I am struggling conceptually with what this question is asking me to do. The question is:
In polar coordinates the Helmholtz equation has solutions of the form:
$$u_m=(AJ_m(\sqrt{\lambda}+(BY_m(\sqrt{\lambda}r)(C\cos{m}\phi+D\sin{m}\phi)$$
where J and Y are bessel functions of the first and second kind. Find solutions to the 2D wave equation that are regular at the origin.
Now, I am happy with separation of variables of the wave equation, into R(r), $\Phi(\phi)$, and T(t). I think I am correct in thinking that the helmholtz solutions are for the time independent part of the wave equation, i.e. $u_m=R(r)\times\Phi(\phi)$ ?
Then my question is, can I just multiply the time dependence of the separated wave equation onto the Helmholtz solution, to get a general solution to the 2D wave equation? i.e.:
$$u(r,\phi,t)=T_0e^{\pm i\lambda t}(AJ_m(\sqrt{\lambda}+(BY_m(\sqrt{\lambda}r)(C\cos{m}\phi+D\sin{m}\phi)$$
Feels like I am missing the point. Thanks,