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Energy considerations and the inhomogeneous wave equation

  1. Dec 5, 2011 #1

    If I have a forced wave equation

    [tex] u_{tt}-c^2u_{xx}= f(x,t) [/tex]

    what is my associated energy law?

    For instance, in the homogeneous case

    [tex]\Box u=0 [/tex]

    I know that

    [tex] E(t)=\frac{1}{2}\int u_t^2 +c^2|u_x|^2 \ dx [/tex]

    which implies that [itex]\frac{d E(t)}{d t}[/itex] is equal to zero. (just use integration by parts)

    I'm not sure if I should just conclude that

    [tex]\frac{d E}{dt}= \int u_t f(x,t) dx [/tex]

    by following the same logic as was used to show that this quantity is zero in the homogeneous case. Or we can more generally see that the forced wave equation has a Lagrangian given by

    [tex] L = \frac{1}{2}\iint -u_t^2 +c^2|u_x|^2 - 2u f(x,t) \ dx \ dt[/tex]

    which means that the associated Hamiltonian, or energy, would be

    [tex] E(t) = \frac{1}{2}\int u_t^2 +c^2|u_x|^2 + 2u f(x,t) \ dx [/tex]

    which leads to a different result than what I quoted above.

    Any help is appreciated!


    Last edited: Dec 5, 2011
  2. jcsd
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