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Steady-State Temperature: Fourier Transform

  1. Nov 16, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the steady-state temperature in a semi-infinite plate covering the region x>0, 0≤y≤1,if the edges along the x axis and y axis are insulated and the top edge is held at
    gif.latex?u(x,1)=\left\{\begin{matrix}100,%20&%200%3Cx%3C1\\0,%20&%20x%3E1%20\end{matrix}\right..gif

    Hint: Look for a solution as a Fourier integral.

    2. Relevant equations
    gif.gif


    3. The attempt at a solution
    All I really need help with is how to set up the integral for u. I know the rough form of it from equations in the book, but they are for very different problems. Usually, the semi-infinite plate covers the entire first quadrant (infinite in positive x and y) but here, it ends abruptly at y=1. I tried,
    [URL]http://latex.codecogs.com/gif.latex?u=\begin{Bmatrix}%20e^{-kx}\\e^{kx}%20\end{Bmatrix}%20\begin{Bmatrix}%20sinky\\cosky%20\end{Bmatrix}[/URL]
    And eliminated the ekx solution, since u should not be infinite as x-->∞. I wasn't sure what trig function to use, and furthermore, when I tried to plug in my initial condition for u(x,1), I got a strange looking integral,
    [URL]http://latex.codecogs.com/gif.latex?\int_{0}^{\infty%20}B(k)e^{-kx}sinkdk[/URL]
    Which didn't seem to be correct or helpful at all. How do I get something which I could use as a Fourier integral pair? Also, I am clueless as to how the insulated sides factor in.

    If anyone could show me how to get the correct integral for u(x,y), I know I can finish the rest of the problem, but this is just unlike any example in the book; usually the edge with the piecewise boundary condition is perpendicular to the axis in which the block goes to infinity, not parallel it. I feel like that's really throwing me off.
     
    Last edited by a moderator: Apr 26, 2017
  2. jcsd
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