Sensitivity and next measurements

Source: supplied project-theory book, Stage 5. The complete bilingual chapter follows, with its derivations and qualifications. Textbook source (Markdown) · Original Markdown.

来源：所给项目理论教材阶段 5。下方完整收录双语章节及推导和限定。教材原文（Markdown） · 原始 Markdown。

Keep the source’s evidence states: optical normalization is an accounting check; activation is uncalibrated; R(t) is prescribed; near-wall loading and cohesive release remain conditional.

保留来源证据状态：光学归一化属于核算检查；激活尚未标定；R(t) 为规定输入；近壁加载及黏聚释放仍有条件。

5. Sensitivity and the next measurements 5.1 What a Spearman ranking measures 5.2 Optical sensitivity follows directly from the temperature scale 5.3 Bubble-load sensitivity follows from acceleration scaling 5.4 Interface toughness enters the availability ratios explicitly 5.5 Why the measurement order matters

5. Sensitivity and the next measurements

敏感性与下一步测量

What the theory tells us to measure next. Corresponding to Theory 05, source page 5.

理论告诉我们下一步应测量什么。对应 Theory 05，原始材料第5页。

The final page uses the calculations to prioritize uncertain inputs rather than to declare a validated transfer threshold. Its reported rankings are calculation-only Spearman correlations. [S1, p. 5]

最后一页利用计算结果对不确定输入进行优先级排序，而不是宣称已经得到经验证的转印阈值。其中报告的排序仅来自计算所得的 Spearman 秩相关。[S1，第5页]

5.1 What a Spearman ranking measures

Spearman 排序衡量什么

For sampled inputs $X$ and outputs $Y$ ,

对于抽样得到的输入 $X$ 和输出 $Y$ ，

\boxed{\rho_s=\operatorname{Corr}\bigl(\operatorname{rank}X,\operatorname{rank}Y\bigr)}.

5.1 What a Spearman ranking measures / Spearman 排序衡量什么; ρ_s: liquid / layer / droplet density; X: sampled input; Y: inferred output / spherical harmonic

It measures whether larger input values tend to accompany larger or smaller outputs. It is not a governing equation, and its magnitude depends on the sampled ranges and assumptions.

它衡量的是：较大的输入值是否倾向于对应较大或较小的输出值。它不是控制方程，其数值大小取决于抽样范围与相关假设。

The directions of several reported relationships can nevertheless be understood from the equations we derived.

不过，原始材料报告的若干关系，其正负方向可以通过已经推导出的方程来理解。

5.2 Optical sensitivity follows directly from the temperature scale

光学敏感性直接来自温升尺度

From

由

\Delta T_{\mathrm{ad},0}=\frac{\eta_{\mathrm{abs}}F_0}{\rho_{\mathrm{abs}}c_p\delta_{\mathrm{abs}}},

5.2 Optical sensitivity follows directly from the temperature scale / 光学敏感性直接来自温升尺度; Δ: change / pressure difference; T_ad,0: temperature / kinetic energy as defined; η_abs: absorbed fraction; F_0: peak beam fluence; ρ_abs: liquid / layer / droplet density; c_p: sound speed / heat capacity; δ_abs: optical absorption depth

take logarithms:

两边取对数：

\ln\Delta T_{\mathrm{ad},0}=\ln\eta_{\mathrm{abs}}+\ln F_0-\ln\delta_{\mathrm{abs}}-\ln(\rho_{\mathrm{abs}}c_p).

Holding the other quantities fixed,

保持其他量不变，可得

\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.

