Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 38A / Solution

Linearized continuity and Euler equations about a uniform resting inviscid fluid are ${\rho }_t^{\prime }+{\rho }_0\nabla \cdot \mathbf{u}=0$ and ${\rho }_0{\mathbf{u}}_t=-\nabla p^{\prime }$. For a velocity potential $\mathbf{u}=\nabla \varphi$ and sound-speed relation $p^{\prime }=c^2{\rho }^{\prime }$, integration of Euler's equation fixes the temporal gauge so $p^{\prime }=-{\rho }_0{\varphi }_t$. Substitution into continuity givesUse complex amplitudes with time factor $e^{i\omega t}$. The outgoing wave in fluid two is ${\Phi }_2(x)=A{e}^{-i{k}_{2}x}$, with $k_i=\omega /c_i$. Write ${\Phi }_1=C\cos (k_{1}x)+D\sin (k_{1}x)$. Continuity of pressure and normal velocity at zero givesThe piston velocity is $i\omega ϵ$, so ${\Phi }_1^{\prime }(-L)=i\omega ϵ$. With $\theta =\omega L/c_1$, this givesand hence the requested amplitude isFor complex pressure and velocity amplitudes, the mean acoustic energy flux is $\frac{1}{2}\mathrm{Re}(P{U}^{\ast })$. Here $P=-i\omega {\rho }_2{\Phi }_2$ and $U=-i{k}_2{\Phi }_2$, soFor ${\rho }_2\ll {\rho }_1$ and comparable sound speeds, weak transmission loads the finite segment. There are narrow large-flux resonances near $\theta =(n+1/2)\pi$, separated by $\Delta L=\pi c_1/\omega$. The off-resonance scale is proportional to ${\rho }_2$, while peak flux is proportional to ${\rho }_1^2/{\rho }_2$; the angular width is of order $({\rho }_2/{\rho }_1)(c_2/c_1)$. These amplified solutions remain within linear acoustics only when the resulting wave amplitudes are small.