Why didn't I see a peak at half the frequency in my FFT analysis of two waves?

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Sam Smith
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I am carrying out FFT analysis to compare two waves. One looks very much like a sine wave the other has an extra dip occurring at half the frequency of the main wave. I have been thinking around how I might expect this to show up in the FFT analysis. At first i was expecting to see a smaller peak at half the main frequency but after a while I thought that it may just result in a few more higher harmonics due to the fact that the wave is becoming more complex. I did the FFT and I didnt see a peak at half frequency. so I guess the second conclusion is correct btu I still can't reconcile why I would expect to see a peak at half frequency of the main peak?
 
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Well the original wave is a two sine waves put together each mirroring the other. The dip is in between them so instead of going back to zero the graph dips instead :)
 
OK, so two sine waves mirroring each other have the same frequency but opposite phase. Each of those sine waves is multiplied by a rectangular wave in the time domain, which means that it is convolved with the transform of the rectangular wave in the frequency domain. So you will expect to see a signal that looks much more like the sum of two Fourier transforms of a rectangular wave. Since they are each modulated by the same frequency but opposite phases the resulting sum is not obvious to me. I would expect it to look "complicated".