I want to find an analytic solution to the 2d wave equation with periodic boundary conditions $$ u_{tt} = c^2(u_{xx} + u_{yy}) $$ $$ u(-L_{1},y,t) = -u(L_{1},y,t) $$ $$ u(x,-L_{2},t) = -u(x,L_{2},t) $$ $$ u(x,y,0) = \psi (x,y) $$ $$ u_{t}(x,y,0) = \phi (x,y) $$
I've gotten the solution so far to be: $$ u_{m,n}(x,y,t) = \left(Acos\left(\frac {n \pi x} {2L_{1}} \right)+Bsin\left(\frac{n \pi x}{2L_{1}}\right)\right ) \left(Ccos\left(\frac{m \pi y}{2L_{2}}\right)+Dsin\left(\frac{m \pi y}{2L_{2}}\right)\right)(\alpha_{m,n}cos(\lambda_{m,n}t)+\beta_{m,n}sin(\lambda_{m,n}t))$$
where $\lambda_{m,n} = \frac{c\pi}{2} \sqrt{\frac{n}{L_{1}^{2}}+\frac{m}{L_{2}^{2}}}$
Now I know usually you use the initial conditions regarding time to find all your coefficients in the resulting fourier series, but I don't see how that's possible since we have basically the multiplication of 3 separate fourier series and as far as I know there is no way to simplify the expression for $u_{m,n}$. Is this even solveable? If so how do I go about doing it?