Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 40A (Waves)

Let ${\rho }^{\prime }$ and $p^{\prime }$ be the density and pressure perturbations. Homentropy gives $p^{\prime }=c_0^2{\rho }^{\prime }$. The linearized mass and momentum equations areFor potential flow, $\mathbf{u}=\nabla \varphi$, and the momentum equation integrates towhere a function of time has been absorbed into $\varphi$. Substitution into mass conservation yieldsso

Multiply the wave equation by ${\rho }_0{\varphi }_t/c_0^2$ and use product rules. This givesThereforeThese are respectively the kinetic-plus-compressional acoustic energy density and its flux.

Choose membrane displacement $\eta$ positive into the fluid. The linear boundary conditions at $z=0$ areA wave localized near the membrane has complex formThe wave equation requiresso localization requires $0<c<c_0$. If $\eta =\mathrm{Re}\{H{e}^{ik(x-ct)}\}$, the kinematic condition and $p^{\prime }=-{\rho }_0{\varphi }_t$ giveEliminating $H$ yieldsOn squaring and writing $s=c/c_0$, we obtain

For real physical fields of period $T=2\pi /(kc)$, defineEquivalently, for complex amplitudes,HereTheir product is purely imaginary, soPressure and normal velocity are in temporal quadrature: energy is stored and returned by the evanescent field, but none is transported away from the membrane on average.

The physical root of the dispersion relation isFor $A\ll 1$,This is a weakly localized, nearly grazing acoustic wave with speed close to the bulk sound speed.

For $A\gg 1$,Now $c\ll c_0$, the decay rate is $\alpha \simeq k$, and the fluid is effectively incompressible. A layer of depth $1/k$ supplies added mass per area of order ${\rho }_0/k$, whose balance with the spring stiffness gives the final scaling.