Acoustic-gravity propagation relation ($k_r^2\simeq ({\omega }^2-S_{\ell }^2)({\omega }^2-N^2)/(c_s^2{\omega }^2)$)

In the Cowling approximation and local short-wavelength limit, the radial wave number satisfies $k_r^2\simeq ({\omega }^2-S_{\ell }^2)({\omega }^2-N^2)/(c_s^2{\omega }^2)$. The Lamb frequency $S_{\ell }$ and stellar buoyancy frequency $N$ delimit acoustic and gravity-wave cavities. This leading interior relation does not replace the near-surface acoustic cutoff or the full gravitational boundary problem.