= Solution
{c}
Each layer has constant <mass density>, initially negligible <velocity> and hence zero <vorticity>. Its <Euler equations for an inviscid fluid> have a conservative gravitational body force, so <Kelvin's circulation theorem> preserves zero bulk <vorticity>. In a simply connected layer, the <irrotational flow> therefore has $\mathbf v_i=\nabla\phi_i$. The <incompressible flow> condition gives
$$
\boxed{\nabla^2\phi_i=0.}
$$
This applies while the ideal layerwise solution is smooth. It allows an interfacial <vortex sheet>; it does not assert that the whole two-fluid flow has zero <vorticity> or that a smooth interface must exist for arbitrarily long times.
The rigid wall <no-penetration boundary conditions> are
$$
\partial_z\phi_1(r,-h,t)=0,\qquad
\partial_z\phi_2(r,h,t)=0,\qquad
\partial_r\phi_i(a,z,t)=0,
$$
with boundedness and $\partial_r\phi_i=0$ at $r=0$ for regular axisymmetric fields. For a small interface displacement and small slope, linearization of the <kinematic boundary condition> about $z=0$ gives
$$
\boxed{\eta_t=\partial_z\phi_1(r,0,t)=\partial_z\phi_2(r,0,t).}
$$
The small-slope condition is an additional requirement for high-<wavenumber> modes: $|\alpha\eta|\ll1$, not just $|\eta|\ll h,a$.
Choose the unperturbed <hydrostatic pressures> as $\bar p_i(z)=p_*-\rho_i gz$. The linearized <Bernoulli equation> gives the <pressure> perturbation $p_i'=-\rho_i\partial_t\phi_i$, after absorbing spatially constant gauges into $\phi_i$. Since <surface tension> and viscous <normal stress> vanish, <pressure> continuity on $z=\eta$ gives
$$
-\rho_1\partial_t\phi_1-\rho_1g\eta
=-\rho_2\partial_t\phi_2-\rho_2g\eta,
$$
or
$$
\boxed{\rho_1\partial_t\phi_1-\rho_2\partial_t\phi_2
=(\rho_2-\rho_1)g\eta\quad\text{at }z=0.}
$$
The initial <velocity> is zero, so $\eta_t(r,0)=0$. Separate <conservation of mass> in each closed layer also requires $\int_0^a\eta(r,t)r\,dr=0$ relative to the initially horizontal interface.
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