Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 40B (Waves)

Write the density and pressure as $\rho ={\rho }_0+{\rho }^{\prime }$ and $p=p_0+p^{\prime }$, and let $\mathbf{u}$ be the small fluid velocity. The linear homentropic acoustic equations arewhere the adiabatic sound speed isIf the flow is irrotational, write $\mathbf{u}=\nabla \varphi$. The momentum equation integrates spatially toa purely time-dependent integration term having been absorbed into $\varphi$. Substitution into continuity gives the wave equationso the wave speed is $c_0$.

To obtain energy conservation, dot the momentum equation with $\mathbf{u}$ and use the product rule:The continuity and pressure-density relations giveTherefore the acoustic energy conservation law iswith acoustic energy density and intensity

Let ${\varphi }_{+}$ and $p_{+}^{\prime }$ denote the fields in $y>0$, and ${\varphi }_{-}$ and $p_{-}^{\prime }$ those in $y<0$. The linearized kinematic boundary condition equates the normal fluid velocity on both faces to the membrane velocity:The linear acoustic pressure relation isPressure below pushes upward and pressure above pushes downward. The transverse force balance for a membrane with mass per area $m$ and tension $T$ is consequentlyor, in terms of the velocity potentials,

Takeand define the positive normal decay rateThe decaying solutions of the acoustic wave equation that satisfy the kinematic conditions areTheir pressures at the membrane areSubstitution into the dynamic boundary condition givesand hence the acoustic wave on a tensioned massive membrane dispersion relation

Writing the phase speed as $c=\omega /k$, the positive added-inertia term givesPhysically, each evanescent acoustic surface wave accelerates a layer of fluid on both sides of the membrane, so the inertia per unit area exceeds $m$. Moreover,and the added mass of an evanescent fluid layer $2{\rho }_0/q$ diverges as $k\to 0$. In that limit $c\ll c_0$, so $q\sim k$ and

The normal acoustic energy flux isIn either half-space, the pressure fields above are proportional to $\cos (kx-\omega t)$, whereas the normal velocities are proportional to $\sin (kx-\omega t)$. They are therefore in temporal quadrature. Averaging over one period givesThus the field is evanescent and stores reactive acoustic energy flux near the membrane without transporting net acoustic energy away from it in the normal direction; the wave's mean energy transport is along the membrane.