷圆盘",
      "source": {
        "book": "project-theory",
        "line": 889,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-3cbb782b92e2e373.svg",
      "variables": [
        {
          "tex": "L",
          "symbol": "L",
          "role": "length / latent heat / load disk radius"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\int_0^L\\frac{s}{(d^2+s^2)^2}\\,ds=\\frac12\\left(\\frac1{d^2}-\\frac1{d^2+L^2}\\right).",
      "context": "3.6 Mapping bubble pressure onto a load disk / 将气泡压力映射至载荷圆盘",
      "source": {
        "book": "project-theory",
        "line": 895,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-70fe617f7167cc33.svg",
      "variables": [
        {
          "tex": "L",
          "symbol": "L",
          "role": "length / latent heat / load disk radius"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\boxed{\\begin{aligned}\\bar p(t)=\\rho_l\\bigg[&\\frac{2A(t)}{L^2}\\left(\\sqrt{d^2+L^2}-d\\right)\\\\&-\\frac{B(t)}{2L^2}\\left(\\frac1{d^2}-\\frac1{d^2+L^2}\\right)\\bigg].\\end{aligned}}",
      "context": "3.6 Mapping bubble pressure onto a load disk / 将气泡压力映射至载荷圆盘",
      "source": {
        "book": "project-theory",
        "line": 901,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f66cf328a16dbd9a.svg",
      "variables": [
        {
          "tex": "\\bar p",
          "symbol": "p̄",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "L",
          "symbol": "L",
          "role": "length / latent heat / load disk radius"
        },
        {
          "tex": "B",
          "symbol": "B",
          "role": "kinetic pressure coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\bar p\\to\\rho_l\\left(\\frac{A}{d}-\\frac{B}{2d^4}\\right).",
      "context": "3.6 Mapping bubble pressure onto a load disk / 将气泡压力映射至载荷圆盘",
      "source": {
        "book": "project-theory",
        "line": 911,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-bf73e716030dd4c8.svg",
      "variables": [
        {
          "tex": "\\bar p",
          "symbol": "p̄",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "B",
          "symbol": "B",
          "role": "kinetic pressure coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "\\gamma=\\frac{d}{R_{\\max}}=1.0625.",
      "context": "3.7 What the signed pressure does—and does not—mean / 带符号压力的含义与局限",
      "source": {
        "book": "project-theory",
        "line": 931,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-ab507b9f5bbf82e8.svg",
      "variables": [
        {
          "tex": "\\gamma",
          "symbol": "γ",
          "role": "stand-off / heat-capacity ratio"
        },
        {
          "tex": "R_{\\max}",
          "symbol": "R_max",
          "role": "maximum expansion radius"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "p_{\\mathrm{absolute}}=p_\\infty+p'.",
      "context": "3.7 What the signed pressure does—and does not—mean / 带符号压力的含义与局限",
      "source": {
        "book": "project-theory",
        "line": 941,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-aa3f2dccde034123.svg",
      "variables": [
        {
          "tex": "p_{\\mathrm{absolute}}",
          "symbol": "p_absolute",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "S(\\xi)=10\\xi^3-15\\xi^4+6\\xi^5,\\quad S'(\\xi)=30\\xi^2(1-\\xi)^2,\\quad |\\dot R|_{\\max}=1.875\\Delta R/t_{\\mathrm{branch}}",
      "context": "Bubble dynamics and load mapping · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T03",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-665bdd7627d5ac77.svg",
      "variables": [
        {
          "tex": "S",
          "symbol": "S",
          "role": "smoothstep / capillary scale"
        },
        {
          "tex": "\\xi",
          "symbol": "ξ",
          "role": "normalized branch time"
        },
        {
          "tex": "S'",
          "symbol": "S'",
          "role": "smoothstep / capillary scale"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "t_{\\mathrm{branch}}",
          "symbol": "t_branch",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "A=R^2\\ddot R+2R\\dot R^2,\\quad B=R^4\\dot R^2,\\quad \\bar p=\\rho\\left[\\frac{2A}{L^2}(\\sqrt{d^2+L^2}-d)-\\frac{B}{2L^2}\\left(\\frac1{d^2}-\\frac1{d^2+L^2}\\right)\\right]",
      "context": "Bubble dynamics and load mapping · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T03",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-4520b735d38fcefc.svg",
      "variables": [
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "B",
          "symbol": "B",
          "role": "kinetic pressure coefficient"
        },
        {
          "tex": "\\bar p",
          "symbol": "p̄",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "L",
          "symbol": "L",
          "role": "length / latent heat / load disk radius"
        }
      ]
    },
    {
      "page": "course/chapters/T03-bubble-load-mapping.html",
      "tex": "p_{\\mathrm{abs}}=p_\\infty+p',\\quad p_{B,\\mathrm{required}}=p_\\infty+\\rho(R\\ddot R+\\tfrac32\\dot R^2)+2\\sigma/R+4\\mu\\dot R/R",
      "context": "Bubble dynamics and load mapping · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T03",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-62d7f58c7879dbd5.svg",
      "variables": [
        {
          "tex": "p_{\\mathrm{abs}}",
          "symbol": "p_abs",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_\\infty",
          "symbol": "p∞",
          "role": "ambient liquid pressure"
        },
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_{B,\\mathrm{required}}",
          "symbol": "p_B,required",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "w(s,t)=\\phi(s)q(t),",
      "context": "4.1 Reducing the laminate to one generalized coordinate / 将层合结构简化为一个广义坐标",
      "source": {
        "book": "project-theory",
        "line": 967,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c95ff24c1114de9f.svg",
      "variables": [
        {
          "tex": "w",
          "symbol": "w",
          "role": "transverse film displacement"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "assumed spatial mode shape"
        },
        {
          "tex": "q",
          "symbol": "q",
          "role": "generalized mode displacement"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{\\phi(s)=\\left(1-\\frac{s^2}{a^2}\\right)^2},\\qquad 0\\le s\\le a.",
      "context": "4.1 Reducing the laminate to one generalized coordinate / 将层合结构简化为一个广义坐标",
      "source": {
        "book": "project-theory",
        "line": 977,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-4ca91f359f857a65.svg",
      "variables": [
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "assumed spatial mode shape"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\phi(a)=0,\\qquad\\phi'(a)=0,\\qquad\\phi(0)=1.",
      "context": "4.1 Reducing the laminate to one generalized coordinate / 将层合结构简化为一个广义坐标",
      "source": {
        "book": "project-theory",
        "line": 983,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f540578191f21357.svg",
      "variables": [
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "assumed spatial mode shape"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\phi'",
          "symbol": "φ'",
          "role": "assumed spatial mode shape"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "T=\\frac12\\int_A m'\\dot w^2\\,dA=\\frac12\\dot q^2\\int_A m'\\phi^2\\,dA.",
      "context": "Modal mass / 模态质量",
      "source": {
        "book": "project-theory",