5.2 Optical sensitivity follows directly from the temperature scale / 光学敏感性直接来自温升尺度; Δ: change / pressure difference; T: temperature / kinetic energy as defined; F_0: peak beam fluence; η_abs: absorbed fraction; δ_abs: optical absorption depth

This explains the positive fluence/absorption dependence and negative absorption-depth dependence in the source’s ranking. It does not reconstruct the numerical rank coefficients without the original uncertainty samples. [S1, p. 5]

这解释了原始材料排序中，温升与能量面密度、吸收比例的正相关，以及与吸收深度的负相关。但如果没有原始不确定性样本，就不能据此复现具体的秩相关系数。[S1，第5页]

5.3 Bubble-load sensitivity follows from acceleration scaling

气泡载荷敏感性来自加速度尺度

For comparable dimensionless trajectories with characteristic radius $R_b$ and time $t_b$ ,

对于无量纲形状相近、特征半径为 $R_b$ 、特征时间为 $t_b$ 的轨迹，

\dot R\sim\frac{R_b}{t_b},\qquad\ddot R\sim\frac{R_b}{t_b^2}.

5.3 Bubble-load sensitivity follows from acceleration scaling / 气泡载荷敏感性来自加速度尺度; Ṙ: radial wall velocity; R_b: bubble wall radius; t_b: physical time; R̈: radial wall acceleration

Consequently,

因此

A=R^2\ddot R+2R\dot R^2\sim\frac{R_b^3}{t_b^2}.

5.3 Bubble-load sensitivity follows from acceleration scaling / 气泡载荷敏感性来自加速度尺度; A: area / volume-acceleration coefficient; R: instantaneous wall radius; R̈: radial wall acceleration; Ṙ: radial wall velocity; R_b: bubble wall radius; t_b: physical time

At fixed observation distance $d$ , the leading pressure scale is

在观测距离 $d$ 固定时，主导压力尺度为

p'\sim\frac{\rho_lR_b^3}{dt_b^2},

5.3 Bubble-load sensitivity follows from acceleration scaling / 气泡载荷敏感性来自加速度尺度; p': pressure at the indicated location; ρ_l: liquid / layer / droplet density; R_b: bubble wall radius; t_b: physical time

and a corresponding impulse scale is

相应的冲量尺度为

\boxed{J_A\sim\frac{\rho_lR_b^3}{dt_b}}.

5.3 Bubble-load sensitivity follows from acceleration scaling / 气泡载荷敏感性来自加速度尺度; J_A: selected areal pressure impulse; ρ_l: liquid / layer / droplet density; R_b: bubble wall radius; t_b: physical time

If stand-off ratio is held fixed instead, so $d=\gamma R_b$ ,

如果保持不变的是距壁比，即 $d=\gamma R_b$ ，则

J_A\sim\frac{\rho_lR_b^2}{\gamma t_b}.

These are explanatory scalings, not substitutes for the disk-average calculation. They show why larger bubbles and shorter motion times tend to increase loading, while also showing that “sensitivity to radius” depends on which geometric quantities are held fixed.

这些属于解释性的尺度关系，不能替代圆盘平均压力计算。它们说明了为什么更大的气泡和更短的运动时间往往会增大载荷，同时也表明，“对半径的敏感性”取决于哪些几何量被保持不变。

The source’s calculated ranking identifies $R_{\max}$ and boundary gain as strong positive influences on growth impulse, and growth time as a negative influence. [S1, p. 5]

原始材料的计算排序表明， $R_{\max}$ 和边界增益对增长阶段冲量具有较强的正向影响，而增长时间具有负向影响。[S1，第5页]

5.4 Interface toughness enters the availability ratios explicitly

界面韧度显式进入可用性比值

For fixed impulse and mass,

当冲量与质量固定时，

\Pi_J=\frac{J_A}{\sqrt{2m'G_c}}\propto G_c^{-1/2}.

5.4 Interface toughness enters the availability ratios explicitly / 界面韧度显式进入可用性比值; Π_J: impulse availability ratio; J_A: selected areal pressure impulse; m': mass or areal mass; G_c: fracture work per release area

For fixed modal energy and release area,

当模态能量与释放面积固定时，

\Pi_E\propto G_c^{-1}.

