it is known that starting from the Maxwell equations it is possible to get the following Helmholtz equations:

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In time domain it corresponds to a sinusoidal EM field, according to the wave equation:

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Now my question is: is the time - behaviour of E/H always sinusoidal? It seems quite strange to me: I did not see any assumption of sinusoidal behaviour of current/charge sources. From this result it seems to me that sources can have any time - behaviour (also constant) and there will be a sine EM wave in time. I think I have missed some part of the reasoning.

  • $\begingroup$ You missed the part where the source is assumed to be sinusoidal. This can be as a direct ansatz, or as a Fourier decomposition of the general solution. The details will depend on where you actually saw this material. $\endgroup$ Jul 6 '19 at 6:38
  • 3
    $\begingroup$ The wave equation does not only have sinusoidal solutions. $\endgroup$
    – G. Smith
    Jul 6 '19 at 6:39
  • $\begingroup$ Where did $k^{2}$ in the first equation come from? $\endgroup$ Jul 6 '19 at 22:21

According to Fourier's Theorem, there are a very broad class of functions that can be approximated to arbitrary accuracy by sums of sinusoidal functions. This means that if you can solve the Helmholtz equation for a sinusoidal source, you can also solve it for any source whose behavior can be described by a Fourier series. In fact, since the Helmholtz wave equation is a linear PDE, you can solve it for almost any arbitrary source $f(r)$ by:

  1. Decomposing $f(r)$ into sinusoidal components,
  2. Solving the Helmholtz wave equation for each component, and
  3. Adding the solutions back together.

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