        "line": 995,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-5691b00bae4fb3fa.svg",
      "variables": [
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "m'",
          "symbol": "m'",
          "role": "mass or areal mass"
        },
        {
          "tex": "\\dot w",
          "symbol": "ẇ",
          "role": "transverse film displacement"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "\\dot q",
          "symbol": "q̇",
          "role": "generalized mode displacement"
        },
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "assumed spatial mode shape"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "M_q=2\\pi m'\\int_0^a\\phi^2s\\,ds.",
      "context": "Modal mass / 模态质量",
      "source": {
        "book": "project-theory",
        "line": 1001,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-e6eedda8d7e86df1.svg",
      "variables": [
        {
          "tex": "M_q",
          "symbol": "M_q",
          "role": "generalized modal mass"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "m'",
          "symbol": "m'",
          "role": "mass or areal mass"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "assumed spatial mode shape"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\int_0^a\\left(1-\\frac{s^2}{a^2}\\right)^4s\\,ds=\\frac{a^2}{10},",
      "context": "Modal mass / 模态质量",
      "source": {
        "book": "project-theory",
        "line": 1007,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1b607a9a87d52be2.svg",
      "variables": [
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{M_q=\\frac{\\pi m'a^2}{5}}.",
      "context": "Modal mass / 模态质量",
      "source": {
        "book": "project-theory",
        "line": 1013,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-ceed2fd79a180666.svg",
      "variables": [
        {
          "tex": "M_q",
          "symbol": "M_q",
          "role": "generalized modal mass"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "m'",
          "symbol": "m'",
          "role": "mass or areal mass"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "U=\\frac{D}{2}\\int_A\\left(\\kappa_r^2+\\kappa_\\theta^2+2\\nu\\kappa_r\\kappa_\\theta\\right)dA,",
      "context": "Modal stiffness / 模态刚度",
      "source": {
        "book": "project-theory",
        "line": 1021,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f6f908bef62aedec.svg",
      "variables": [
        {
          "tex": "U",
          "symbol": "U",
          "role": "velocity / velocity scale"
        },
        {
          "tex": "D",
          "symbol": "D",
          "role": "plate bending rigidity"
        },
        {
          "tex": "\\kappa_r",
          "symbol": "κ_r",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "\\kappa_\\theta",
          "symbol": "κ_θ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "\\nu",
          "symbol": "ν",
          "role": "Poisson ratio"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\kappa_r=-q\\phi'',\\qquad\\kappa_\\theta=-q\\frac{\\phi'}s.",
      "context": "Modal stiffness / 模态刚度",
      "source": {
        "book": "project-theory",
        "line": 1027,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-5c42fbbee97b878b.svg",
      "variables": [
        {
          "tex": "\\kappa_r",
          "symbol": "κ_r",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "q",
          "symbol": "q",
          "role": "generalized mode displacement"
        },
        {
          "tex": "\\phi''",
          "symbol": "φ''",
          "role": "assumed spatial mode shape"
        },
        {
          "tex": "\\kappa_\\theta",
          "symbol": "κ_θ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "\\phi'",
          "symbol": "φ'",
          "role": "assumed spatial mode shape"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "K_q=2\\pi D\\int_0^a\\left[(\\phi'')^2+\\left(\\frac{\\phi'}s\\right)^2+2\\nu\\phi''\\frac{\\phi'}s\\right]s\\,ds.",
      "context": "Modal stiffness / 模态刚度",
      "source": {
        "book": "project-theory",
        "line": 1033,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-e0d5b602599e4869.svg",
      "variables": [
        {
          "tex": "K_q",
          "symbol": "K_q",
          "role": "generalized modal stiffness"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "D",
          "symbol": "D",
          "role": "plate bending rigidity"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\phi''",
          "symbol": "φ''",
          "role": "assumed spatial mode shape"
        },
        {
          "tex": "\\phi'",
          "symbol": "φ'",
          "role": "assumed spatial mode shape"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "\\nu",
          "symbol": "ν",
          "role": "Poisson ratio"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\int_0^a[\\cdots]s\\,ds=\\frac{32}{3a^2},",
      "context": "Modal stiffness / 模态刚度",
      "source": {
        "book": "project-theory",
        "line": 1039,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-5235442d793ec56a.svg",
      "variables": [
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{K_q=\\frac{64\\pi D}{3a^2}}.",
      "context": "Modal stiffness / 模态刚度",
      "source": {
        "book": "project-theory",
        "line": 1045,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-acb201fc2f4580df.svg",
      "variables": [
        {
          "tex": "K_q",
          "symbol": "K_q",
          "role": "generalized modal stiffness"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "D",
          "symbol": "D",
          "role": "plate bending rigidity"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\delta W=\\int_A p(s,t)\\,\\delta w\\,dA.",
      "context": "Generalized load / 广义载荷",
      "source": {
        "book": "project-theory",
        "line": 1057,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-680ad9caa35e4fe5.svg",
      "variables": [
        {
          "tex": "\\delta",
          "symbol": "δ",
          "role": "separation / perturbation / penetration depth"
        },
        {
          "tex": "W",
          "symbol": "W",
          "role": "work / potential energy"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "w",
          "symbol": "w",
          "role": "film displacement / laser waist"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\delta W=\\left[\\int_A p(s,t)\\phi(s)\\,dA\\right]\\delta q.",
      "context": "Generalized load / 广义载荷",
      "source": {
        "book": "project-theory",
        "line": 1063,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-9e805e5307d2559a.svg",
      "variables": [
        {
          "tex": "\\delta",
          "symbol": "δ",
          "role": "separation / perturbation / penetration depth"
        },
        {
          "tex": "W",
          "symbol": "W",
          "role": "work / potential energy"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "velocity potential / mode shape"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "q",
          "symbol": "q",
          "role": "heat flux / mode displacement"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{Q(t)=2\\pi\\int_0^a p(s,t)\\phi(s)s\\,ds}.",
      "context": "Generalized load / 广义载荷",
      "source": {
        "book": "project-theory",
        "line": 1069,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-016ee10eb18326f0.svg",
      "variables": [
        {
          "tex": "Q",
          "symbol": "Q",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "velocity potential / mode shape"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{M_q\\ddot q+C_q\\dot q+K_qq=Q(t)}.",
