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Electromagnetic Wave Equation

  1. Oct 5, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that the solution [itex]\textbf{E}=E(y,z)\textbf{n}\cos(\omega t-k_xx)[/itex] substituted into the wave equation yields

    [itex]\frac{\partial^2 E(y,z)}{\partial y^2}+\frac{\partial^2 E(y,z)}{\partial z^2}=-k^2E(y,z)[/itex]

    where [itex]k^2=\frac{\omega^2}{c^2}-k_x^2[/itex]
    2. Relevant equations
    See above.

    3. The attempt at a solution
    I plugged the given solution into [itex]\frac{\partial^2 \textbf{E}}{\partial y^2}+\frac{\partial^2 \textbf{E}}{\partial z^2}=\frac{1}{c^2}\frac{\partial^2 \textbf{E}}{\partial t^2}[/itex] and got:

    [itex]\textbf{n}\cos(\omega t-k_xx)[\frac{\partial^2 E(y,z)}{\partial y^2}+\frac{\partial^2 E(y,z)}{\partial z^2}]=-\frac{\omega^2}{c^2}E(y,z)\textbf{n}\cos(\omega t-k_xx)[/itex]

    Now, canceling like terms I get:

    [itex]\frac{\partial^2 E(y,z)}{\partial y^2}+\frac{\partial^2 E(y,z)}{\partial z^2}=-\frac{\omega^2}{c^2}E(y,z)[/itex]

    But I'm missing a [itex]k_x^2[/itex] term on the RHS, and cannot figure out where this could/would have come from...can someone please explain?
  2. jcsd
  3. Oct 5, 2014 #2
    I was able to get the [itex]k_x^2[/itex] term by determining [itex]\nabla^2\textbf{E}[/itex] and rearranging, thus obtaining the desired relation.

    However, I'm not entirely sure why it's necessary to determine [itex]\nabla^2[/itex]. Can someone please explain this to be?
  4. Oct 5, 2014 #3


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    Homework Helper

    You had to evaluate the ##\nabla^2## operator because that is the definition of the wave function. ## \nabla^2 \vec{E} = \frac{\partial^2 \vec{E}}{\partial t^2}## Adding an ##x## dependence into your function for ##\vec{E}## meant you had to fully evaluate the Laplacian.
  5. Oct 5, 2014 #4
    I see...I was considering this equation as only a two-dimensional one...for some reason I was overlooking the x component in the cosine function. Not entirely sure why, perhaps because of the E(y,z) term, but I now realize this is simply a coefficient corresponding to the amplitude.

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