Start at a bubble's maximum radius R_max, where its wall is momentarily at rest. For this benchmark, assume a spherical cavity in an infinite incompressible liquid, a constant ambient pressure p_∞, and a constant vapor pressure p_v. Neglect noncondensable gas, surface tension, viscosity, acoustic radiation, and nearby walls. The inward pressure gap is positive:

从气泡达到最大半径 R_max、壁面瞬时静止的时刻开始。为建立这一基准模型，假设空腔保持球形，位于无限延伸的不可压缩液体中；环境压力 p_∞ 和蒸气压 p_v 均保持恒定。忽略不凝性气体、表面张力、黏性、声辐射及邻近壁面。向内的压力差为正：

Δp = p_∞ − p_v > 0

Why omit noncondensable gas? The simple Rayleigh formula needs a constant inward pressure gap. A fixed amount of gas instead becomes compressed as R falls. Under the Lesson 6 polytropic law:

为什么忽略不凝性气体？简单的 Rayleigh 公式要求向内的压力差恒定。若泡内保留固定量的气体，R 减小时气体会受压。按照第六课的多方规律：

p_g(R) = p_g,max(R_max/R)^3κ

Its rising pressure reduces the inward drive and may eventually cause rebound. Neglecting gas isolates the liquid-inertia benchmark; it is not a rule for all cavitation bubbles.

不断升高的气体压力会减小向内驱动力，并可能最终使气泡回弹。忽略气体是为了单独研究液体惯性的基准问题，并非所有空化气泡都必须这样处理。

Δp_net(R) = p_∞ − p_v − p_g(R)

Pressure drives; liquid inertia delays

压力驱动塌缩，液体惯性决定耗时

With those assumptions, Rayleigh–Plesset reduces to an equation with a constant inward pressure drive. At the start, Ṙ = 0, so the initial wall acceleration is inward. As the cavity shrinks, pressure work becomes liquid kinetic energy and the inward wall speed rises.

在上述假设下，Rayleigh–Plesset 方程化为向内压力驱动力恒定的方程。起始时 Ṙ = 0，因此壁面初始加速度指向内部。随着空腔缩小，压力所做的功转化为液体动能，壁面向内的速度逐渐增大。

R R̈ + (3/2)Ṙ² = −Δp/ρ

The collapse-time scale

塌缩时间的尺度

A pressure difference has units of energy per volume. Over a cavity of size R_max, available pressure work therefore scales as Δp R_max³. The surrounding liquid's kinetic energy scales as its density times a volume R_max³ times a characteristic speed squared. If collapse takes time t, that speed is of order R_max/t. Equating these scales gives the proportional relation, without its numerical coefficient:

压差的单位是单位体积的能量。对于尺度为 R_max 的空腔，可释放的压力功因此按 Δp R_max³ 缩放。周围液体的动能按其密度、体积尺度 R_max³ 与特征速度平方的乘积缩放。若塌缩历时为 t，该速度的量级为 R_max/t。平衡这两个尺度即可得到比例关系，但尚得不到数值系数：

Δp R_max³ ∼ ρR_max³(R_max/t)²
⇒ t ∝ R_max√(ρ/Δp)

Integrate the radial equation

对径向方程积分

Now recover the coefficient from the exact ideal equation above. Multiply it by 4πρR²Ṙ. The left-hand side becomes the time derivative of the surrounding liquid's kinetic energy from Lesson 5; the right-hand side is pressure power as the cavity volume changes:

现在从上面的精确理想方程求出数值系数。将方程两边乘以 4πρR²Ṙ。左边成为第五课所求周围液体动能的时间导数；右边是空腔体积变化时压力所做功的功率：

d[2πρR³Ṙ²]/dt = −Δp · d[(4π/3)R³]/dt

At the starting maximum radius, the wall is at rest, so its kinetic energy is zero. Integrating from that state to a smaller radius R makes the liquid's kinetic energy equal the pressure work released by the lost cavity volume:

在起始最大半径处，壁面静止，因此液体动能为零。从该状态积分至较小半径 R，可得液体动能等于空腔体积减小所释放的压力功：

2πρR³Ṙ² = (4π/3)Δp(R_max³ − R³)

Solving this equality for wall speed gives two algebraic signs. During collapse, R decreases, so take the negative root:

由这一等式解出壁面速度时，代数上有正负两个根。塌缩期间 R 减小，因此取负根：

Ṙ = −√{[2Δp/(3ρ)] [(R_max/R)³ − 1]}

To accumulate elapsed time, divide each small decrease in radius by the magnitude of that speed. Let x = R/R_max, so the starting radius is x = 1 and the ideal zero-radius endpoint is x = 0. The change of variables separates all physical parameters from a dimensionless integral:

要累加历时，就用每一小段半径缩小量除以该处速度的大小。令 x = R/R_max，则起始半径对应 x = 1，理想的零半径终点对应 x = 0。变量代换使全部物理参数与一个无量纲积分分离：

t_c = ∫₀^R_maxdR/|Ṙ|
= R_max√[3ρ/(2Δp)] ∫₀¹x^3/2/√(1 − x³) dx

The remaining integral is a pure number. Set z = x³; then dx = (1/3)z^−2/3dz. This converts it into a standard beta-function integral. Using B(a,b) = Γ(a)Γ(b)/Γ(a+b) gives:

剩余的积分是一个纯数。令 z = x³，于是 dx = (1/3)z^−2/3dz。它由此转为标准的 Beta 函数积分。利用 B(a,b) = Γ(a)Γ(b)/Γ(a+b)，得到：

I = (1/3)∫₀¹z^−1/6(1 − z)^−1/2dz
= B(5/6, 1/2)/3 = Γ(5/6)Γ(1/2)/[3Γ(4/3)] ≈ 0.7468342;
√(3/2)I ≈ 0.9146814

The coefficient is therefore a consequence of the spherical, incompressible, constant-pressure model and its initial condition, not an empirical fit. Substituting it into the time expression yields:

因此，该系数来自球对称、不可压缩、压差恒定模型及其初始条件，并非经验拟合。将它代入时间表达式可得：

t_c ≈ 0.915 R_max√(ρ/Δp)

At the same pressure gap and liquid density, doubling R_max doubles this ideal time. A stronger inward pressure gap shortens it according to an inverse square root. The formula is a benchmark for judging the order of magnitude of a measured collapse time; it is not a complete model of a gas-containing bubble.

在压力差与液体密度不变时，R_max 加倍会使这一理想时间加倍。更大的向内压力差则按平方根的倒数缩短时间。该公式可作为判断实测塌缩时间数量级的基准，但不是含气气泡的完整模型。