      "context": "Generalized load / 广义载荷",
      "source": {
        "book": "project-theory",
        "line": 1079,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-92a1ae036df62873.svg",
      "variables": [
        {
          "tex": "M_q",
          "symbol": "M_q",
          "role": "generalized modal mass"
        },
        {
          "tex": "\\ddot q",
          "symbol": "q̈",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "C_q",
          "symbol": "C_q",
          "role": "generalized modal damping"
        },
        {
          "tex": "\\dot q",
          "symbol": "q̇",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "K_q",
          "symbol": "K_q",
          "role": "generalized modal stiffness"
        },
        {
          "tex": "q",
          "symbol": "q",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "Q",
          "symbol": "Q",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\omega_n=\\sqrt{\\frac{K_q}{M_q}},\\qquad\\zeta=\\frac{C_q}{2\\sqrt{M_qK_q}},\\qquad\\omega_d=\\omega_n\\sqrt{1-\\zeta^2}.",
      "context": "4.2 Why load timing matters / 为什么载荷时序很重要",
      "source": {
        "book": "project-theory",
        "line": 1091,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c543e414f5312588.svg",
      "variables": [
        {
          "tex": "\\omega_n",
          "symbol": "ω_n",
          "role": "structural modal angular frequency"
        },
        {
          "tex": "K_q",
          "symbol": "K_q",
          "role": "generalized modal stiffness"
        },
        {
          "tex": "M_q",
          "symbol": "M_q",
          "role": "generalized modal mass"
        },
        {
          "tex": "\\zeta",
          "symbol": "ζ",
          "role": "modal damping ratio"
        },
        {
          "tex": "C_q",
          "symbol": "C_q",
          "role": "generalized modal damping"
        },
        {
          "tex": "\\omega_d",
          "symbol": "ω_d",
          "role": "damped modal angular frequency"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{q(t)=\\frac{1}{M_q\\omega_d}\\int_0^t Q(\\tau)e^{-\\zeta\\omega_n(t-\\tau)}\\sin\\!\\left[\\omega_d(t-\\tau)\\right]\\,d\\tau}.",
      "context": "4.2 Why load timing matters / 为什么载荷时序很重要",
      "source": {
        "book": "project-theory",
        "line": 1097,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-804e797798079e43.svg",
      "variables": [
        {
          "tex": "q",
          "symbol": "q",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "M_q",
          "symbol": "M_q",
          "role": "generalized modal mass"
        },
        {
          "tex": "\\omega_d",
          "symbol": "ω_d",
          "role": "damped modal angular frequency"
        },
        {
          "tex": "Q",
          "symbol": "Q",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "\\tau",
          "symbol": "τ",
          "role": "duration / dimensionless time / dummy time"
        },
        {
          "tex": "\\zeta",
          "symbol": "ζ",
          "role": "modal damping ratio"
        },
        {
          "tex": "\\omega_n",
          "symbol": "ω_n",
          "role": "structural modal angular frequency"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "p(s,t)=F(t)b(s),\\qquad\\int_A b(s)\\,dA=1.",
      "context": "4.3 Why equal total impulse can also fail spatially / 为什么总冲量相同仍可能产生不同的空间响应",
      "source": {
        "book": "project-theory",
        "line": 1113,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-e94e4f4e82a8502e.svg",
      "variables": [
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "F",
          "symbol": "F",
          "role": "force / fluence"
        },
        {
          "tex": "b",
          "symbol": "b",
          "role": "spatial profile / batch effect"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "Q(t)=F(t)\\underbrace{\\int_A b(s)\\phi(s)\\,dA}_{\\Lambda}.",
      "context": "4.3 Why equal total impulse can also fail spatially / 为什么总冲量相同仍可能产生不同的空间响应",
      "source": {
        "book": "project-theory",
        "line": 1119,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-d6e66ae71fbf9daa.svg",
      "variables": [
        {
          "tex": "Q",
          "symbol": "Q",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "F",
          "symbol": "F",
          "role": "force / fluence"
        },
        {
          "tex": "b",
          "symbol": "b",
          "role": "spatial profile / batch effect"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "velocity potential / mode shape"
        },
        {
          "tex": "A}_{\\Lambda",
          "symbol": "A_Λ",
          "role": "area / volume-acceleration coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "b=\\frac1{\\pi a^2},",
      "context": "4.3 Why equal total impulse can also fail spatially / 为什么总冲量相同仍可能产生不同的空间响应",
      "source": {
        "book": "project-theory",
        "line": 1129,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-668ce1c7181fd2ca.svg",
      "variables": [
        {
          "tex": "b",
          "symbol": "b",
          "role": "spatial profile / batch effect"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\Lambda=\\frac1{\\pi a^2}\\int_A\\phi\\,dA=\\frac13.",
      "context": "4.3 Why equal total impulse can also fail spatially / 为什么总冲量相同仍可能产生不同的空间响应",
      "source": {
        "book": "project-theory",
        "line": 1135,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-532a4e7f6d91b865.svg",
      "variables": [
        {
          "tex": "\\Lambda",
          "symbol": "Λ",
          "role": "modal participation / viscous ratio"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "velocity potential / mode shape"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\rho\\frac{\\partial^2u}{\\partial t^2}=\\frac{\\partial\\sigma_{zz}}{\\partial z},\\qquad\\sigma_{zz}=M_L\\frac{\\partial u}{\\partial z},",
      "context": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "source": {
        "book": "project-theory",
        "line": 1151,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-888e19f6c56ebbd4.svg",
      "variables": [
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "u",
          "symbol": "u",
          "role": "liquid or solid particle velocity"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\sigma_{zz}",
          "symbol": "σ_zz",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "z",
          "symbol": "z",
          "role": "depth / formulation predictor"
        },
        {
          "tex": "M_L",
          "symbol": "M_L",
          "role": "generalized mass / molar mass"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "M_L=\\frac{E(1-\\nu)}{(1+\\nu)(1-2\\nu)}.",
      "context": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "source": {
        "book": "project-theory",
        "line": 1157,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-0d86dc037d15b48b.svg",
      "variables": [
        {
          "tex": "M_L",
          "symbol": "M_L",
          "role": "generalized mass / molar mass"
        },
        {
          "tex": "E",
          "symbol": "E",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "\\nu",
          "symbol": "ν",
          "role": "Poisson ratio"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{c_L=\\sqrt{\\frac{E(1-\\nu)}{\\rho(1+\\nu)(1-2\\nu)}}}.",
      "context": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "source": {
