The best method should be using separation of variables rather than using Laplace transform.

The general solution is of the form $u(x,y,t)=\int_0^\infty\int_0^\infty C_1(r,s)\sin xr\sin ys\sin(ct\sqrt{r^2+s^2})~dr~ds+\int_0^\infty\int_0^\infty C_2(r,s)\sin xr\sin ys\cos(ct\sqrt{r^2+s^2})~dr~ds+\int_0^\infty\int_0^\infty C_3(r,s)\sin xr\cos ys\sin(ct\sqrt{r^2+s^2})~dr~ds+\int_0^\infty\int_0^\infty C_4(r,s)\sin xr\cos ys\cos(ct\sqrt{r^2+s^2})~dr~ds+\int_0^\infty\int_0^\infty C_5(r,s)\cos xr\sin ys\sin(ct\sqrt{r^2+s^2})~dr~ds+\int_0^\infty\int_0^\infty C_6(r,s)\cos xr\sin ys\cos(ct\sqrt{r^2+s^2})~dr~ds+\int_0^\infty\int_0^\infty C_7(r,s)\cos xr\cos ys\sin(ct\sqrt{r^2+s^2})~dr~ds+\int_0^\infty\int_0^\infty C_8(r,s)\cos xr\cos ys\cos(ct\sqrt{r^2+s^2})~dr~ds$

Now substitute the conditions $u(0,y,t)=\sin n\pi y\sin\omega t$ , $u(l,y,t)=0$ , $u(x,0,t)=0$ and $u(x,1,t)=0$ for eliminating some of the arbitrary functions.

This post imported from StackExchange Mathematics at 2014-06-09 19:14 (UCT), posted by SE-user doraemonpaul