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Seperation of variables in the 2 dimensional wave equation

[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. Homework Statement
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. Homework 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!
 

Answers and Replies

[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. Homework Statement
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

I would have left the relevant equations and attempted solutions parts, but for some reason I can't get it to post with that and I didn't have very much to say in those anyway. If possible, well explained steps would be best because I'm kind of lost, and then I can figure out what I'm doing.

Thanks for your help!
 
Dang it, it was posting but it came up with an error. Um.... I don't know how to delete this right off hand, but give me a minute I'll see if I can figure it out.
 
It doesn't seem I can delete. If an admin reads this please delete this and one of the other copies.
 
I'll set this one and one other to solved to try to remove the amount of people opening it.
 
Sorry for the copies.
 
Please delete this one, setting it to solved to try to help remove additional readers of the copy.
 

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