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Write the density and pressure as ${\rho }_0+{\rho }^{\prime }$ and $p_0+p^{\prime }$, and let the small fluid velocity be the gradient of an acoustic velocity potential, $\mathbf{u}=\nabla \varphi$. The linear homentropic acoustic equations arewhere the adiabatic sound speed of a perfect gas is $c_0^2=\gamma p_0/{\rho }_0$. Absorbing a function of time into $\varphi$, the momentum equation givesEliminating $p^{\prime }$ and ${\rho }^{\prime }$ gives the acoustic wave equation

Let the shell displacement be $a(t)-a_0=\mathrm{Re}(\eta e^{-i\omega t})$ and put $k=\omega /c_0$, $\theta =k{a}_0$. The spherically symmetric solution regular at the centre isThe kinematic boundary condition ${\varphi }_r(a_0,t)=\dot{a}(t)$ givesSince $p^{\prime }=-{\rho }_0{\varphi }_t$, the pressure amplitude on the shell is thereforeThe radial Newton's second law per unit area isso its complex amplitude satisfiesFor a nonzero mode, cancellation of $\eta$ and multiplication by $a_0^2/(m{c}_0^2)$ yield the frequency equation for spherically symmetric vibration of a gas-filled elastic shell:The spectrum is discrete and contains infinitely many positive solutions. Indeed, for sufficiently large $n$, $\theta \cot \theta -1$ runs from $+\infty$ to $-\infty$ on $(n\pi ,(n+1)\pi )$, forcing at least one root there. These high modes are predominantly standing acoustic modes of the gas, while the lowest mode is predominantly the shell's radial breathing mode; the coupling shifts both families.

Use the acoustic velocity potential conventionPutFor a spherically symmetric harmonic solution, writeThe wave equation andgive the radial Helmholtz equation

The rigid inner sphere requiresA form satisfying this condition automatically isIndeed, if the bracket is $h(r)$, then $h(R)=1$ and $h^{\prime }(R)=1/R$, so $(h/r{)}^{\prime }=0$ at $R$.

At the outer sphere,soSincewe obtainThereforeThe assumption $\alpha \le \pi /2$ keeps the displayed denominator positive.

Use the acoustic velocity potential conventionFor a spherically symmetric outgoing wave with time dependence $e^{i\omega t}$ and $k=\omega /c_0$, writeLinearizing the boundary condition at the mean surface $r=a$ givesSincewe findThus the outgoing acoustic field of a pulsating sphere is