= Atmospheric acoustic cutoff transformation
{title2=$\delta\rho=e^{-z/(2H)}\Psi,\quad\omega_c=c/(2H)$}
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> $\omega^2=c^2k^2+\omega_c^2$, 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.
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