A rectangular box measuring a times b times c has all its wa

A rectangular box measuring a times b times c has all its walls at temperature T_1 except for the one at z = c which has a temperature T_2. When the box comes to equilibrium, the temperature function T(x, y, z) satisfies the diffusion equation with a time derivative of zero (in equilibrium the temperature doesn\'t change with time): k/cp nabla^2T (x, y, z) = 0 Find the temperature in the box in the form: T(x, y, z) = T_1 + tau (x, y, z) where: tau (x, y, z) = sigma_n, m a_nm sin n pi x/a sin m pi y/b f(z) Find the function f(z) and the coefficients a_nm.

Solution

Use separation of variables:

Let T= X(x) Yy) F(z)

Obtain from Laplace eqn X\"/X + Y\"/Y = - F\"/F = K ( constant)

F eqn can be solved along with X and Y

let X\"/X = -A²

Y\"/Y =-B²

then F\"/F = (A²+B²)

applying the b.c x=a, y=b, z=c

and integrating over the space, orthogonality gives the coeffs a_mn

find A= npi/a

B =mpi/b

F is also of the form sin( lpi/c) where l² /c²= -m2 /a² -n² /b²

amn found by intergating and using the BC