5.4 Interface toughness enters the availability ratios explicitly / 界面韧度显式进入可用性比值; Π_E: energy availability ratio; G_c: fracture work per release area

Thus a larger nominal fracture energy decreases both availability ratios. This is consistent with the negative toughness ranking reported for the release surrogate, while remaining conditional on the particular screening calculation. [S1, p. 5]

因此，更大的名义断裂能会使两个可用性比值都减小。这与原始材料中释放替代量对韧度呈负相关的排序一致，但仍然以具体筛查计算的假设为条件。[S1，第5页]

5.5 Why the measurement order matters

为什么测量顺序很重要

The source proposes the following order:

原始材料提出了如下顺序：

\begin{gathered} \text{Optical deposition / 光学沉积}\\\downarrow\\ R(t)\ \text{and bubble geometry / 气泡历程与几何}\\\downarrow\\ \text{Boundary pressure and shear / 边界压力与剪切}\\\downarrow\\ \text{Rate-dependent mixed-mode interface properties}\\ \text{速率相关的混合模态界面性质} \end{gathered}

5.5 Why the measurement order matters / 为什么测量顺序很重要; t: physical time

This order follows the direction of the model dependencies. [S1, p. 5]

这一顺序遵循模型之间的依赖方向。[S1，第5页]

For example, fitting interface toughness while the incident pressure is unknown allows the toughness parameter to compensate for an incorrect loading amplitude. A successful displacement fit would then not identify the true interface property.

例如，在入射压力未知的情况下拟合界面韧度，就可能使韧度参数去补偿错误的载荷幅值。这样一来，即使成功拟合了位移，也不能说明识别出了真实的界面性质。

Likewise, obtaining the correct absorbed energy does not identify activation coefficients, and choosing an $8\ \mu\mathrm m$ bubble does not establish that the laser will generate one. The source explicitly warns against using downstream parameters to compensate for unknown upstream inputs. [S1, p. 5]

同样，得到正确的吸收能量并不意味着识别出了激活系数；选取一个 $8\ \mu\mathrm m$ 的气泡，也不能证明激光会产生这样的气泡。原始材料明确提醒，不应利用下游参数去补偿未知的上游输入。[S1，第5页]

The complete picture

完整图景

You can now read the theory as a sequence of different questions:

现在可以将这套理论理解为一系列彼此不同、逐级衔接的问题：

\begin{aligned} &F(r),\ g(t),\ \mu_a\\ &\qquad\longrightarrow Q'''(\mathbf x,t),\ D_{\mathrm{abs}},\ E_{\mathrm{abs}}\\[6pt] &D_{\mathrm{abs}},\ \dot q'''_{\max},\ \text{droplet state / 液滴状态}\\ &\qquad\longrightarrow p_1,\ P_{\ge1},\ P_{\mathrm{det}}\\[6pt] &R(t),\ \dot R(t),\ \ddot R(t)\\ &\qquad\longrightarrow p'(\mathbf x,t)\\[6pt] &p'(\mathbf x,t),\ \text{structural properties / 结构性质}\\ &\qquad\longrightarrow q(t),\ \text{transmitted stress field / 传递应力场}\\[6pt] &\text{interface tractions and separations / 界面牵引与分离量}\\ &\qquad\longrightarrow\text{damage and possible release / 损伤与可能的释放}. \end{aligned}

The diagram locates the physical system; its key lists the symbols in the formula. Read the definition in the surrounding derivation when a letter has different roles.

图示标出物理体系，图例列出公式中的符号。若某字母有多种作用，应以邻近推导中的定义为准。

The key missing connection in the present source is that activation and laser heating do not yet generate $R(t)$ through a closed forward model. The radius history is prescribed. Likewise, the resulting pressure screen does not yet provide validated near-wall loading or resolved cohesive tractions. Those are stated limitations of the supplied theory, not steps that can be filled by algebra alone. [S1, pp. 3–4]

当前原始材料中缺失的关键连接是：激活与激光加热尚未通过一个闭合的正向模型生成 $R(t)$ 。半径历程是人为规定的。同样，由此得到的压力筛查也尚未提供经验证的近壁载荷，或经过空间解析的黏聚牵引。这些都是所提供理论明确说明的限制，而不是仅靠代数运算就能补齐的步骤。[S1，第3–4页]

The most important conceptual separation is:

最重要的概念区分是：

Energy accounting tells you what is available. Activation statistics tell you whether an event occurs. Bubble mechanics tells you how liquid is accelerated. Structural dynamics tells you how that loading moves the film. Fracture mechanics tells you whether the interface actually separates.

