= Solution
For an axisymmetric separated amplitude $\widehat\phi(r,z)=R(r)Z(z)$, <Laplace equation in cylindrical coordinates> becomes
$$
\frac1R\frac1r(rR')'
=-\frac{Z''}{Z}
=-k^2.
$$
The radial equation is
$$
R''+\frac1rR'+k^2R=0.
$$
Its solution regular at the axis is the <Bessel function> $J_0(kr)$. The vertical equation is $Z''-k^2Z=0$, and the bottom condition $Z'(-h)=0$ selects
$$
\boxed{Z(z)=\cosh(k(z+h))}
$$
up to an irrelevant constant multiplier. Finally, the sidewall condition gives
$$
0=R'(R)=kJ_0'(kR).
$$
Therefore the allowed positive wavenumbers are
$$
\boxed{k_n=\frac{x_n}{R},
\qquad J_0'(x_n)=0},
$$
and
$$
\boxed{\widehat\phi(r,z)
=A J_0(k_nr)\cosh(k_n(z+h))}.
$$
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