What to expect from a spectra...

Seems like a simple question. The main thing I'm trying to figure out is SHOULD we see the frequency of the long period wave group modulations in a spectrum? If we have a WSE time series that exhibits groupiness and we take the fft we do NOT see the envelope frequency. Now lets say you started with the wave group time series (so all positive) and took the fft...do you see modulations of the wave groups? Well the answer seems to be yes although I can't say I know why. I've made a series of simple cases and all give the same result. One is posted here. Start with 2 waves at different frequencies and add them together (BLUE). Then I square it so that it looks like the wave group time series we are using in the vortex research (RED). I also, for comparison, compute the hilbert transform of the original WSE times series (GREEN). Then I take the FFT of all these.

NOTE: The modulations of the wave groups occur with period 2*pi/ds (ds = difference in sigma). In this particular case 2*pi/ds = 25 sec = 0.04 Hz. Both the hilbert transform and the FFT of eta^2 give this frequency in the spectrum. So it seems that in our model results, if we look at an f-ky spectrum of the offshore wave group energy, we SHOULD see the longer period modulations in the spectum and these may correspond to long period fluctuations in the velocity time series (hence forced repsonse motions, not free oscillations).