Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 40D / b

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 $\omega$.