Steady State Temperature in Semi-Infinite Plate with Discontinuity

[chill_factor]

Homework Statement

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.

Homework Equations

∂2T/∂x2 + ∂2T/∂y2 = 0

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?

 

[mighty2000]

Thank you for posting your problem from Mary Boas' Mathematical Methods in Physical Sciences. It seems like you have made good progress in solving the problem using separation of variables. However, you have encountered a difficulty with the boundary conditions, specifically the discontinuity at y=0.

In this case, you can use the method of images to solve the problem. This method involves creating a mirror image of the given boundary conditions at y=0, and using the superposition principle to find the solution.

For your specific problem, you can create a mirror image of the boundary condition T=100 at 10 ≤ x ≤ 20, which will then give you a boundary condition of T=100 at -10 ≤ x ≤ 0. You can then combine this with the original boundary condition of T=0 at 0 ≤ x ≤ 10, and use the superposition principle to solve for the temperature distribution.

I hope this helps you solve the problem. If you have any further questions, please feel free to ask. Good luck!