= Poincaré equation for inertial waves
{c}
{title2=$\partial_t^2\nabla^2w+4\Omega^2\partial_z^2w=0$}
Linear incompressible motion about rest in a frame rotating with $\boldsymbol\Omega=\Omega\mathbf k$ obeys $\mathbf u_t+2\boldsymbol\Omega\times\mathbf u=-\nabla P/\rho$. If $\zeta=v_x-u_y$ is its vertical <vorticity>, taking <curl> gives $\zeta_t=2\Omega w_z$, while eliminating <pressure> using continuity gives $\partial_t\nabla^2w+2\Omega\zeta_z=0$. Differentiating in time yields the displayed scalar equation. All three <velocity> components must be retained, even when the fields depend on only two spatial coordinates. The <Coriolis acceleration> generally generates a transverse <velocity> that is absent in a strictly two-component description.
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