= First integral of polytropic spherical-bubble motion
{title2=$R^3\dot R^2=\frac2\rho\int_{R_0}^R s^2[p_g(s)-p_\infty]ds$}
For gas pressure depending only on radius and initial speed zero, multiplying the <Rayleigh equation for an inviscid spherical bubble> by $2R^2\dot R$ integrates to the displayed identity. A polytropic pressure $p_g=K/R^{3\kappa}$ gives elementary power integrals when $\kappa\ne1$, and a logarithm when $\kappa=1$. Turning radii are the positive zeros of this first integral. At $\kappa=2$ and $p_g(R_0)=p_\infty/2$, they are $R_0$ and $R_0/2^{1/3}$; the inward and outward accelerations at them have opposite signs, giving bounded undamped oscillation in the <ideal> model.
Back to article page
