= One-sided fluid-loaded membrane radiation
{title2=$D=(Tk^2-m\omega^2)\gamma-\rho_0\omega^2$}
For one acoustic half-space and vacuum on the other side, a harmonic point <force> gives displacement amplitude $F\gamma/D$ and fluid <acoustic velocity potential> amplitude $i\omega F e^{-\gamma y}/D$. For time dependence $e^{-i\omega t}$, the <outgoing acoustic square-root branch> has $\gamma>0$ for $|k|>k_0$ and $\gamma=-i\sqrt{k_0^2-k^2}$ for $|k|<k_0$. The causal limit $\omega\mapsto\omega+i0$ places the positive guided-mode pole above the inversion contour and the negative pole below. The <saddle point> generates cylindrical radiation, whereas guided residues decay normally to the membrane.
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