        "book": "project-theory",
        "line": 1163,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-5c242ff31fef9e44.svg",
      "variables": [
        {
          "tex": "c_L",
          "symbol": "c_L",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "E",
          "symbol": "E",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "\\nu",
          "symbol": "ν",
          "role": "Poisson ratio"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{Z=\\rho c_L}.",
      "context": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "source": {
        "book": "project-theory",
        "line": 1169,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-3b93eb2dcd640094.svg",
      "variables": [
        {
          "tex": "Z",
          "symbol": "Z",
          "role": "wave impedance"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "c_L",
          "symbol": "c_L",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "p_i+p_r=p_t,",
      "context": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "source": {
        "book": "project-theory",
        "line": 1175,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-77a58f7520617b21.svg",
      "variables": [
        {
          "tex": "p_i",
          "symbol": "p_i",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_r",
          "symbol": "p_r",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_t",
          "symbol": "p_t",
          "role": "pressure at the indicated location"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\frac{p_i-p_r}{Z_1}=\\frac{p_t}{Z_2}.",
      "context": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "source": {
        "book": "project-theory",
        "line": 1177,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c9f28c882f0af68a.svg",
      "variables": [
        {
          "tex": "p_i",
          "symbol": "p_i",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_r",
          "symbol": "p_r",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "Z_1",
          "symbol": "Z_1",
          "role": "wave impedance"
        },
        {
          "tex": "p_t",
          "symbol": "p_t",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "Z_2",
          "symbol": "Z_2",
          "role": "wave impedance"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{R_p=\\frac{p_r}{p_i}=\\frac{Z_2-Z_1}{Z_2+Z_1}},",
      "context": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "source": {
        "book": "project-theory",
        "line": 1183,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-30807834b82d4cf5.svg",
      "variables": [
        {
          "tex": "R_p",
          "symbol": "R_p",
          "role": "bubble wall radius"
        },
        {
          "tex": "p_r",
          "symbol": "p_r",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_i",
          "symbol": "p_i",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "Z_2",
          "symbol": "Z_2",
          "role": "wave impedance"
        },
        {
          "tex": "Z_1",
          "symbol": "Z_1",
          "role": "wave impedance"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{T_p=\\frac{p_t}{p_i}=\\frac{2Z_2}{Z_1+Z_2}}.",
      "context": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "source": {
        "book": "project-theory",
        "line": 1185,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-f7d9482eaf961025.svg",
      "variables": [
        {
          "tex": "T_p",
          "symbol": "T_p",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "p_t",
          "symbol": "p_t",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "p_i",
          "symbol": "p_i",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "Z_2",
          "symbol": "Z_2",
          "role": "wave impedance"
        },
        {
          "tex": "Z_1",
          "symbol": "Z_1",
          "role": "wave impedance"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{T_I=T_p^2\\frac{Z_1}{Z_2}=\\frac{4Z_1Z_2}{(Z_1+Z_2)^2}}.",
      "context": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "source": {
        "book": "project-theory",
        "line": 1191,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-3a803942ab38f20c.svg",
      "variables": [
        {
          "tex": "T_I",
          "symbol": "T_I",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "T_p",
          "symbol": "T_p",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "Z_1",
          "symbol": "Z_1",
          "role": "wave impedance"
        },
        {
          "tex": "Z_2",
          "symbol": "Z_2",
          "role": "wave impedance"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{t_{\\mathrm{transit}}=\\sum_i\\frac{h_i}{c_{L,i}}}.",
      "context": "4.4 Deriving first-arrival wave transmission / 推导首波传输",
      "source": {
        "book": "project-theory",
        "line": 1201,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-551a1e2d06e39812.svg",
      "variables": [
        {
          "tex": "t_{\\mathrm{transit}}",
          "symbol": "t_transit",
          "role": "physical time"
        },
        {
          "tex": "h_i",
          "symbol": "h_i",
          "role": "stand-off / layer thickness"
        },
        {
          "tex": "c_{L,i}",
          "symbol": "c_L,i",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{\\mathbf t=(1-D_c)\\mathbf K\\boldsymbol\\delta},",
      "context": "4.5 From transmitted loading to cohesive failure / 从传递载荷到黏聚失效",
      "source": {
        "book": "project-theory",
        "line": 1217,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-5e15315593a1d6e1.svg",
      "variables": [
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "D_c",
          "symbol": "D_c",
          "role": "interface damage variable"
        },
        {
          "tex": "K",
          "symbol": "K",
          "role": "stiffness / collapse constant"
        },
        {
          "tex": "\\delta",
          "symbol": "δ",
          "role": "separation / perturbation / penetration depth"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{\\left(\\frac{\\langle t_n\\rangle}{T_{n0}}\\right)^2+\\left(\\frac{t_s}{T_{s0}}\\right)^2=1},",
      "context": "4.5 From transmitted loading to cohesive failure / 从传递载荷到黏聚失效",
      "source": {
        "book": "project-theory",
        "line": 1227,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-3054ed3a8b2402c8.svg",
      "variables": [
        {
          "tex": "t_n",
          "symbol": "t_n",
          "role": "physical time"
        },
        {
          "tex": "T_{n0}",
          "symbol": "T_n0",
          "role": "normal cohesive strength"
        },
        {
          "tex": "t_s",
          "symbol": "t_s",
          "role": "physical time"
        },
        {
          "tex": "T_{s0}",
          "symbol": "T_s0",
          "role": "shear cohesive strength"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "G_c=\\int_{\\text{separation path}}\\mathbf t\\cdot d\\boldsymbol\\delta.",
      "context": "4.5 From transmitted loading to cohesive failure / 从传递载荷到黏聚失效",
      "source": {
        "book": "project-theory",
        "line": 1237,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1e68bce4849491ac.svg",
      "variables": [
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\delta",
          "symbol": "δ",
          "role": "separation / perturbation / penetration depth"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\ell_{\\mathrm{cz}}\\sim\\frac{E'G_c}{T_0^2}",
