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Irrotational flow permits $\mathbf{u}=\nabla \varphi$, and incompressible flow imposes $\nabla \cdot \mathbf{u}=0$, hence $\Delta \varphi =0$. In the laboratory frame, the no-penetration condition at the translating sphere is ${\partial }_r\varphi (a,\theta )=U\cos \theta$.

For separation of variables, take the angular factor $g(\theta )=\cos \theta$, which satisfies $\frac{1}{\sin \theta }(\sin \theta g^{\prime }{)}^{\prime }=-2g$. The radial equation iswith solutions $r$ and $r^{-2}$. Rest at infinity excludes the $r$ term. The normal boundary condition fixes the remaining coefficient, giving the potential flow around a translating sphereThere is no azimuthal velocity. With fluid density ${\rho }_f$, the kinetic energy isThe angular integral is $2\pi$, and the radial integral is $1/(3a^3)$, so $K_f={\rho }_f\pi a^3{U}^2/3$. Since $M=(4\pi a^3/3){\rho }_s$ and $k={\rho }_f/{\rho }_s$, $K_f=kM{U}^2/4$. Thus the added mass of a sphere is $kM/2$.

During a fall through $h$, gravity does work $Mgh$ and buoyancy does work $-kMgh$. Starting from rest, conservation of energy givesConsequently $U=\sqrt{4(1-k)gh/(2+k)}$. The mass cancels: the reduced speed comes from both buoyancy and the fluid's added mass.

The physical root of the dispersion relation isFor $A\ll 1$,This is a weakly localized, nearly grazing acoustic wave with speed close to the bulk sound speed.

For $A\gg 1$,Now $c\ll c_0$, the decay rate is $\alpha \simeq k$, and the fluid is effectively incompressible. A layer of depth $1/k$ supplies added mass per area of order ${\rho }_0/k$, whose balance with the spring stiffness gives the final scaling.

Unsteady Bernoulli gives the surface pressure from $p+\rho {\partial }_t\varphi +\rho |\nabla \varphi {|}^2/2=C(t)$. The velocity-squared contribution integrates to zero net force, while the unsteady term gives $F=-m_a\dot{V}$ with added mass $m_a=\frac{1}{2}(4\pi \rho a^3/3)=2\pi \rho a^3/3$. Hence $(m+m_a)a=G$, so $m^{\ast }=m+2\pi \rho a^3/3$: accelerating the bubble also accelerates surrounding fluid.