- #1
rmjmu507
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Homework Statement
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]
Homework Equations
See above.
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?