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Apply the Unsteady Bernoulli equation at $r=a(t)$ and $r=b(t)$ and subtract, thereby eliminating its time-dependent constant. For $A=a\dot{a}$ and $\varphi =A\log r+C(t)$,where each partial derivative is taken at fixed $r$. Since $u_r=A/r$ and $p(b,t)=p_{\infty }$,Equivalently, $\dot{A}={\dot{a}}^2+a\ddot{a}$. This is the pressure in radially symmetric annular potential flow.

For irrotational flow, $\mathbf{u}=\nabla \varphi$. Incompressibility gives the Laplace equationThe rigid side walls impose no penetration:and decay in the infinitely deep fluid requires $\nabla \varphi \to 0$ as $z\to -\infty$.

Linearizing the kinematic condition from part (a) about $z=0$ givesThe Unsteady Bernoulli equation, evaluated at the surface and linearized about hydrostatic equilibrium, gives the dynamic conditionThese are the linearized free-surface boundary conditions. Initial values for $\eta$ and $\varphi$ specify the transient; below we also record the initially quiescent solution.

Write the velocity potentials in the upper, middle and lower domains as ${\varphi }_{+}$, ${\varphi }_0$ and ${\varphi }_{-}$. Irrotational flow and incompressible flow imply the Laplace equationThe far-field boundary conditions are

Every shear layer is a material interface. The exact kinematic boundary condition at $y=h+\eta$ isand at $y=-h-\eta$ it isHere each equation is evaluated on the indicated moving interface.

Finally, pressure continuity is the dynamic boundary condition for an inviscid interface. The Unsteady Bernoulli equation givesafter absorbing spatially constant functions of time into the potentials.

The inner base flow is at rest, so its linearized Unsteady Bernoulli equation gives $p_1^{\prime }=-\sigma {\varphi }_1$. The outer base potential is ${\Gamma }_0\theta$, hence its cross term with the perturbation givesThe base outer pressure satisfies $d{p}_{20}/dr={\Gamma }_0^2/r^3$, whereas the inner base pressure is constant. Expanding pressure continuity on the displaced sheet therefore givesAfter multiplication by $-1$ this isAn additive function of time in Bernoulli appears as the stated constant; it vanishes for every nonaxisymmetric Fourier mode.

If the pressure at $r=b(t)$ is $p_{\infty }$, the Unsteady Bernoulli equation gives