Internal-wave polarization

For a propagating, non-cutoff two-dimensional internal gravity wave proportional to $e^{i(kx+mz-\omega t)}$, incompressibility gives $U=-mW/k$. Thus the oscillatory velocity field is perpendicular to the wavevector and parallel to the group velocity, whereas the phase velocity is parallel to the wavevector.

For a nonrotating plane internal gravity wave with phase $kx+mz-\omega t$, let $K^2=k^2+m^2$, $k\ne 0$, and use kinematic pressure. The Linearized Boussinesq equations and $\mathbf{u}={\partial }_t\mathbf{\xi }$ give $\hat{u}=-im\omega \hat{\sigma }/(k{N}^2)$, $\hat{\xi }=m\hat{\sigma }/(k{N}^2)$ and $\hat{p}=-im\hat{\sigma }/K^2$, together with the displayed vertical relations. Buoyancy and displacement field are in phase or opposite phase; velocity and pressure are in quadrature with them. Incompressibility makes both the oscillating velocity and displacement field perpendicular to the wavevector.