Fluid-loaded membrane edge scattering

A pinned semi-infinite membrane scatters an incoming guided mode into reflected guided modes and outgoing sound. For unit incident displacement the scattered endpoint value is $-1$. Its Half-range Fourier transform therefore has nonzero endpoint terms even though the total displacement vanishes. The Wiener-Hopf equation kernel is $2/(m{\omega }^2-T{k}^2)+\gamma /({\rho }_0{\omega }^2)$ with the outgoing branch ${\gamma }^2=k^2-{\omega }^2/c_0^2$.

Reconstructing the left scattered displacement in fluid-loaded membrane edge scattering divides by $m{\omega }^2-T{k}^2$. The pole at the lower bare-membrane wavenumber $b=\omega \sqrt{m/T}$ must cancel to preserve outgoing half-plane analyticity. This gives a linear condition determining the otherwise unknown endpoint slope $A$. The prescribed incident pole in the total field is a different singularity.