- #1
Logarythmic
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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?
[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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