= Inviscid warp equations of a nearly Keplerian disk
{title2=$\Sigma R^2\Omega(W_t-i\omega_1W)=R^{-1}G_R,\quad G_t-i\omega_2G=\tfrac14 PR^3\Omega W_R$}
Here $G$ is the complex internal horizontal torque, $P$ the vertically integrated pressure, and $\omega_1,\omega_2$ the approximate <nodal precession> and <apsidal precession> rates. In the exactly Keplerian case, local plane waves obey $\omega^2=k^2P/(4\Sigma)=k^2c_s^2/4$, so tilt disturbances propagate at $c_s/2$. The secular thin-disk approximation requires wavelengths large relative to the height while the local approximation requires them small relative to the orbital radius.
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