What causes electromagnetic waves to propagate in free space?

In free space, $rho=0$ and $J=0$, so there are no electromagnetic sources/sinks. Maxwell’s equations thus reduce to:

$nablacdot E = 0$

$nablacdot B = 0$

$nablatimes E = -frac{partial B}{partial t}$

$nablatimes B = mu_0epsilon_0 frac{partial E}{partial t} $

Suppose I was writing a simple simulation to visualize the electromagnetic field in free space. I have seen people talk about waves propagating in free space, and I know that there is no such thing as electromagnetic waves being created from nothing — it is usually assumed that such electromagnetic waves propagating in free space are plane waves that originated in a charged source extremely far away.

However, when I actually implement the “oscillation” in the EM field, where does that oscillation come from — speaking practically from a coding point of view? Do you just hardwire, e.g., a sinusoidal source at the location you are interested in probing the EM field?

And if there were no such “magical” waves propagating in free space, would the EM field just remain smooth, without any oscillations, vibrations, sinusoids, etc.? In other words, can you have a completely stationary electromagnetic field, or would the last two of Maxwell’s equations above prevent such stationary EM fields? But then what, in free space, would cause the initial change in the electric or magnetic field to get the oscillations going?

To put it one last way: suppose I wrote a simulation involving the 4 Maxwell’s equations above (free space). Would the EM field be stationary for all time, and the only way a propagating wave would appear is if I perturbed, say, the electric field which activated a never-ending loop of the curl equations? So if the initial values of E and B were both 0 in my simulation, then they would stay 0 for all time. But if one or both initial values of E and B were non-zero, then the curl equations would be “activated” and result in a never-ending loop of oscillations?