= Plane-wave forcing of a pinned elastic half-sheet
{title2=$K=\gamma/2+\omega^2/(m\omega^2-Tk^2)$}
For identical unit-density, unit-sound-speed fluids above and below a sheet on $x<0$, use time $e^{i\omega t}$ and transform $\int e^{ikx}\phi\,dx$. The <outgoing acoustic square-root branch> obeys $\gamma^2=k^2-\omega^2$. If $J$ is the pressure jump upper minus lower, $s=\eta'(0)$ and $d^+$ the right <Half-range Fourier transform> of the scattered normal derivative, the <Wiener-Hopf equation> is
$$
KJ+d^+=\frac{\omega\sin\theta_0}{k+\omega\cos\theta_0}+\frac{T\omega^2s}{m\omega^2-Tk^2}.
$$
The total displacement is pinned, so its transformed second derivative is $s-k^2\eta^-$. The bare-sheet pole $b=\omega\sqrt{m/T}$ in reconstructed displacement must cancel: $J(b)=Ts$. This is an <incoming-pole cancellation at a pinned membrane edge> condition, distinct from the outgoing coupled-mode pole at the zero of $K$.
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