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Laplace's equation

  • #1
281
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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:

Answers and Replies

  • #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.
 
  • #3
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So the X- and the Z-part are correct, but not the Y-part?
 
  • #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.
 
  • #5
281
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Sorry, I do not understand.
 

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