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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.

A deforming free swimmer must satisfy total linear and angular momentum balances. The unknown lateral translation and yaw are its recoil correction. If the reference translation is the true centre of mass, the prescribed deformation must first have its mass-weighted mean removed. Otherwise the reference translation differs from actual centre of mass displacement. The body mass and its hydrodynamic added mass contribute with the same sign to its effective inertial coefficient.

For a nonseparating small body in a smooth nearly inviscid eddy of speed $U$ and length $\ell$, the fluctuating resultant is an added mass or pressure-gradient force $F\sim {\rho }_0{a}^3{U}^2/\ell$. A uniform pressure integrates to zero over a closed surface. With time $\ell /U$, its acoustic dipole power divided by the eddy's acoustic quadrupole power is $(a/\ell {)}^6/M^2$, where $M=U/c_0$ is the Mach number. Equal powers therefore give the displayed threshold. An area-scaled separated drag force with the same eddy time instead gives $a/\ell \sim \sqrt{M}$; a body-scale time gives a different criterion. The threshold depends on the specified loading and time models.