 Solution

Use complex amplitudes and take their real parts at the end. Write the radial surface displacement as $\eta =ϵe^{i(kz-\omega t)}$ and the combined potential asThe linear kinematic boundary condition is $u_r(R)={\partial }_t\eta$. Differentiate it once more in time and use the radial momentum equation, ${\partial }_t{u}_r=-{\partial }_{r}W$, evaluated at the unperturbed radius $R$:With $\kappa =kR$, the mode amplitudes therefore satisfyThe prime differentiates the Modified Bessel function of the first kind with respect to its dimensionless argument, rather than with respect to $r$. The result has units $\mathrm{m}{\mathrm{s}}^{-2}$, as required because $ϵ{\Pi }_0$ is a potential. It applies to $k>0$; the uniform $k=0$ limit requires separate treatment.