= Buoyancy polarization of a plane internal gravity wave
{title2=$\widehat\zeta=-\widehat\sigma/N^2,\quad\widehat w=i\omega\widehat\sigma/N^2$}
For a nonrotating <plane internal gravity wave> with phase $kx+mz-\omega t$, let $K^2=k^2+m^2$, $k\ne0$, and use <kinematic pressure>. The <Linearized Boussinesq equations> and $\mathbf u=\partial_t\boldsymbol\xi$ give $\widehat u=-im\omega\widehat\sigma/(kN^2)$, $\widehat\xi=m\widehat\sigma/(kN^2)$ and $\widehat p=-im\widehat\sigma/K^2$, 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>.
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