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Steady State Temperature in Semi-Infinite Plate with Discontinuity

  1. Jan 22, 2012 #1
    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?
     
  2. jcsd
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