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Energy flow in the wave equation (PDE)

  1. May 4, 2007 #1
    1. The problem statement, all variables and given/known data

    I have a problem that I'm trying to make sense of. Note y_t is the partial derivative of y with respect to t and y_tt is the second order partial derivative of y with respect to t, etc. The complete problem statement is the following:

    Show that for the equation y_tt - c^2 y_xx = 0
    the quantity E = 1/2(y_t^2 + c^2 y_x^2)
    satisfies a conservation law dE/dt + dJ/dx = 0

    2. Relevant equations



    3. The attempt at a solution

    I calculated dE/dt to = y_t * y_tt + c^2 y_x * y_xt so dJ/dx must equal the negation of dE/dt. But I'm not sure where J comes from. I'm guessing that it is somehow related to y through the wave equation but I'm not sure how. Also it is unclear to me how I could integrate the negation of dE/dt to arrive at J. I gave the complete problem statement. In that chapter of the book the function J is used but I don't think it applies to this problem. It involves a function J in terms of other variables. Hoping what I gave makes sense.

    Brian
     
  2. jcsd
  3. Aug 18, 2010 #2
    I think this is just a mathematical trick. After you have differentiated E with respect to t then you
    can observe that the two terms may also be obtained by differentiating another function with respect
    to x. This function turns out to be y_t y_x and we can call it -J. Making this definition gives the result.

     
  4. Aug 18, 2010 #3

    hunt_mat

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    Homework Helper

    You know E, you know the conservation relation that E and J satisfy, can't you use these two to try and compute what J has to be?

    Mat
     
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