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I'm trying to derive the cut off frequency for a wave guide. I found a derivation on wikipedia, but I don't understand the first step where we go from the wave equation to the helmholtz equation. Why does only considering $\psi(x,y,z,t)=\psi(x,y,z)e^{i\omega t}$ mean you can go from the wave equation to the helmholtz equation?

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By guessing a time dependence of the form $e^{i\omega t}$ you remove the time dependence from the equation. specifically, it transforms to the Helmholz equation.

I recommend you set $\psi(x,y,z,t)=u(x,y,z)e^{i\omega t}$ and substitute into the wave equation and do the algebra and see for yourself. You'll get the Helmholz equation on the spatial part $u(x,y,z)$.

I suggest you read on separation of variables

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  • $\begingroup$ I think I see it now, let $\psi (x,y,z,t)e^{i\omega t}$ in the wave equation, then expand the brackets. I see where the $\omega^2 $ comes from now in the helmholtz equation, but where does the $-e^{i\omega t}$ go when you take the second derivative wrt t? Oh, does it cancel out with the other $-e^{i\omega t}$ from the lapace? $\endgroup$
    – David
    Commented Mar 16, 2021 at 13:34
  • $\begingroup$ sorry for the delay. the factor $e^{i\omega t}$ indeed cancels out with the one left over from applying the Laplacian . @David $\endgroup$
    – Tomka
    Commented Mar 17, 2021 at 5:37

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