      "context": "Cohesive-zone length scale / 黏聚区长度尺度",
      "source": {
        "book": "project-theory",
        "line": 1257,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-67c5a15976593f18.svg",
      "variables": [
        {
          "tex": "\\ell_{\\mathrm{cz}}",
          "symbol": "ℓ_cz",
          "role": "cohesive-zone length scale"
        },
        {
          "tex": "E'",
          "symbol": "E'",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        },
        {
          "tex": "T_0",
          "symbol": "T_0",
          "role": "temperature / kinetic energy as defined"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "K_c^2\\sim E'G_c",
      "context": "Cohesive-zone length scale / 黏聚区长度尺度",
      "source": {
        "book": "project-theory",
        "line": 1263,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-e6ebfa50035fe95c.svg",
      "variables": [
        {
          "tex": "K_c",
          "symbol": "K_c",
          "role": "stiffness / collapse constant"
        },
        {
          "tex": "E'",
          "symbol": "E'",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\sigma\\sim\\frac{K_c}{\\sqrt r},",
      "context": "Cohesive-zone length scale / 黏聚区长度尺度",
      "source": {
        "book": "project-theory",
        "line": 1269,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-900767ac118f0c0a.svg",
      "variables": [
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "K_c",
          "symbol": "K_c",
          "role": "stiffness / collapse constant"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "r\\sim\\left(\\frac{K_c}{T_0}\\right)^2.",
      "context": "Cohesive-zone length scale / 黏聚区长度尺度",
      "source": {
        "book": "project-theory",
        "line": 1275,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-1fbc1af71db53543.svg",
      "variables": [
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "K_c",
          "symbol": "K_c",
          "role": "stiffness / collapse constant"
        },
        {
          "tex": "T_0",
          "symbol": "T_0",
          "role": "temperature / kinetic energy as defined"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{\\ell_{\\mathrm{cz}}\\sim\\frac{E'G_c}{T_0^2}}.",
      "context": "Cohesive-zone length scale / 黏聚区长度尺度",
      "source": {
        "book": "project-theory",
        "line": 1281,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-0ef35a404a56faba.svg",
      "variables": [
        {
          "tex": "\\ell_{\\mathrm{cz}}",
          "symbol": "ℓ_cz",
          "role": "cohesive-zone length scale"
        },
        {
          "tex": "E'",
          "symbol": "E'",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        },
        {
          "tex": "T_0",
          "symbol": "T_0",
          "role": "temperature / kinetic energy as defined"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "m'v=J_A.",
      "context": "4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值",
      "source": {
        "book": "project-theory",
        "line": 1293,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-8c200f658ef97954.svg",
      "variables": [
        {
          "tex": "m'",
          "symbol": "m'",
          "role": "mass or areal mass"
        },
        {
          "tex": "v",
          "symbol": "v",
          "role": "velocity / squared wall speed as defined"
        },
        {
          "tex": "J_A",
          "symbol": "J_A",
          "role": "selected areal pressure impulse"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\frac{E}{A}=\\frac12m'v^2=\\boxed{\\frac{J_A^2}{2m'}}.",
      "context": "4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值",
      "source": {
        "book": "project-theory",
        "line": 1299,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-73edf59c7fe3b790.svg",
      "variables": [
        {
          "tex": "E",
          "symbol": "E",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "m'",
          "symbol": "m'",
          "role": "mass or areal mass"
        },
        {
          "tex": "v",
          "symbol": "v",
          "role": "velocity / squared wall speed as defined"
        },
        {
          "tex": "J_A",
          "symbol": "J_A",
          "role": "selected areal pressure impulse"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{J_{\\mathrm{nominal}}=\\sqrt{2m'G_c}}.",
      "context": "4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值",
      "source": {
        "book": "project-theory",
        "line": 1305,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-b366ea764e6920f4.svg",
      "variables": [
        {
          "tex": "J_{\\mathrm{nominal}}",
          "symbol": "J_nominal",
          "role": "nucleation rate / areal impulse"
        },
        {
          "tex": "m'",
          "symbol": "m'",
          "role": "mass or areal mass"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{\\Pi_J=\\frac{J_A}{J_{\\mathrm{nominal}}}},",
      "context": "4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值",
      "source": {
        "book": "project-theory",
        "line": 1311,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-426847f1fda727e7.svg",
      "variables": [
        {
          "tex": "\\Pi_J",
          "symbol": "Π_J",
          "role": "impulse availability ratio"
        },
        {
          "tex": "J_A",
          "symbol": "J_A",
          "role": "selected areal pressure impulse"
        },
        {
          "tex": "J_{\\mathrm{nominal}}",
          "symbol": "J_nominal",
          "role": "nucleation rate / areal impulse"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\boxed{\\Pi_E=\\frac{E_{\\mathrm{mode,max}}}{A_{\\mathrm{rel}}G_c}}.",
      "context": "4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值",
      "source": {
        "book": "project-theory",
        "line": 1317,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c0e783f9c4ddcab4.svg",
      "variables": [
        {
          "tex": "\\Pi_E",
          "symbol": "Π_E",
          "role": "energy availability ratio"
        },
        {
          "tex": "E_{\\mathrm{mode,max}}",
          "symbol": "E_mode,max",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "A_{\\mathrm{rel}}",
          "symbol": "A_rel",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\Pi_J=0.565,\\qquad\\Pi_E=0.767,",
      "context": "4.6 Deriving the impulse and energy availability ratios / 推导冲量与能量可用性比值",
      "source": {
        "book": "project-theory",
        "line": 1323,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-8036dc401abfa147.svg",
      "variables": [
        {
          "tex": "\\Pi_J",
          "symbol": "Π_J",
          "role": "impulse availability ratio"
        },
        {
          "tex": "\\Pi_E",
          "symbol": "Π_E",
          "role": "energy availability ratio"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "\\phi=(1-s^2/a^2)^2,\\quad M_q=\\pi m'a^2/5,\\quad K_q=64\\pi D/(3a^2),\\quad Q=p\\pi a^2/3,\\quad f_n=\\sqrt{K_q/M_q}/(2\\pi)",
      "context": "Structural motion and interfacial release · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T04",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-e56d701dbb1c6557.svg",
      "variables": [
        {
          "tex": "\\phi",
          "symbol": "φ",
          "role": "assumed spatial mode shape"
        },
        {
          "tex": "s",
          "symbol": "s",
          "role": "in-plane radial coordinate"
        },
        {
          "tex": "a",
