This is from the lecture notes of MIT.
I am confused about the sentence "where the $a_{nm}$ and $b_{nm}$ are found from the ICs". It is from problem 3 in this pdf.
I tried to use orthogonality, but I don't know how to deal with the $J$ and the double summation.
Could anyone kindly help how to do that? Thanks so much!
Hint
Use the following orthogonality
$$\eqalign{ & {\varphi _{mn}}(r,\theta ) = {J_{3n}}\left( {\sqrt {{\lambda _{nm}}} r} \right)\cos \left( {3n\theta } \right) \cr & \int_{r = 0}^R {\int_{\theta = 0}^{2\pi } {{\varphi _{mn}}{\varphi _{pq}}rdrd\theta } } = {\delta _{mp}}{\delta _{nq}}\int_{r = 0}^R {\int_{\theta = 0}^{2\pi } {\varphi _{mn}^2rdrd\theta } } \cr} $$
where $\delta_{ij}$ is the Kronecker's delta.