I have a problem solving(adsbygoogle = window.adsbygoogle || []).push({});

[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?

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# Homework Help: Laplace's equation

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