A certain oscillation results from the addition of coherent oscillations of the same direction results from the addition of coherent where `k` is the number of the oscillation `(k = 1, 2, ………., N), varphi` is the phase difference between the `kth` and `~(k -1)th` oscillations. Find the amplitude of the resultant oscillation.

Explanation

We use the method of complex amplitudes. Then the ampulitudes are
`A_(1) = a, A_(2) = ae^(i varphi), ..A_(N) = ae^(i(N - 1)varphi)`
and the resultant complex amplitude is
`A = A_(1) + A_(2) +………+A_(N) = a (1 -e^(i varphi) + e^(2i varphi)+.......+e^(i(N - 1)varphi))`
`= a(1 - e^(iN varphi))/(1 -e^(i varphi))`
The correspnding orbinary amplitude is
`|A| = a |(1 - e^(iN varphi))/(1 - e^(i varphi))| = a[(1 - e^(iNvarphi))/(1 - e^(i varphi)) xx (1 - e^(-iN varphi))/(1 -e^(-i varphi))]^(1//2)`
`= a[(2- 2 cos N varphi)/(2 - 2 cos varphi)]^(1//2) = a (sin((Nvarphi)/2))/(sin((varphi)/(2)))`.