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Solution of wave equation

  1. Jul 30, 2011 #1
    1. The problem statement, all variables and given/known data

    The wave equation is
    [itex]\nabla^2 \mathbf{A}(\mathbf{r},t) = \frac{1}{c^2} \frac{\partial^2 \mathbf{A}(\mathbf{r},t)}{\partial t^2}[/itex]

    I want to get a solution for the vector potential A.

    2. Relevant equations
    we can use the Fourier transformation

    [itex] \mathbf{A}(\mathbf{k},\omega) = \int{d\mathbf{r}}\int{dt \mathbf{A}(\mathbf{r},t) \exp{[-i(\mathbf{k \cdot r} - \omega t)]}} [/itex]

    [itex] \mathbf{A}(\mathbf{r},t) = \int{\frac{d\mathbf{k}}{(2\pi)^3}} \int{\frac{d\omega}{2 \pi} \mathbf{A}(\mathbf{k},\omega) \exp{[-i(\mathbf{k \cdot r} - \omega t)]}} [/itex]

    to get

    [itex] \left( \mathbf{k}^2 - \frac{\omega^2}{c^2}\right) \mathbf{A}(\mathbf{k},\omega) =0 [/itex]

    (so [itex] \omega = c k [/itex] )

    3. The attempt at a solution

    Now the the solution is (formally)

    [itex] \mathbf{A}(\mathbf{r},t) = \sum_{\lambda =1,2} \int{\frac{d\mathbf{k}}{(2\pi)^3}} \int{\frac{d\omega}{2 \pi} A_\lambda(\mathbf{k},\omega) \hat{\epsilon}_\lambda(k) \cos{(\mathbf{k \cdot r} - \omega t + \varphi_\omega)} \delta(\omega -ck)} [/itex]

    but i don't understand well why we have 2 polarization vector and form where they come...

    thanks all,
  2. jcsd
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