Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 38C (Waves)

For a plane acoustic wave propagating along $\hat{\mathbf{k}}$, the linear momentum equation and dispersion $\omega =c_{0}k$ give $\tilde{p}={\rho }_0{c}_0{u}_{\parallel }$ and $\mathbf{u}=u_{\parallel }\hat{\mathbf{k}}$. Therefore the instantaneous acoustic intensity isFor harmonic complex amplitudes the time average instead has the factor one half, $⟨\mathbf{I}⟩=\frac{1}{2}{\rho }_0{c}_0|\hat{\mathbf{u}}{|}^2\hat{\mathbf{k}}$.

For a radial velocity potential, set $v=r\varphi$. The spherical wave equation reduces to $v_{tt}=c_0^2{v}_{rr}$, soNo incoming radiation from infinity means $G=0$. Linearizing the moving-surface condition at $r=a$ gives ${\varphi }_r(a,t)=i\omega aϵe^{i\omega t}$, with real parts understood. Write the outgoing solution as $\varphi =A{e}^{i\omega (t-(r-a)/c_0)}/r$. Its radial derivative at $a$ is $-A(1+i\omega a/c_0)e^{i\omega t}/a^2$. Therefore the outgoing acoustic field of a pulsating sphere isIn the far field the velocity amplitude has magnitude $(\omega /c_0)|A|/r$. Multiply its average plane-wave intensity by the spherical area to obtain the mean acoustic power of a pulsating sphere:This is positive outward radiated power, evaluated to leading quadratic order in the small real displacement amplitude.