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Fourier/heat problem involving hyperbolic sine

  1. Mar 14, 2017 #1
    1. The problem statement, all variables and given/known data
    A rectangular box measuring a x b x c has all its walls at temperature T1 except for the one at z=c which is held at temperature T2. When the box comes to equilibrium, the temperature function T(x,y,z) satisfies ∂T/∂t =D∇2T with the time derivative on the left equal to zero. Find the temperature T in the box in the form T(x,y,z) = T1 + τ(x,y,z) where τ is the fourier series τ(x,y,z)=∑∑anmsin(nπx/a)sin(mπy/b)f(z).

    Find f(z) and find anm

    2. Relevant equations
    D=kA/mc
    All others listed in the question

    3. The attempt at a solution
    so I see that it's sin functions in x and y in the fourier series because there are zero's at 0 and a, b for both. However since there's only one 0 for z at z=0, I'm assuming that f(z) has to be sinh. But I don't think I can just take the argument there to be jπz/c.

    If someone can bump me forward I'm sure I can figure the rest of the question out.
     
  2. jcsd
  3. Mar 14, 2017 #2

    TSny

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    What partial differential equation must τ(x,y,z) satisfy?
     
  4. Mar 14, 2017 #3
    T(x,y,z) satisfies ∂T/∂t =D∇2T and T(x,y,z) = T1 + τ(x,y,z)
     
  5. Mar 14, 2017 #4

    TSny

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    What differential equation does the "tau" function τ(x,y,z) satisfy under equilibrium conditions?
    Use this to determine the form of f(z).
     
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