能量核算告诉你有多少资源可用；激活统计告诉你事件是否发生；气泡力学告诉你液体如何被加速；结构动力学告诉你这些载荷如何使薄膜运动；断裂力学告诉你界面是否真正发生分离。

Keeping those questions distinct is what turns a collection of equations into a defensible cavitation-transfer theory.

只有始终区分这些问题，才能把一组方程转化为一套依据清晰、边界明确的空化转印理论。

Source notes

来源说明

[S1] Primary supplied source. Laser_Triggered_Cavitation_Theories_20260925.pdf, five pages, Theory 01–05 (slides 07–11 of the original 17-slide sequence). The pages bear the footer “Laser-Triggered Cavitation Transfer — Detailed bilingual visual notes” and the date 19 September 2026. The filename date and the printed footer date are retained as supplied. The five pages are included unchanged in Appendix A.

[S1] 用户提供的主要材料。 Laser_Triggered_Cavitation_Theories_20260925.pdf，共五页，内容为 Theory 01–05（原17页幻灯片序列中的第07–11页）。各页页脚标为“Laser-Triggered Cavitation Transfer — Detailed bilingual visual notes”，日期为2026年9月19日。文件名日期与页脚日期均按原样保留。五页原文在附录 A 中原样收录。

[R1] Supplementary link carried over from the preceding explanation. Spherical radial-flow assumptions and the Rayleigh–Plesset framework; publisher page in The Journal of the Acoustical Society of America. No new bibliographic verification or literature expansion was performed for this translation.

[R1] 沿用前文讲解中的补充链接。 用于球对称径向流动假设与 Rayleigh–Plesset 框架；链接指向 The Journal of the Acoustical Society of America 的出版页面。本次翻译未另外进行书目信息核验或文献扩展。

https://pubs.aip.org/asa/jasa/article/155/2/1593/3266983/A-unifying-Rayleigh-Plesset-type-equation-for

[R2] Supplementary link carried over from the preceding explanation. The qualification concerning negative pressure and metastable liquids; arXiv record 2410.17626. This preserves the reference already attached to that qualification in the preceding explanation.

[R2] 沿用前文讲解中的补充链接。 用于关于负压与亚稳态液体的限定说明；arXiv 记录号为2410.17626。此处保留了前文该项说明所附的参考链接。

https://arxiv.org/abs/2410.17626

Inspect all five original graphical theory pages.

查看全部五页原始图文理论。

Integrated calculation and model check

综合计算与模型检查

Use the declared parameter set in each part. The values are teaching scenarios unless specifically identified as inherited project inputs. Each reference panel follows its question.

