Starting from Euler's equations for an inviscid incompressible fluid of density $\rho$ with no body force, undergoing irrotational motion, show that the pressure $p$ is given by

$$\frac{p}{\rho}+\frac{\partial \phi}{\partial t}+\frac{1}{2}(\nabla \phi)^{2}=F(t)$$

for some function $F(t)$, where $\phi$ is the velocity potential.

The fluid occupies an infinite domain and contains a spherical gas bubble of radius $R(t)$ in which the pressure is $p_{g}$. The pressure in the fluid at infinity is $p_{\infty}$.

Show that

$$\ddot{R} R+\frac{3}{2} \dot{R}^{2}=\frac{p_{g}-p_{\infty}}{\rho}$$

The bubble contains a fixed mass $M$ of gas in which

$$p_{g}=C\left(M / R^{3}\right)^{2}$$

for a constant $C$. At time $t=0, R=R_{0}, \dot{R}=0$ and $p_{g}=p_{\infty} / 2$. Show that

$$\dot{R}^{2} R^{3}=\frac{p_{\infty}}{\rho}\left[R_{0}^{3}-\frac{R_{0}^{6}}{3 R^{3}}-\frac{2}{3} R^{3}\right]$$

and deduce that the bubble radius oscillates between $R_{0}$ and $R_{0} / 2^{1 / 3}$.