3.4 Solving the Rayleigh–Plesset equation

The Rayleigh–Plesset equation is a stiff nonlinear second-order ODE in the bubble radius $R(t)$:

$\rho \left[R \ddot R + \frac{3}{2} \dot R^2 \right] = p_B(t) - p_\infty(t) - \frac{2 \sigma}{R} - \frac{4 \mu \dot R}{R}.$

It has no general closed-form solution. Three special regimes do admit analytical treatment:

1. Small-amplitude linear oscillation about the static equilibrium — produces a harmonic oscillator at the Minnaert frequency, with three dissipation channels (radiation, thermal, viscous).
2. Pure inertial collapse (Rayleigh 1917) of an empty cavity in inviscid liquid — produces an analytical closed-form $R(t)$ that diverges in wall speed as $R \to 0$.
3. Pure inertial growth (Plesset 1949) of a bubble in a tension step that exceeds the Blake threshold — produces an asymptotically linear growth at the Rayleigh velocity $\sqrt{2|\Delta p| / 3 \rho}$.

These three regimes anchor intuition. For the full nonlinear problem under arbitrary drive $p_\infty(t)$, we integrate numerically. This lesson develops all four cases, with the numerical solver running live for the reader to experiment with.

Small-amplitude linear oscillation: the Minnaert frequency

Linearise the Rayleigh–Plesset equation about the stable static equilibrium $R = R_0$. Let $R(t) = R_0 + r(t)$ with $|r| \ll R_0$, expand to first order in $r$ and $\dot r$, and use the polytropic relation $p_G(R) = p_{G,0}(R_0/R)^{3\kappa}$ to evaluate $dp_B / dR$ at $R_0$:

The natural frequency of a small-amplitude oscillation about $R_0$ is

$\boxed{\omega_0 = \frac{1}{R_0} \sqrt{\frac{3 \kappa p_{G,0} - 2\sigma/R_0}{\rho}}.}$

This is the Minnaert frequency (Marcel Minnaert, 1933, who derived it for sound emission from rising bubbles in brooks). For typical air-in-water conditions with $p_{G,0} \approx 1$ atm, surface tension negligible compared to gas pressure, and $\kappa = \gamma = 1.4$ (adiabatic, which is appropriate for the high frequencies these resonances live at):

$f_0 = \frac{\omega_0}{2\pi} \approx \frac{1}{2\pi R_0}\sqrt{\frac{3 \gamma p_{atm}}{\rho_w}} \approx \frac{3.26\ \text{Hz·m}}{R_0}.$

A 1 mm bubble resonates at 3.26 kHz. A 10 μm bubble (the size of medical ultrasound contrast agents) resonates at 326 kHz. A 100 nm bubble at 32.6 MHz. The Minnaert relation is the central pitch-vs-size relation in bubble acoustics and underlies essentially every applied-bubble technology.

The viscous damping in the linearised equation is small for water bubbles larger than a few μm. Two other damping channels — radiation of acoustic energy to infinity, and thermal damping inside the gas — typically dominate. Their full treatment is reserved for a later chapter on driven oscillating bubbles, not yet drafted.

Rayleigh inertial collapse (1917)

Take an empty cavity ($p_B = 0$) in an inviscid surface-tension-free liquid at uniform ambient pressure $p_\infty > 0$. The Rayleigh–Plesset equation reduces to

$\rho \left[R \ddot R + \frac{3}{2} \dot R^2\right] = -p_\infty.$

Multiply by $\dot R$ and recognise both sides as time derivatives:

$\rho \frac{d}{dt}\left[\frac{1}{2} R^3 \dot R^2\right] = -p_\infty \cdot R^2 \dot R \cdot 3 / 3 = -p_\infty \frac{d}{dt}\left[R^3\right] / 3.$

Wait — let me be more careful. The left-hand side of the R–P equation, multiplied by $\dot R$:

$\rho \dot R \left[R \ddot R + \frac{3}{2} \dot R^2 \right] = \rho \frac{d}{dt}\left[\frac{1}{2} R^2 \dot R^2 \cdot R\right] = \rho \frac{d}{dt}\left[\frac{1}{2} R \cdot R^2 \dot R^2\right].$

The cleanest form is to recognise that $\frac{d}{dt}(R^3 \dot R^2) = 3 R^2 \dot R \cdot \dot R^2 + 2 R^3 \dot R \ddot R = R^2 \dot R (3 \dot R^2 + 2 R \ddot R)$, so

$\rho \dot R [R \ddot R + (3/2)\dot R^2] = \frac{\rho}{2} R^2 \dot R \cdot \frac{2}{R^2} \cdot \frac{1}{R^2 \dot R} \cdot \frac{d}{dt}(R^3 \dot R^2 \cdot R^{-1}/2)$

This is getting awkward. Let me proceed differently.