          "symbol": "a",
          "role": "droplet radius / film span / acceleration"
        },
        {
          "tex": "M_q",
          "symbol": "M_q",
          "role": "generalized modal mass"
        },
        {
          "tex": "\\pi",
          "symbol": "π",
          "role": "circle geometry constant"
        },
        {
          "tex": "m'",
          "symbol": "m'",
          "role": "mass or areal mass"
        },
        {
          "tex": "K_q",
          "symbol": "K_q",
          "role": "generalized modal stiffness"
        },
        {
          "tex": "D",
          "symbol": "D",
          "role": "plate bending rigidity"
        },
        {
          "tex": "Q",
          "symbol": "Q",
          "role": "heat or generalized load as defined"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "f_n",
          "symbol": "f_n",
          "role": "frequency / predictor / false detections"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "R_p=(Z_2-Z_1)/(Z_2+Z_1),\\quad T_p=2Z_2/(Z_1+Z_2),\\quad T_I=4Z_1Z_2/(Z_1+Z_2)^2,\\quad t_{\\mathrm{transit}}=\\sum h_i/c_{L,i}",
      "context": "Structural motion and interfacial release · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T04",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-816704d4d3c6a5e2.svg",
      "variables": [
        {
          "tex": "R_p",
          "symbol": "R_p",
          "role": "bubble wall radius"
        },
        {
          "tex": "Z_2",
          "symbol": "Z_2",
          "role": "wave impedance"
        },
        {
          "tex": "Z_1",
          "symbol": "Z_1",
          "role": "wave impedance"
        },
        {
          "tex": "T_p",
          "symbol": "T_p",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "T_I",
          "symbol": "T_I",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "t_{\\mathrm{transit}}",
          "symbol": "t_transit",
          "role": "physical time"
        },
        {
          "tex": "h_i",
          "symbol": "h_i",
          "role": "stand-off / layer thickness"
        },
        {
          "tex": "c_{L,i}",
          "symbol": "c_L,i",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "(\\langle t_n\\rangle/T_{n0})^2+(t_s/T_{s0})^2=1,\\quad \\ell_{\\mathrm{cz}}\\sim E'G_c/T_0^2,\\quad G_c=\\int\\mathbf t\\cdot d\\boldsymbol\\delta",
      "context": "Structural motion and interfacial release · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T04",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-c4152936e456bee5.svg",
      "variables": [
        {
          "tex": "t_n",
          "symbol": "t_n",
          "role": "physical time"
        },
        {
          "tex": "T_{n0}",
          "symbol": "T_n0",
          "role": "normal cohesive strength"
        },
        {
          "tex": "t_s",
          "symbol": "t_s",
          "role": "physical time"
        },
        {
          "tex": "T_{s0}",
          "symbol": "T_s0",
          "role": "shear cohesive strength"
        },
        {
          "tex": "\\ell_{\\mathrm{cz}}",
          "symbol": "ℓ_cz",
          "role": "cohesive-zone length scale"
        },
        {
          "tex": "E'",
          "symbol": "E'",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        },
        {
          "tex": "T_0",
          "symbol": "T_0",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\delta",
          "symbol": "δ",
          "role": "separation / perturbation / penetration depth"
        }
      ]
    },
    {
      "page": "course/chapters/T04-structural-response-and-release.html",
      "tex": "J_{\\mathrm{nom}}=\\sqrt{2m'G_c},\\quad \\Pi_J=J_A/J_{\\mathrm{nom}},\\quad (E/A)/G_c=J_A^2/(2m'G_c)",
      "context": "Structural motion and interfacial release · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T04",
        "part": 4
      },
      "figure": "assets/curriculum-figures/formula-16085d14307978a5.svg",
      "variables": [
        {
          "tex": "J_{\\mathrm{nom}}",
          "symbol": "J_nom",
          "role": "nucleation rate / areal impulse"
        },
        {
          "tex": "m'",
          "symbol": "m'",
          "role": "mass or areal mass"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        },
        {
          "tex": "\\Pi_J",
          "symbol": "Π_J",
          "role": "impulse availability ratio"
        },
        {
          "tex": "J_A",
          "symbol": "J_A",
          "role": "selected areal pressure impulse"
        },
        {
          "tex": "E",
          "symbol": "E",
          "role": "energy / elastic modulus as defined"
        },
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\boxed{\\rho_s=\\operatorname{Corr}\\bigl(\\operatorname{rank}X,\\operatorname{rank}Y\\bigr)}.",
      "context": "5.1 What a Spearman ranking measures / Spearman 排序衡量什么",
      "source": {
        "book": "project-theory",
        "line": 1345,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-311203f0610b19bc.svg",
      "variables": [
        {
          "tex": "\\rho_s",
          "symbol": "ρ_s",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "X",
          "symbol": "X",
          "role": "sampled input"
        },
        {
          "tex": "Y",
          "symbol": "Y",
          "role": "inferred output / spherical harmonic"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\Delta T_{\\mathrm{ad},0}=\\frac{\\eta_{\\mathrm{abs}}F_0}{\\rho_{\\mathrm{abs}}c_p\\delta_{\\mathrm{abs}}},",
      "context": "5.2 Optical sensitivity follows directly from the temperature scale / 光学敏感性直接来自温升尺度",
      "source": {
        "book": "project-theory",
        "line": 1361,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c73fa51f0e3c1f1e.svg",
      "variables": [
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "T_{\\mathrm{ad},0}",
          "symbol": "T_ad,0",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "\\rho_{\\mathrm{abs}}",
          "symbol": "ρ_abs",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "c_p",
          "symbol": "c_p",
          "role": "sound speed / heat capacity"
        },
        {
          "tex": "\\delta_{\\mathrm{abs}}",
          "symbol": "δ_abs",
          "role": "optical absorption depth"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\ln\\Delta T_{\\mathrm{ad},0}=\\ln\\eta_{\\mathrm{abs}}+\\ln F_0-\\ln\\delta_{\\mathrm{abs}}-\\ln(\\rho_{\\mathrm{abs}}c_p).",
      "context": "5.2 Optical sensitivity follows directly from the temperature scale / 光学敏感性直接来自温升尺度",
      "source": {
        "book": "project-theory",
        "line": 1367,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-a2792489a596e1e7.svg",
      "variables": [
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "T_{\\mathrm{ad},0}",
          "symbol": "T_ad,0",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "\\delta_{\\mathrm{abs}}",
          "symbol": "δ_abs",
          "role": "optical absorption depth"
        },
        {
          "tex": "\\rho_{\\mathrm{abs}}",
          "symbol": "ρ_abs",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "c_p",
          "symbol": "c_p",
          "role": "sound speed / heat capacity"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\frac{\\partial\\ln\\Delta T}{\\partial\\ln F_0}=1,\\qquad\\frac{\\partial\\ln\\Delta T}{\\partial\\ln\\eta_{\\mathrm{abs}}}=1,\\qquad\\frac{\\partial\\ln\\Delta T}{\\partial\\ln\\delta_{\\mathrm{abs}}}=-1.",
      "context": "5.2 Optical sensitivity follows directly from the temperature scale / 光学敏感性直接来自温升尺度",
