So you mean that the current of the battery and the current reaction of the ballast cancel and then many sinewaves start to oscillate by their own causing a mixture? But I'm a bit confused. How can the current generated by the ballast cancel the current of the battery that already passed? Would be good if you post a drawing of how you think the signal would look like, for understand it better.
It is an aid to think about the math behind, to help you to grab the math more intuitively.
Because it is the impedance after the switch off what matter, but then there is no current. Quite unintuitive to think about what causes the current once it does not flow anymore.
So the aid works like this:
If you have two current sources in parallel, their currents sum up.
If the first source is a constant, lets say +1A,
and the second source is a step from 0 to -1A,
their sum would be a step from 1A to zero, so matching what really your battery and switch are doing.
Now because we have a linear system (voltages are proportional to currents exciting them), we may analyze the system separately for each virtual source and sum up the resulting voltage waveforms and we get the same response:
The constant 1A makes nothing on an inductor except the DC shift onthe wire resistance, so its response is a DC voltage. But that wont radiate, so we may quite easily neglect it at all.
The second source then forms 0 -> -1A step, flowing into the bunch of wire with R, L, C all along it. So for evaluating the radiation, we need to evaluate how the circuit (with the switch open) responds to this step.
So you may form a bunch of differential equations and solve them,
Or you may transform everyrhing into a frequency domain and just multiply the impedance/transfer/radiation impedances (depends what you want to calculate) at each frequency by the given frequency component of the source signal.
Now we know the source is a (unity) step, so has a 1/freq spectrum.
The system itself is just a bunch of resonances, so a bunch of spikes with their respective Q and peak value.
By multiplying these two functions you get the spectrum of either the coil voltage or of what is being radiated.
Because there is always one frequency with the highest impedance, sticking above the others (the fundamental resonant frequency), its response will be what plays the dominant role on the output, so dictates the decaying oscillations, similar as on the scope picture you presented.
And if caring about radiation, the picture of both E, as well as H field component will look very alike.