Hermite vertical mode of an isothermal disk (${\xi }_{xx}-x{\xi }_x+n\xi =0$)

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 $n=0,1,\dots$ and the local dispersion relation $({\hat{\omega }}^2-n{\Omega }_K^2)({\hat{\omega }}^2-{\Omega }_K^2)=k^2{c}_s^2{\hat{\omega }}^2$, where $\hat{\omega }=\omega -m{\Omega }_K$. 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.

For the $n=1$, $m=1$ Hermite vertical mode of an isothermal disk, the local dispersion relation is $[(\omega -{\Omega }_K{)}^2-{\Omega }_K^2{]}^2=k^2{c}_s^2(\omega -{\Omega }_K{)}^2$. Its low inertial-frame frequency branches satisfy $\omega \simeq \pm k{c}_s/2$. These waves transport a changing tilt of successive annuli, or a warp, with group velocity $\pm c_s/2$. The vertical velocity is approximately independent of height. A rigid tilt is the zero-wavenumber, zero-pattern-frequency limit; the gas still executes vertical orbital motion at frequency ${\Omega }_K$ in its rotating frame.