I am trying to find the solution to the Problem of a vibrating, damped circular Membrane with periodic, uniform forcing and vanishing deflection at a radius $r = a$. The damping term is proportional to velocity. $$ \partial_{tt} u(r, \theta, t) - c^2 \nabla^2 u + d \partial_t u = -c^2 f(t)$$ Writing the time dependence as $u, f \sim e^{-i \omega t}$, one arrives at $k^2 U(r, \theta) + \nabla^2 U = F$ with $k^2 = \frac{\omega^2}{c^2} + i \omega \frac{d}{c^2}$. Using separation of variables $U = R(r) \Theta(\theta)$, one can transform the equation to $$k^2 \Theta R + \frac{1}{r^2} R \Theta^{\prime\prime} + \Theta R^{\prime\prime} + \frac{1}{r} \Theta R^{\prime} = F .$$ The solution is a combination of a homogoeneous solution where the right-hand side is Zero and a particular solution. The particular solution can be $U = - \frac{F}{k^2}$ and the homogeneous solution is subject to the differential equations $\frac{\Theta^{\prime\prime}}{\Theta} = -m^2$ and $R^{\prime\prime} {r^*}^2 + {r^*} R^{\prime} + ({r^*}^2 - m^2) R = 0$ with ${r^*} = k r$ and $m \in N_0$. The boundary condition is $U(r=a) = 0$. In a typical analysis, the Bessel function $J_m(r^*)$ would be used for the radial function.
Question: How do I adjust the homogeneous solution to the boundary condition considering that the solution of the Bessel equation would have to have a complex value at the perimeter?
Update: The solution is $$U = \frac{F}{k^2} \left( 1 - \frac{J_0(k r)}{J_0(k a)} \right) .$$