# Helmholtz equation and sources

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

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

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.

• 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. – Emilio Pisanty Jul 6 at 6:38
• The wave equation does not only have sinusoidal solutions. – G. Smith Jul 6 at 6:39
• Where did $k^{2}$ in the first equation come from? – Cinaed Simson Jul 6 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,