= Solution
Use complex amplitudes and take their <real parts> at the end. Write the radial surface displacement as $\eta=\epsilon e^{i(kz-\omega t)}$ and the combined potential as
$$
W=\epsilon\Pi_0I_0(kr)e^{i(kz-\omega t)}.
$$
The linear <kinematic boundary condition> is $u_r(R)=\partial_t\eta$. Differentiate it once more in time and use the radial momentum equation, $\partial_tu_r=-\partial_rW$, evaluated at the unperturbed radius $R$:
$$
\partial_t^2\eta=-\left.\partial_rW\right|_R.
$$
With $\kappa=kR$, the mode amplitudes therefore satisfy
$$
-\omega^2\epsilon=-\epsilon\Pi_0kI'_0(\kappa),
\qquad
\boxed{\Pi_0=\frac{\omega^2}{kI'_0(\kappa)}.}
$$
The 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 $\mathrm{m\,s^{-2}}$, as required because $\epsilon\Pi_0$ is a potential. It applies to $k>0$; the uniform $k=0$ limit requires separate treatment.
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