Question

Demonstrate that when two harmonic oscillations are added, the time-averaged energy of the resultant oscillation is equal to the sum of the energies of the constituent oscillations, if both of them

(a) have the same direction and are incoherent, and all the values of the phase difference between the oscillations are equally probable;

(b) are mutually perpendicular, have the same frequency and an arbitrary phase difference.

Answer 1

(a) In this case the net vibration is given by

Where δ is the phase difference between the two vibrations which varies rapidly and randomly in the interval (0, 2π). (This is what is meant by incoherence.)

The total energy will be taken to be proportional to the time average of the square of the displacement.

In the same units the energies of the two oscillations are a21 and a22 respectively so the proposition is proved.