= Bending wave of a Keplerian disk
{title2=$\omega\simeq\pm kc_s/2\quad(m=1,\ |k|H\ll1)$}
= Bending waves of a Keplerian disk
{synonym}
For the $n=1$, $m=1$ <Hermite vertical mode of an isothermal disk>, the local dispersion relation is $[(\omega-\Omega_K)^2-\Omega_K^2]^2=k^2c_s^2(\omega-\Omega_K)^2$. Its low inertial-frame frequency branches satisfy $\omega\simeq\pm kc_s/2$. 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.
Back to article page
