First integral of polytropic spherical-bubble motion ($R^3{\dot{R}}^2=\frac{2}{\rho }{\int }_{R_0}^R{s}^2[p_g(s)-p_{\infty }]ds$)

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 $\kappa \ne 1$, 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.