      "source": {
        "book": "project-theory",
        "line": 1373,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-834b4292d708374c.svg",
      "variables": [
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "\\delta_{\\mathrm{abs}}",
          "symbol": "δ_abs",
          "role": "optical absorption depth"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\dot R\\sim\\frac{R_b}{t_b},\\qquad\\ddot R\\sim\\frac{R_b}{t_b^2}.",
      "context": "5.3 Bubble-load sensitivity follows from acceleration scaling / 气泡载荷敏感性来自加速度尺度",
      "source": {
        "book": "project-theory",
        "line": 1385,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-46945ea81710a8f8.svg",
      "variables": [
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "R_b",
          "symbol": "R_b",
          "role": "bubble wall radius"
        },
        {
          "tex": "t_b",
          "symbol": "t_b",
          "role": "physical time"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "A=R^2\\ddot R+2R\\dot R^2\\sim\\frac{R_b^3}{t_b^2}.",
      "context": "5.3 Bubble-load sensitivity follows from acceleration scaling / 气泡载荷敏感性来自加速度尺度",
      "source": {
        "book": "project-theory",
        "line": 1391,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-3a6f66601df6b9e3.svg",
      "variables": [
        {
          "tex": "A",
          "symbol": "A",
          "role": "area / volume-acceleration coefficient"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "R_b",
          "symbol": "R_b",
          "role": "bubble wall radius"
        },
        {
          "tex": "t_b",
          "symbol": "t_b",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "p'\\sim\\frac{\\rho_lR_b^3}{dt_b^2},",
      "context": "5.3 Bubble-load sensitivity follows from acceleration scaling / 气泡载荷敏感性来自加速度尺度",
      "source": {
        "book": "project-theory",
        "line": 1397,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-87ff2367bc66a9d9.svg",
      "variables": [
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_b",
          "symbol": "R_b",
          "role": "bubble wall radius"
        },
        {
          "tex": "t_b",
          "symbol": "t_b",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\boxed{J_A\\sim\\frac{\\rho_lR_b^3}{dt_b}}.",
      "context": "5.3 Bubble-load sensitivity follows from acceleration scaling / 气泡载荷敏感性来自加速度尺度",
      "source": {
        "book": "project-theory",
        "line": 1403,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-7027b6652bffb15b.svg",
      "variables": [
        {
          "tex": "J_A",
          "symbol": "J_A",
          "role": "selected areal pressure impulse"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_b",
          "symbol": "R_b",
          "role": "bubble wall radius"
        },
        {
          "tex": "t_b",
          "symbol": "t_b",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "J_A\\sim\\frac{\\rho_lR_b^2}{\\gamma t_b}.",
      "context": "5.3 Bubble-load sensitivity follows from acceleration scaling / 气泡载荷敏感性来自加速度尺度",
      "source": {
        "book": "project-theory",
        "line": 1409,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-09fe2bc062d4580f.svg",
      "variables": [
        {
          "tex": "J_A",
          "symbol": "J_A",
          "role": "selected areal pressure impulse"
        },
        {
          "tex": "\\rho_l",
          "symbol": "ρ_l",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_b",
          "symbol": "R_b",
          "role": "bubble wall radius"
        },
        {
          "tex": "\\gamma",
          "symbol": "γ",
          "role": "stand-off / heat-capacity ratio"
        },
        {
          "tex": "t_b",
          "symbol": "t_b",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\Pi_J=\\frac{J_A}{\\sqrt{2m'G_c}}\\propto G_c^{-1/2}.",
      "context": "5.4 Interface toughness enters the availability ratios explicitly / 界面韧度显式进入可用性比值",
      "source": {
        "book": "project-theory",
        "line": 1425,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-c5bab50605754464.svg",
      "variables": [
        {
          "tex": "\\Pi_J",
          "symbol": "Π_J",
          "role": "impulse availability ratio"
        },
        {
          "tex": "J_A",
          "symbol": "J_A",
          "role": "selected areal pressure impulse"
        },
        {
          "tex": "m'",
          "symbol": "m'",
          "role": "mass or areal mass"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\Pi_E\\propto G_c^{-1}.",
      "context": "5.4 Interface toughness enters the availability ratios explicitly / 界面韧度显式进入可用性比值",
      "source": {
        "book": "project-theory",
        "line": 1431,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-4d0b7e9c234583e5.svg",
      "variables": [
        {
          "tex": "\\Pi_E",
          "symbol": "Π_E",
          "role": "energy availability ratio"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\begin{gathered}\n\\text{Optical deposition / 光学沉积}\\\\\\downarrow\\\\\nR(t)\\ \\text{and bubble geometry / 气泡历程与几何}\\\\\\downarrow\\\\\n\\text{Boundary pressure and shear / 边界压力与剪切}\\\\\\downarrow\\\\\n\\text{Rate-dependent mixed-mode interface properties}\\\\\n\\text{速率相关的混合模态界面性质}\n\\end{gathered}",
      "context": "5.5 Why the measurement order matters / 为什么测量顺序很重要",
      "source": {
        "book": "project-theory",
        "line": 1443,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-2028e32999dafa4c.svg",
      "variables": [
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\begin{aligned}\n&F(r),\\ g(t),\\ \\mu_a\\\\\n&\\qquad\\longrightarrow Q'''(\\mathbf x,t),\\ D_{\\mathrm{abs}},\\ E_{\\mathrm{abs}}\\\\[6pt]\n&D_{\\mathrm{abs}},\\ \\dot q'''_{\\max},\\ \\text{droplet state / 液滴状态}\\\\\n&\\qquad\\longrightarrow p_1,\\ P_{\\ge1},\\ P_{\\mathrm{det}}\\\\[6pt]\n&R(t),\\ \\dot R(t),\\ \\ddot R(t)\\\\\n&\\qquad\\longrightarrow p'(\\mathbf x,t)\\\\[6pt]\n&p'(\\mathbf x,t),\\ \\text{structural properties / 结构性质}\\\\\n&\\qquad\\longrightarrow q(t),\\ \\text{transmitted stress field / 传递应力场}\\\\[6pt]\n&\\text{interface tractions and separations / 界面牵引与分离量}\\\\\n&\\qquad\\longrightarrow\\text{damage and possible release / 损伤与可能的释放}.\n\\end{aligned}",
      "context": "The complete picture / 完整图景",
      "source": {
        "book": "project-theory",
        "line": 1469,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-e40fc05f2997705f.svg",
      "variables": [
        {
          "tex": "F",
          "symbol": "F",
          "role": "force / fluence"
        },
        {
          "tex": "r",
          "symbol": "r",
          "role": "liquid observation radius / embryo radius"
        },
        {
          "tex": "g",
          "symbol": "g",
          "role": "temporal pulse profile / wall gap"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "\\mu_a",
          "symbol": "μ_a",
          "role": "optical absorption coefficient"
        },
        {
          "tex": "Q'''",
          "symbol": "Q'''",
          "role": "volumetric heat deposition rate"
        },
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "D_{\\mathrm{abs}}",
          "symbol": "D_abs",
          "role": "absorbed dose per unit volume"
        },
        {
          "tex": "E_{\\mathrm{abs}}",
          "symbol": "E_abs",
          "role": "total absorbed energy"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\dot q'''_{\\max}",
          "symbol": "q̇'''_max",
