# Homework Help: Steady State Temperature in Semi-Infinite Plate with Discontinuity

1. Jan 22, 2012

### chill_factor

1. The problem statement, all variables and given/known data

This is problem 13.2.2 in Mathematical Methods in Physical Sciences by Mary Boa.

Find the steady state temperature distribution using the Laplace Equation on a semi-infinite plate extending in the y direction with the following boundary conditions:

On the lines y = infinity, T = 0. x = 0 , T = 0. x = 20, T=0.

On the line x=0, T=0 at 0 ≤ x ≤ 10, and T=100 at 10 ≤ x ≤ 20.

2. Relevant equations

∂2T/∂x2 + ∂2T/∂y2 = 0

3. The attempt at a solution

Seperation of variables: T(x,y)=U(x)V(y)

∂2(UV)/∂x2 + ∂2(UV)/∂y2 = 0

V*∂2(U)/∂x2 + U*∂2(V)/∂y2 = 0

Divide both sides by UV.

(1/U)*∂2(U)/∂x2 + (1/V)*∂2(V)/∂y2 = 0

Rearrange and let both equal a constant

(1/U)*∂2(U)/∂x2 = -(1/V)*∂2(V)/∂y2 = -k^2

Write as 2 ODEs and rearrange

d2(U)/dx2 = -k^2 * U
d2(V)/dy2 = k^2 * V

Use boundary conditions: x=0, U=0. x=20, U=20. U is likely to be U = sin(n∏x/20) where n=0,1,2,...

Boundary conditions for V: y approach infinite, V = 0, V form likely to be V = e^(-n∏y/20).

T= UV = Ʃ Tn * e^(-n∏y/20)sin(n∏x/20)

Usually we plug in the function for T at y=0 for T, set y= 0 on the right hand side of the equation and get

T(y=0) = Ʃ Tn * sin(n∏x/20)

we then solve for Tn, write it as a series, then we're done.

But here, T(y=0) is two expressions with a discontinuity. How do I proceed?