使用每小题明确给出的参数。除明确标为沿用项目输入外，数值均为教学情景。各参考解答紧随题目。

Double F0 and halve absorption depth, holding ηabs and heat capacity fixed. Calculate the temperature multiplier. Double Rb while holding distance d and bubble time fixed; repeat with fixed stand-off ratio.
ηabs 与热容固定，F0 加倍、吸收深度减半，求温升倍率。泡时间固定，Rb 加倍时分别计算固定距离 d 与固定离壁比下的冲量尺度倍率。
Reference answer / 参考答案
$\Delta T\propto\eta_{\mathrm{abs}}F_0/\delta_{\mathrm{abs}},\quad J_A\sim\rho R_b^3/(dt_b),\quad J_A|_{d=\gamma R_b}\sim\rho R_b^2/(\gamma t_b)$
$Sensitivity and next measurements · original governing laws; Δ: change / pressure difference; T: temperature / kinetic energy as defined; η_abs: absorbed fraction; F_0: peak beam fluence; δ_abs: optical absorption depth; J_A: selected areal pressure impulse; ρ: liquid / layer / droplet density; R_b: bubble wall radius; t_b: physical time; γ: stand-off / heat-capacity ratio$
The diagram locates the physical system; its key lists the symbols in the formula. Read the definition in the surrounding derivation when a letter has different roles.
图示标出物理体系，图例列出公式中的符号。若某字母有多种作用，应以邻近推导中的定义为准。
Temperature increases by 4. At fixed d, impulse scale increases by 8; at fixed γ, by 4. State what is held fixed before calling something “radius sensitivity.” These are conditional scalings, not reconstructed source rank coefficients.
温升增加 4 倍。固定 d 时冲量尺度增加 8 倍；固定 γ 时增加 4 倍。讨论“半径敏感性”前须说明固定量。这些是有条件尺度关系，并非重建来源排序系数。
If Gc increases by four, holding impulse, modal energy and release area fixed, compute the changes in ΠJ and ΠE. Explain the meaning of a Spearman rank coefficient.
Gc 增至四倍，冲量、模态能量及释放面积固定，求 ΠJ 与 ΠE 变化，并解释 Spearman 秩相关系数。
Reference answer / 参考答案
$\Pi_J\propto G_c^{-1/2},\quad \Pi_E\propto G_c^{-1},\quad \rho_s=\mathrm{Corr}(\mathrm{rank}X,\mathrm{rank}Y)$
$Sensitivity and next measurements · original governing laws; Π_J: impulse availability ratio; G_c: fracture work per release area; Π_E: energy availability ratio; ρ_s: liquid / layer / droplet density; X: sampled input; Y: inferred output / spherical harmonic$
The diagram locates the physical system; its key lists the symbols in the formula. Read the definition in the surrounding derivation when a letter has different roles.
图示标出物理体系，图例列出公式中的符号。若某字母有多种作用，应以邻近推导中的定义为准。
ΠJ halves, ΠE becomes one quarter. Spearman measures rank association within the sampled assumptions/ranges; it is not causation, a constitutive law, or a universal material sensitivity. Numerical source rankings cannot be recreated without the original samples.
ΠJ 减半，ΠE 变为四分之一。Spearman 衡量所采样假设和范围内的秩关联，不是因果、本构规律或普适材料敏感性。没有原始样本，不能重建来源数值排序。
Write a measurement sequence that makes the project chain identifiable. Which stage remains prescribed, and where must hydrogel and pressure-limit extensions enter?
写出使项目链可辨识的测量顺序，指出仍规定输入的阶段，以及水凝胶和压力极限扩展应在哪里进入。
Reference answer / 参考答案
$E_{\mathrm{abs}}\longrightarrow (p_1,R(t))\longrightarrow (p,\boldsymbol\tau)\longrightarrow (q,\mathbf t,\boldsymbol\delta)\longrightarrow \mathrm{release}$
The diagram locates the physical system; its key lists the symbols in the formula. Read the definition in the surrounding derivation when a letter has different roles.
图示标出物理体系，图例列出公式中的符号。若某字母有多种作用，应以邻近推导中的定义为准。
Measure optical deposition first, then activation and bubble geometry/history, then boundary pressure/shear, then rate-dependent mixed-mode interface properties. R(t) remains prescribed in the supplied theory. Read Appendix A (hydrogel) and Appendix B (pressure limits) after this core route; neither closes missing target measurements by itself.
先测光能沉积，再测激活与气泡几何历程，随后测边界压力和剪切，最后测速率相关混合模态界面性质。所给理论仍规定 R(t)。完成主路线后阅读附录 A（水凝胶）与 B（压力极限）；二者均不能自行补齐目标测量。

Brief conceptual synthesis

简短概念综合

After using the reference panels, give a short explanation of the physical mechanism, the strongest assumption, and one observation that could falsify this chapter’s model. Preparation of later chapters does not require waiting for that reply.

核对参考解答后，简述物理机制、最强假设，以及一个可证伪本章模型的观测。后续章节编写无需等待该回答。