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Energy and Wave Equation

  1. Nov 10, 2014 #1

    joshmccraney

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    Gold Member

    Hi PF!

    SO we have defined energy per unit mass as $$E(t) = \int_0^L \frac{1}{2} u_t^2 + \frac{c^2}{2} u_x^2 dx$$. We are given a vibrating string that exhibits ##u_x(0,t) = 0## and ##u(L,t)=0##. I am trying to figure out what is happening with total energy, ##E(t)##. My work is $$\int_0^L \frac{1}{2} u_t^2 + \frac{c^2}{2} u_x^2 dx = \int_0^L \frac{1}{2} u_t^2 dx + \int_0^L \frac{c^2}{2} u_x^2 dx=\int_0^L \frac{1}{2} u_t^2 dx + \frac{c^2}{2} u_x u \bigg|_0^L - \int_0^L u u_{xx} dx\\
    =\int_0^L \frac{1}{2} u_t^2 dx - \frac{c^2}{2}\int_0^L u u_{xx} dx = \int_0^L \frac{1}{2} u_t^2 - \frac{1}{2} u u_{tt} dx$$ where the last quantity appears from ##u## satisfying the wave eq, ##u_{tt} = c^2 u_{xx}##. Is there anything more I can/should do?

    Thanks!
     
  2. jcsd
  3. Nov 16, 2014 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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