Gravity Waves

When gravity is the only restoring force, Eqn. 41, after taking the dot product with k and with , and some rearrangement becomes,

where k_z is the component of k along z, and which further simplifies to,

where θ_z is the angle between k and z, and ω_BV is the Brunt-Väisälä frequency,

It is clear from Eqn. 45 that gravity waves are dispersive (v_p varies with wavelength), and anisotropic (v_p varies with k). In addition gravity waves can only propagate with angular frequencies less than ω_BV and cannot propagate vertically (θ_z ≠ 0). The group velocity is also perpendicular to the phase velocity such that when waves are upwardly-propagating, energy is downwardly-propagating.