Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 40D (Waves)

Let $\tilde{\rho }$, $\tilde{p}$, and $u$ be first-order perturbations. Linearized mass and momentum conservation and the homentropic relation areDifferentiate the first equation in time and use the divergence of the second to obtainThe perturbation is irrotational, so write $u=\nabla \varphi$. Integrating the momentum equation spatially, with a purely time-dependent constant absorbed into $\varphi$, gives

Use complex fields with time factor $e^{i\omega t}$ and put $k_{\pm }=\omega /c_{\pm }$. The outgoing potential in $x>0$ isContinuity of pressure and velocity at $x=0$ givesHenceThe piston condition ${\Phi }_{-}^{\prime }(-L)=i\omega ϵ$, with $\lambda =k_{-}L$, givesThis is the transmitted velocity potential from a piston through an acoustic interface.

For the outgoing wave, the pressure and velocity amplitudes are $-i\omega {\rho }_{+}T$ and $-i{k}_{+}T$. Its time-averaged acoustic energy flux is thereforeIf $\lambda =n\pi$, then $|T|=ϵc_{+}$ andIf $\lambda =(n+\frac{1}{2})\pi$, thenandWhen ${\rho }_{+}\ll {\rho }_{-}$ and $c_{+}\simeq c_{-}$, the half-integer family is strongly resonantly enhanced relative to the integer family by approximately $({\rho }_{-}/{\rho }_{+}{)}^2$ at equal $\omega$.