I am trying to solve a problem that has been set for me. I haven't come across a problem like this like, so i need some help getting through it. It is used to model the vibrations of a rectangular plate of width $L$ and height $H$ and edges are fixed.
$$u_{tt}-c^2(u_{xx}+u_{yy})=0$$
we are asked to state appropriate boundary conditions, so i assumed they would be $$u(x,0)=0, u(0,y)=0$$ $$u(L,y)=0, u(x,H)=0$$ as the edges are fixed. Am i right in assuming so?
Next it asks us to seek separable solutions $u(x,t)=X(x)Y(y)T(t)$ to obtain ODE's for all 3, and state BC's on X and Y. Consequently going on to finding the general solution where a double Fourier Series should be obtained.
Any help with this will be highly appreciated, as I really want to get my head around this.
Thanks
You are missing a $t$ slot for $u$ when you specified the boundary conditions, but otherwise they are correct if you wanted clamped edges all the way around your rectangle, i.e. for $u=u(x,y,t)$ you have $$ u(x,0,t)=u(x,H,t)=0,\ 0<x<L,\ t>0,\\ u(0,y,t)=u(L,y,t)=0,\ 0<y<H,\ t>0. $$
From there, plug $uX,y,t)=X(x)Y(y)T(t)$ back into the PDE. You should get ODE problems for $X$, $Y$, and $T$ similar to what you got in spatially one-dimensional problems.