Acoustic cutoff frequency (${\omega }_{\mathrm{ac}}\simeq c_s/(2H_{\rho })$)

A stratified atmosphere can reflect low-frequency acoustic waves while permitting sufficiently high-frequency waves to escape. In a plane-parallel isothermal atmosphere, the angular acoustic cutoff is approximately $c_s/(2H_{\rho })$, where $H_{\rho }$ is the density scale height. Its detailed form depends on the atmospheric profile and on the wave-variable approximation.

In an isothermal atmosphere with constant density scale height $H$, the equation $\delta {\rho }_{tt}-c^2\delta {\rho }_{zz}-(c^2/H)\delta {\rho }_z=0$ loses its first spatial derivative under $\delta \rho =e^{-z/(2H)}\Psi$. Direct differentiation gives ${\Psi }_{tt}-c^2{\Psi }_{zz}+c^2\Psi /(4H^2)=0$. Thus a plane wave has dispersion relation ${\omega }^2=c^2{k}^2+{\omega }_c^2$, with real acoustic cutoff frequency ${\omega }_c=c/(2H)$. The cutoff term is positive; a minus sign would instead predict growing sufficiently long spatial modes. Vertical propagation requires $|\omega |\ge {\omega }_c$ within this ideal constant-coefficient model.