Matched chromospheric interfacial mode (${\epsilon }^{-1}\simeq 2k^2{I}_a{I}_b$)

For a high-sound-speed, low-density atmosphere over a denser stellar envelope, set ${\omega }^2=gk(1-\epsilon )$ 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 $\Pi /\xi =-2\epsilon gk{I}_b$ at the match. Above, write $\rho =\epsilon \hat{\rho }$, $\Pi =\epsilon \psi$, $\xi =\mu \psi$. The upward-decaying solution satisfies ${\mu }^{\prime }+2k\mu =-k/(g\hat{\rho })$, so $\mu (z_0)=-(k/g)\epsilon I_a$. Writing $f=\Pi /(\epsilon \xi )$ 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 $\epsilon \simeq 2{\rho }_a/{\rho }_b$, the expansion of the exact interfacial gravity-wave dispersion relation.