= Inertial-wave dispersion relation
{title2=$\omega^2=4(\boldsymbol\Omega\cdot\mathbf k)^2/|\mathbf k|^2$}
Insert a transverse <plane wave> $\mathbf u=\mathbf u_0e^{i(\mathbf k\cdot\mathbf r-\omega t)}$ into the <inertial-wave vorticity equation>, with $\mathbf k\cdot\mathbf u_0=0$. It gives $\omega\mathbf k\times\mathbf u_0=2i(\boldsymbol\Omega\cdot\mathbf k)\mathbf u_0$. Applying $\mathbf k\times$ again and using transversality gives the title <dispersion relation>. If $\theta$ is the angle to the rotation axis, $|\omega|=2|\Omega\cos\theta|$. At fixed direction the <angular frequency> does not depend on <wavelength>. Parallel wavevectors give the maximum $2|\Omega|$; perpendicular ones have zero frequency and belong to the nonoscillating balanced limit.
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