= Hermite vertical mode of an isothermal disk
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{title2=$\xi_{xx}-x\xi_x+n\xi=0$}
= Hermite vertical modes of an isothermal disk
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In an inviscid locally isothermal <Keplerian disk>, a radial <normal mode> of <wavenumber> $k$ has pressure variable $\xi$ proportional to a <Probabilists' Hermite polynomial> in $x=z/H$. Gaussian-weighted finite energy excludes the exponentially growing vertical branch. Polynomial termination gives $n=0,1,\ldots$ and the local dispersion relation $(\widehat\omega^2-n\Omega_K^2)(\widehat\omega^2-\Omega_K^2)=k^2c_s^2\widehat\omega^2$, where $\widehat\omega=\omega-m\Omega_K$. At singular values introduced by eliminating velocity components, the original equations must also be checked. The $n=0$ propagating branch is an <inertial-acoustic wave>; $n=1$ has vertically uniform vertical velocity and odd pressure perturbation, permitting bending disturbances.
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