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

  1. Apr 6, 2008 #1
    [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!
     
  2. jcsd
  3. Apr 6, 2008 #2
    [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

    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!
     
  4. Apr 6, 2008 #3
    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.
     
  5. Apr 6, 2008 #4
    It doesn't seem I can delete. If an admin reads this please delete this and one of the other copies.
     
  6. Apr 6, 2008 #5
    I'll set this one and one other to solved to try to remove the amount of people opening it.
     
  7. Apr 6, 2008 #6
    Sorry for the copies.
     
  8. Apr 6, 2008 #7
    Please delete this one, setting it to solved to try to help remove additional readers of the copy.
     
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