# Fourier/heat problem involving hyperbolic sine

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1. Mar 14, 2017

### danmel413

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. Mar 14, 2017

### TSny

What partial differential equation must τ(x,y,z) satisfy?

3. Mar 14, 2017

### danmel413

T(x,y,z) satisfies ∂T/∂t =D∇2T and T(x,y,z) = T1 + τ(x,y,z)

4. Mar 14, 2017

### TSny

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