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Homework Help: Laplace's equation

  1. Oct 2, 2006 #1
    I have a problem solving

    [tex]\nabla^2 T(x,y,z) = 0[/tex]
    [tex]T(0,y,z)=T(a,y,z)=0 [/tex]
    [tex]T(x,0,z)=T(x,b,z)=T_0 \sin{\frac{\pi x}{a} [/tex]
    [tex]T(x,y,0)=T(x,y,c)=const.[/tex]

    I use separation of variables and get

    [tex]X_n (x) = \sin{\frac{n \pi x}{a} [/tex]
    [tex]Y_n (y) = \cosh{\sqrt{\frac{n^2 \pi^2}{c^2} + \frac{n^2 \pi^2}{a^2}}y} + \sinh{\sqrt{\frac{n^2 \pi^2}{c^2} + \frac{n^2 \pi^2}{a^2}}y} [/tex]
    [tex]Z_n (z) = \cos{\frac{n \pi z}{c} [/tex]
    [tex]T(x,y,z) = \sum_{n=1}^\infty a_n X_n (x) Y_n (y) Z_n (z)[/tex]

    where I have used the boundary conditions for x and z. Is this correct?
    If it is, I'm having problems to wrap this up. I suppose I can use the condition for T(x,0,z) to get the constants. My calculations gives me

    [tex]a_n = \frac{T_0}{\cos{\frac{\pi z}{c}}}[/tex]

    but then I can't get it toghether with the condition for T(x,b,z)...
    Any ideas?
     
    Last edited: Oct 2, 2006
  2. jcsd
  3. Oct 2, 2006 #2

    dextercioby

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    Well, first of all, the "y" part of the solution must be periodic, but i'm afraid sinh & cosh are not...The same with the "z" & "x" part.

    Daniel.
     
  4. Oct 2, 2006 #3
    So the X- and the Z-part are correct, but not the Y-part?
     
  5. Oct 2, 2006 #4

    Meir Achuz

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    That is easiest to do as 4 separate problems, each having 5 sides grounded.
    Then add the 4 solutions.
     
  6. Oct 2, 2006 #5
    Sorry, I do not understand.
     
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