Do you have a set of modes of the rectangle? These are solutions to the wave equation without regard to the initial conditions and are of the form $f(x,y)e^{i\omega t}$. If not, do you have a set for the square?
If you do, you need to expand your $u(x,y,0)$ in these modes. They tend to be numbered $u_{mn}(x,y)$ where $m$ is the number of half waves in the $x$ direction and $n$ is the number in the $y$ direction. You are looking for a set of coefficients $a_{mn}$ so the solution $u(x,y,t)=a_{mn}u_{mn}e^{i \omega_{mn}t}$There is probably a proof that the modes are orthogonal, in which case you can just do like a Fourrier expansion. $a_{mn}=\int_{rectangle}u(x,y,0)u_{mn}(x,y)dxdy$ This only works if the modes are orthogonal.