I have the following wave equation $\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=\dfrac{1}{c^2}\dfrac{\partial^2 u}{\partial t^2}$ with boundary conditions at $x=0,\ u=\sin(\omega t )\sin(n\pi y)$ at $x=l,\ u=0$ and at $y=0,\ u=0;\ y=1, u=0$. I derived results taking Laplace transform in time $\left(\dfrac{\partial^2 U}{\partial x^2}+\dfrac{\partial^2 U}{\partial y^2}=\dfrac{s^2 U}{c^2}\right)$, and it has the solution $\dfrac{\sin(k(l-x))}{\sin(kl)}\sin(\omega t)\sin(n\pi y)+\mbox{transients}$. The result doesn't seem to give traveling wave solution but a standing wave solution in $x$ and $y$.
I'm wondering if I should take a double Laplace transform in $x$ and $t$. Can someone please let me know if that approach will help? Also, will I be able to capture traveling waves, say before it reaches the boundary at $x=l$?
PS: $U$ is Laplace transform of $u$ in $t$. The initial conditions are 0.