= Inertial modes in a rigid rotating sphere
{title2=$\nabla_h^2W+(1-4\Omega^2/s^2)W_{zz}=0$}
In a uniform-density <incompressible flow> inside a smooth rigid sphere rotating at angular speed $\Omega$, a <normal mode> with phase $e^{i\sigma t+im\phi}$ has intrinsic frequency $s=\sigma+m\Omega$. For $s\ne0,\pm2\Omega$, eliminating <velocity> from the <linearized Euler equations> gives the displayed pressure equation, with $W=p'/\rho$. Impermeability gives $R W_R+(2m\Omega/s)W+(1-4\Omega^2/s^2)zW_z=0$ on the sphere. There is no condition $W=0$: the rigid wall supplies variable normal <pressure>. These <inertial waves> are restored by <Coriolis acceleration>, and their pressure equation is hyperbolic for $|s|<2|\Omega|$.
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