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Trying to find dispersion relation

  1. Sep 4, 2011 #1
    1. The problem statement, all variables and given/known data
    [itex]\imath\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2}=0[/itex]

    [itex]\left(x,t\right) = \int^{\infty}_{-\infty}A\left(k\right)e^{\imath\left(kx-wt\right)}dk[/itex]

    [itex]u\left(x,0\right)=\delta\left(x\right) [/itex]


    2. Relevant equations
    Not sure how to get w(k)



    3. The attempt at a solution
    [itex]A\left(k\right) = \frac{1}{2\pi}\int^{\infty}_{-\infty}\delta\left(x\right)e^{-\imath\left(kx\right)}dx = \frac{1}{2\pi}[/itex]

    plugging this in to u(x,t) do I work with


    [itex] u\left(x,t\right) =\frac{1}{2\pi}\int^{\infty}_{-\infty}e^{\imath\left(kx-wt\right)}dk[/itex]

    This is where I am stuck. I know w(k) is the dispersion relation. If I put in the pde do I just deal with

    [itex] \imath \left(-\imath w\right) + \frac{d^{2}u}{dt^{2}} = w +\frac{d^{2}u}{dt^{2}}=0 [/itex]

    ???

    Not sure if this is what I even want to do. Any guidance would be appreciated.
     
  2. jcsd
  3. Sep 4, 2011 #2
    I was making the problem too hard on myself. I got

    [itex] w=k^2 [/itex]
     
    Last edited: Sep 4, 2011
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