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An elastic membrane is a thin deformable sheet whose transverse restoring force is supplied by in-plane tension. For uniform tension $T$ and mass per area $m$, a two-dimensional small-displacement model has $m{\eta }_{tt}=T{\eta }_{xx}-[p]$, where the pressure jump is defined from the lower fluid to the upper fluid. Fluid loading modifies its traveling modes through the acoustic wave on a tensioned massive membrane.

Use the Fourier transform convention $\hat{p}(k,y)={\int }_{\mathbb{R}}{e}^{ikx}p(x,y)dx$. Let $\gamma (k{)}^2=k^2-k_0^2$, choosing $\mathrm{Re}\gamma >0$ on the real contour with $\mathrm{Im}\omega <0$. For time dependence $e^{i\omega t}$, this is the decaying continuation of the outgoing Sommerfeld radiation condition.

The transformed Helmholtz equation has upper and lower solutions $P(k)e^{-\gamma y}$ and $Q(k)e^{\gamma y}$. The two kinematic traces give $-\gamma P={\rho }_0{\omega }^2\hat{\eta }=\gamma Q$, henceThe elastic membrane equation gives $[\hat{p}]=(m{\omega }^2-T{k}^2)\hat{\eta }$. Therefore the acoustic wave on a tensioned massive membrane has dispersion relationEquivalently, $T{k}^2={\omega }^2(m+2{\rho }_0/\gamma )$. The fluid on both sides supplies a positive added mass of an evanescent fluid layer, which is a useful independent check on the sign. In particular, the incident pressure isThe symbol $m$ here denotes elastic membrane mass per area, rather than the fluctuating Mach number used in Question 1.

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