= Adiabatic vertical modes of a Gaussian atmosphere
{title2=$\sigma_n^2=\Omega^2[1+\gamma(2n+1)]$}
A <plane-parallel atmosphere> with <isothermal sound speed> $c_s$ in gravity $-\Omega^2z$ has <mass density> $\rho_0e^{-z^2/(2H^2)}$, $H=c_s/\Omega$. Above a rigid base, vertically directed <adiabatic fluid perturbations> satisfy $w''-zw'/H^2+(\sigma^2/\Omega^2-1)w/(\gamma H^2)=0$ 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 $-y''+x^2y/4=(\lambda+1/2)y$. 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.
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