Pressure projection in a displaced stratified blob (${\omega }^2=N^2{k}_h^2/(k_h^2+m^2)$)

Incompressibility makes pressure adjust throughout the surrounding fluid. Projecting the linear Boussinesq equations perpendicular to a Fourier wavevector gives $w_t=[k_h^2/(k_h^2+m^2)]b$ and $b_t=-N^{2}w$. A localized spherical disturbance contains many wavevector directions and therefore many internal gravity wave frequencies, rather than one oscillator at $N$. Its shape evolves and energy radiates into the surroundings; a rigid-sphere added mass analogy illustrates extra inertia but does not give an exact coherent frequency for a free material blob.