Linear homentropic acoustic equations

For perturbations $({\rho }^{\prime },p^{\prime },\mathbf{u})$ about a uniform fluid of density ${\rho }_0$, the linearized continuity equation, momentum equation, and homentropic pressure perturbation relation areFor irrotational flow $\mathbf{u}=\nabla \varphi$, they imply $p^{\prime }=-{\rho }_0{\varphi }_t$ and ${\varphi }_{tt}=c_0^2{\nabla }^2\varphi$.

Eliminating the velocity and density perturbations from the linear homentropic acoustic equations gives

For harmonic convention $e^{i\omega t-ikx}$, choose $\mathrm{Re}\gamma >0$ by continuation from $\mathrm{Im}\omega <0$ and impose vertical factors $e^{-\gamma y}$ above the interface and $e^{\gamma y}$ below it. At positive real frequency, $\gamma =i\sqrt{k_0^2-k^2}$ for propagating Fourier components and is positive real for evanescent components. This analytic prescription fixes the radiation branch and how the Fourier contour passes poles.