Bandwidth of second harmonic two times that of the fundamental

Hello all,

I was curious on why the bandwidth of the second harmonic of a PA or switch or any non linear device is twice that of the fundamental and the 3rd harmonic three times that of the fundamental tone.

Thanks

Say the fundamental is modulated waveform with a lower frequency of f.1 and higher frequency of f.2, so the bandwidth is:
BW = f.2-f.1.
All the frequencies are doubled for the second harmonic, so the lower frequency of the 2nd harmonic is 2*f.1, and the upper frequency is 2*f.2, so the bandwidth of the 2nd harmonic is:
2*f.2-2*f.1 = 2*(f.2-f.1) = 2*BW of fundamental.