= Matched chromospheric interfacial mode
{title2=$\varepsilon^{-1}\simeq2k^2I_aI_b$}
For a high-sound-speed, low-density atmosphere over a denser stellar envelope, set $\omega^2=gk(1-\varepsilon)$ in the <plane-parallel Cowling displacement-pressure equations>. Define $I_a=\int_{-\infty}^{z_0}e^{2k(z-z_0)}\rho^{-1}dz$ and $I_b=\int_{z_0}^{\infty}e^{-2k(z-z_0)}\rho\,dz$. The lower leading solution has $\Pi/\xi=-2\varepsilon gk I_b$ at the match. Above, write $\rho=\varepsilon\widehat\rho$, $\Pi=\varepsilon\psi$, $\xi=\mu\psi$. The upward-decaying solution satisfies $\mu'+2k\mu=-k/(g\widehat\rho)$, so $\mu(z_0)=-(k/g)\varepsilon I_a$. Writing $f=\Pi/(\varepsilon\xi)$ below, continuity of displacement and Lagrangian pressure requires $\mu(z_0)f(z_0)=1$, giving the displayed condition. The integrals must converge and the upper displacement must actually decay. For two constant-density incompressible layers it gives $\varepsilon\simeq2\rho_a/\rho_b$, the expansion of the exact <interfacial gravity-wave dispersion relation>.
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