Demonstrate that when two harmonic oscillations are added, the time-averged energy of the resultant oscillation is equal to the sun of the energies of the consituent osciallations, if both of them
(a) have the same direction and are incoherent, and all the values of the phase difference between the oscillations are equal probable,
(b) are mutually perpendicular, ahve teh same frequency and an arbitary phase difference.

Explanation

(a) In this case the net vibration is given by
`x = a_(1) cos omega t + a_(2) cos (omegat + del)`
where `del` is the phase difference between the two vibrations whicgh varies rapidly and randomly in the interval `(0, 2pi)`. (This is what is meant by incoherence.)
Then `x = (a_(1) + a_(2) cos del) cos omegat + a_(2) sin del sin omegat`
The total energy will be taken to be proportional to the time average of the square of the displacement.
Thus `E = lt (a_(1) +a_(2) cos del)^(2) + a_(2)^(2) del gt = a_(1)^(2) + a_(2)^(2)`
as `lt cos del gt = 0` and we have put `lt cos^(2) omegat gt = lt sin^(2) omegat gt = (1)/(2)` and has been absorbed in the overall constant of proportinality.
In the same units the energies of the two oscillations are `a_(1)^(2)` and `a_(2)^(2)` respectively so the proposition is proved.
(b) Here `oversetrarr(r ) = a_(1) cos omegat hati + a_(2) cos (omegat + del) hatj`
and the mean square displacement is `alpha a_(1)^(2) + a_(2)^(2)`
if `del` is fixed but arbitrary. Then as in (a) we see that `E = E_(1) + E_(2)`.