= Wiener-Hopf solution for an open parallel-plate waveguide
{c}
{title2=$L(k)=\gamma\sinh(\gamma b)e^{-\gamma b}$}
For rigid plates $x>0$, $y=\pm b$, an even incident plane wave travelling toward the opening can be scattered using <Wiener-Hopf factorization>. With time dependence $e^{i\omega t}$ and transform $\Phi(k)=\int\phi(x)e^{ikx}dx$, take $\gamma=\sqrt{k^2-k_0^2}$ on the outgoing sheet. The common normal derivative on $y=b$ is supported on $x<0$; its transform is $D^-=iL^+(-k_0)L^-(k)/(k+k_0)$. The interior transform is $D^-\cosh(\gamma y)/(\gamma\sinh(\gamma b))$, while the exterior transform is $-D^-e^{-\gamma(|y|-b)}/\gamma$. These follow from continuity of normal velocity and the half-line pressure jump, followed by <pole subtraction in a Wiener-Hopf equation> and a finite-energy edge condition.
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