Adiabatic vertical modes of a Gaussian atmosphere (${\sigma }_n^2={\Omega }^2[1+\gamma (2n+1)]$)

A plane-parallel atmosphere with isothermal sound speed $c_s$ in gravity $-{\Omega }^{2}z$ has mass density ${\rho }_0{e}^{-z^2/(2H^2)}$, $H=c_s/\Omega$. Above a rigid base, vertically directed adiabatic fluid perturbations satisfy $w^{\prime \prime }-z{w}^{\prime }/H^2+({\sigma }^2/{\Omega }^2-1)w/(\gamma H^2)=0$ and $w(0)=0$. Set $x=z/H$ and $\lambda =({\sigma }^2/{\Omega }^2-1)/\gamma$. The substitution $y=e^{-x^2/4}w$ converts finite Gaussian-weighted kinetic energy into square integrability for $-y^{\prime \prime }+x^{2}y/4=(\lambda +1/2)y$. Odd extension across the base and the oscillator lowering operator force $\lambda$ to be an odd positive integer. Thus $w_n$ is proportional to the Probabilists' Hermite polynomial ${\mathrm{He}}_{2n+1}(z/H)$. These are vertical acoustic waves; a general stably stratified atmosphere also admits internal gravity waves with horizontal structure.