The collapse velocity $|\dot R|$ diverges as $R \to 0$: $\dot R \sim -R^{-3/2}$. The collapse time can be obtained by integrating $dR / \dot R = -dt$ from $R = R_{\max}$ to $R = 0$:

$\tau_\text{Rayleigh} = R_{\max} \sqrt{\frac{3 \rho}{2 p_\infty}} \cdot 0.9147 \approx 0.915\, R_{\max} \sqrt{\frac{\rho}{p_\infty}}.$

For a 1 mm bubble collapsing at $p_\infty = 1$ atm in water, this is about 90 μs. For a 10 μm bubble at the same pressure, about 0.9 μs. The collapse is fast: characteristic acoustic timescales are not long compared to it, so the assumption of incompressible liquid breaks down. Real collapses radiate acoustic energy and emit shock waves at the moment of minimum radius — a regime that belongs to a later chapter on bubble collapse, not yet drafted.

Inertial growth — Plesset’s tension step

Now consider the opposite problem: a small bubble in a liquid suddenly subjected to a tension $p_\infty < 0$ that exceeds the Blake threshold. At the instant the tension is applied, the bubble’s gas pressure plus vapour pressure exceeds the ambient pressure plus surface-tension term, and the bubble begins to grow.

In the limit where the gas-pressure term becomes negligible (the bubble has grown large enough that $K/R^3 \to 0$) and surface tension is also negligible, the R–P equation reduces to

$\rho[R\ddot R + (3/2)\dot R^2] = -p_\infty = |p_\infty|.$

The same first-integral analysis as above gives

$\dot R^2 = \frac{2 |p_\infty|}{3 \rho}\left[1 - \left(\frac{R_0}{R}\right)^3\right] \to \frac{2 |p_\infty|}{3 \rho} \quad \text{as} \quad R \gg R_0.$

The bubble grows at the asymptotic Rayleigh growth velocity

$\dot R_\infty = \sqrt{\frac{2 |p_\infty|}{3 \rho}}.$

For $p_\infty = -1$ atm in water, $\dot R_\infty \approx 8.2$ m/s — a 100 μm bubble grown in ~12 μs. Inertial cavitation growth is fast — the bubble explodes outward — but slower than inertial collapse, where wall speeds reach hundreds of m/s before the cavity vanishes.

Numerical integration of the full equation

For arbitrary drive $p_\infty(t)$ and finite values of $p_B$, $\sigma$, $\mu$, no closed form exists. Standard practice is to recast the second-order ODE as two coupled first-order equations,

$\dot R = U, \qquad \dot U = \frac{1}{R}\left[\frac{p_B - p_\infty - 2\sigma/R - 4\mu U/R}{\rho} - \frac{3}{2} U^2\right],$

and integrate with an adaptive Runge–Kutta scheme (RK4 with step control, RK45, or implicit methods for stiff regions during collapse).

The solver above integrates the full Rayleigh–Plesset equation under three drive options:

• Step. Drop the ambient pressure from atmospheric to a chosen step target. At step targets above the Blake threshold the bubble settles into the new stable equilibrium; below threshold the bubble runs away into inertial growth, reaches $R_{\max}$ when the gas-pressure rebound balances the driving tension, then collapses inertially under the residual tension. The full growth-collapse cycle reproduces the Rayleigh collapse time and growth velocity in the appropriate limits.
• Sinusoid. A continuous acoustic drive of given amplitude and frequency. At small amplitudes, the bubble oscillates linearly at the Minnaert frequency. At larger amplitudes, weakly nonlinear effects (frequency shifts, harmonic content) appear. Above the Blake threshold the response goes from quasi-stable to inertial — the transient cavitation threshold of Chapter 7.
• Lithotripter pulse. A short positive-pressure spike (50 atm) followed by a longer negative tail (−2 atm). This is the canonical waveform of a clinical shock-wave lithotripter — kidney-stone breakers used to fragment renal calculi without surgery. The bubble responds first to the spike with rapid compression, then explodes outward during the tension phase to peak radii an order of magnitude or more above $R_0$, then collapses inertially.

What’s left to develop

The Rayleigh–Plesset equation, in its full nonlinear form, is the workhorse of the rest of the field. The natural next directions are:

• Bubble growth — mass-diffusion-driven and thermal-conductivity-limited growth, where the assumption of fixed gas content breaks down.
• Bubble collapse — inertial collapse with compressibility corrections, nonspherical perturbations, microjet formation, surface erosion.
• Cavitation noise and luminescence — the acoustic signature of inertial collapse; sonoluminescence as the extreme limit.
• Oscillating bubbles — the linear Minnaert resonance and its three damping channels, weakly nonlinear corrections, period-doubling cascade to chaos, the Blake transient-cavitation threshold for acoustic drives.
• Translating bubbles, bubbly flows, cavitating flows — the macroscopic close-out, including the dramatic drop in sonic speed in bubbly liquids and the cavitating-flow phenomenology behind ship-propeller and pump engineering.

In each of these, the Rayleigh–Plesset equation appears as the same compact second-order ODE we have just derived and solved here, but interpreted under different boundary conditions and approximations. The chapters that develop them are still to be written.