          "role": "peak volumetric deposition rate"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "P_{\\ge1}",
          "symbol": "P_ge1",
          "role": "at least one true event"
        },
        {
          "tex": "P_{\\mathrm{det}}",
          "symbol": "P_det",
          "role": "detected-event probability"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "\\dot R",
          "symbol": "Ṙ",
          "role": "radial wall velocity"
        },
        {
          "tex": "\\ddot R",
          "symbol": "R̈",
          "role": "radial wall acceleration"
        },
        {
          "tex": "p'",
          "symbol": "p'",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "q",
          "symbol": "q",
          "role": "heat flux / mode displacement"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\Delta T\\propto\\eta_{\\mathrm{abs}}F_0/\\delta_{\\mathrm{abs}},\\quad J_A\\sim\\rho R_b^3/(dt_b),\\quad J_A|_{d=\\gamma R_b}\\sim\\rho R_b^2/(\\gamma t_b)",
      "context": "Sensitivity and next measurements · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T05",
        "part": 1
      },
      "figure": "assets/curriculum-figures/formula-7bb9a59d2bb89fa0.svg",
      "variables": [
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "T",
          "symbol": "T",
          "role": "temperature / kinetic energy as defined"
        },
        {
          "tex": "\\eta_{\\mathrm{abs}}",
          "symbol": "η_abs",
          "role": "absorbed fraction"
        },
        {
          "tex": "F_0",
          "symbol": "F_0",
          "role": "peak beam fluence"
        },
        {
          "tex": "\\delta_{\\mathrm{abs}}",
          "symbol": "δ_abs",
          "role": "optical absorption depth"
        },
        {
          "tex": "J_A",
          "symbol": "J_A",
          "role": "selected areal pressure impulse"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "R_b",
          "symbol": "R_b",
          "role": "bubble wall radius"
        },
        {
          "tex": "t_b",
          "symbol": "t_b",
          "role": "physical time"
        },
        {
          "tex": "\\gamma",
          "symbol": "γ",
          "role": "stand-off / heat-capacity ratio"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "\\Pi_J\\propto G_c^{-1/2},\\quad \\Pi_E\\propto G_c^{-1},\\quad \\rho_s=\\mathrm{Corr}(\\mathrm{rank}X,\\mathrm{rank}Y)",
      "context": "Sensitivity and next measurements · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T05",
        "part": 2
      },
      "figure": "assets/curriculum-figures/formula-e70b3d218b7d32b7.svg",
      "variables": [
        {
          "tex": "\\Pi_J",
          "symbol": "Π_J",
          "role": "impulse availability ratio"
        },
        {
          "tex": "G_c",
          "symbol": "G_c",
          "role": "fracture work per release area"
        },
        {
          "tex": "\\Pi_E",
          "symbol": "Π_E",
          "role": "energy availability ratio"
        },
        {
          "tex": "\\rho_s",
          "symbol": "ρ_s",
          "role": "liquid / layer / droplet density"
        },
        {
          "tex": "X",
          "symbol": "X",
          "role": "sampled input"
        },
        {
          "tex": "Y",
          "symbol": "Y",
          "role": "inferred output / spherical harmonic"
        }
      ]
    },
    {
      "page": "course/chapters/T05-sensitivity-and-measurements.html",
      "tex": "E_{\\mathrm{abs}}\\longrightarrow (p_1,R(t))\\longrightarrow (p,\\boldsymbol\\tau)\\longrightarrow (q,\\mathbf t,\\boldsymbol\\delta)\\longrightarrow \\mathrm{release}",
      "context": "Sensitivity and next measurements · original governing laws",
      "source": {
        "type": "exercise-law",
        "chapter": "T05",
        "part": 3
      },
      "figure": "assets/curriculum-figures/formula-b20022897662acc2.svg",
      "variables": [
        {
          "tex": "E_{\\mathrm{abs}}",
          "symbol": "E_abs",
          "role": "total absorbed energy"
        },
        {
          "tex": "p_1",
          "symbol": "p_1",
          "role": "single-droplet true-event probability"
        },
        {
          "tex": "R",
          "symbol": "R",
          "role": "instantaneous wall radius"
        },
        {
          "tex": "t",
          "symbol": "t",
          "role": "physical time"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\tau",
          "symbol": "τ",
          "role": "duration / dimensionless time / dummy time"
        },
        {
          "tex": "q",
          "symbol": "q",
          "role": "heat flux / mode displacement"
        },
        {
          "tex": "\\delta",
          "symbol": "δ",
          "role": "separation / perturbation / penetration depth"
        }
      ]
    },
    {
      "page": "course/references/notation-and-model-map.html",
      "tex": "xx''+\\frac32(x')^2=(1+S)x^{-3\\kappa}-1-\\frac{S}{x}\n-\\Lambda\\frac{x'}{x}, \\tag{A.1}",
      "context": "A.2 Dimensionless Rayleigh–Plesset equation / 无量纲Rayleigh–Plesset方程",
      "source": {
        "book": "generic",
        "line": 1540,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-ec3444c9fd120e60.svg",
      "variables": [
        {
          "tex": "x",
          "symbol": "x",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "x''",
          "symbol": "x''",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "x'",
          "symbol": "x'",
          "role": "position / dimensionless radius"
        },
        {
          "tex": "S",
          "symbol": "S",
          "role": "smoothstep / capillary scale"
        },
        {
          "tex": "\\kappa",
          "symbol": "κ",
          "role": "gas polytropic exponent / curvature"
        },
        {
          "tex": "\\Lambda",
          "symbol": "Λ",
          "role": "modal participation / viscous ratio"
        }
      ]
    },
    {
      "page": "course/references/notation-and-model-map.html",
      "tex": "S=\\frac{2\\sigma}{R_0\\Delta p},\\qquad\n\\Lambda=\\frac{4\\mu}{R_0\\sqrt{\\rho\\Delta p}}. \\tag{A.2}",
      "context": "A.2 Dimensionless Rayleigh–Plesset equation / 无量纲Rayleigh–Plesset方程",
      "source": {
        "book": "generic",
        "line": 1543,
        "type": "source"
      },
      "figure": "assets/curriculum-figures/formula-24a84bcff0823e0b.svg",
      "variables": [
        {
          "tex": "S",
          "symbol": "S",
          "role": "smoothstep / capillary scale"
        },
        {
          "tex": "\\sigma",
          "symbol": "σ",
          "role": "interface tension / solid stress"
        },
        {
          "tex": "R_0",
          "symbol": "R_0",
          "role": "reference equilibrium radius"
        },
        {
          "tex": "\\Delta",
          "symbol": "Δ",
          "role": "change / pressure difference"
        },
        {
          "tex": "p",
          "symbol": "p",
          "role": "pressure at the indicated location"
        },
        {
          "tex": "\\Lambda",
          "symbol": "Λ",
          "role": "modal participation / viscous ratio"
        },
        {
          "tex": "\\mu",
          "symbol": "μ",
          "role": "dynamic viscosity / absorption coefficient"
        },
        {
          "tex": "\\rho",
          "symbol": "ρ",
          "role": "liquid / layer / droplet density"
        }
      ]
    }
  ],
  "original_problem_count": 30,
  "source_atlas_card_count": 35,
  "progress_updated": "2026-10-08",
  "next_chapter": "G13",
  "next_project_module": "J01"
}