(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Seperation of variables in the 2 dimensional wave equation

I'd like to apologize right away for the terrible formatting. I was trying to make it pretty and easy to read but I guess I'm just not used the system yet and I had one problem after another. As you'll see at one point the formatting pretty much went out the window. I hope you can still figure out what I'm trying to say. Sorry!

1. The problem statement, all variables and given/known data

Solve the wave equation in 2 dimensions by separation of variables.

([tex]\delta[/tex][tex]^{2}[/tex]u)/([tex]\delta[/tex]t[tex]^{2}[/tex])=4(([tex]\delta[/tex][tex]^{2}[/tex]u)/([tex]\delta[/tex]x[tex]^{2}[/tex])+([tex]\delta[/tex][tex]^{2}[/tex]u)/([tex]\delta[/tex]y[tex]^{2}[/tex]))

a=[tex]\pi[/tex]/2 , b=[tex]\pi[/tex]

V(0,y,t)=V(a,y,t)=0

V(x,0,t)=V(x,b,t)=0

V(x,y,0)=0

([tex]\delta[/tex]V)/([tex]\delta[/tex]t)(x,y,0)=g(x,y)=([tex]\pi[/tex]/2-x)([tex]\pi[/tex]-y)

Show that

V(x,y,t)=Sigma(n=1,[tex]\infty[/tex])Sigma(m=1,[tex]\infty[/tex])(((sin(2sqrt(lambda sub(nm))t))/(nm*sqrt(lambda sub(nm)))*sin(2nx)sin(m)y)

where

lambda sub(nm)=4n^2+m^2

2. Relevant equations

All I know is it's going to be semi-related to the general solution of the wave equation:

([tex]\delta[/tex][tex]^{2}[/tex]u)/([tex]\delta[/tex]t[tex]^{2}[/tex])=(k^2)*([tex]\delta[/tex][tex]^{2}[/tex]u)/([tex]\delta[/tex]x[tex]^{2}[/tex])

3. The attempt at a solution

I know that k^2 is going to be equal to the 4. Other then that I'm lost. If possible, well explained steps would be best so I can figure out what I'm doing.

Thanks for your help!

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# Homework Help: Seperation of variables in the 2 dimensional wave equation

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