Atmospheric acoustic cutoff transformation ($\delta \rho =e^{-z/(2H)}\Psi ,{\omega }_c=c/(2H)$)

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.