All Problems Interference of Light
Problem 5.66
A certain oscillation results from the addition of coherent oscillations of the same direction \(\xi_{k}=a \cos [\omega t+(k-1) \varphi]\), where \(k\) is the number of the osciliation \((k=1,2, \ldots, N), \varphi\) is the phase difference between the kth and \((k-1)\) th oscillations. Find the amplitude of the resultant oscillation.
\(5.66 .\) Let us represent the \(k\) th oscillation in the complex form \[ \xi_{k}=a \theta^{i[\omega t+(k-1) \varphi]}=a_{k}^{*} \mathrm{e}^{i \omega t} \] where \(a_{k}^{*}=a e^{i(k-1) \varphi}\) is the complex amplitude. Then the complex amplitude of the resulting oscillation is \[ \begin{aligned} A^{*}=\sum_{k=1}^{N} a \theta^{i(k-1)} &=a\left[1+\mathrm{e}^{i \Phi}+\mathrm{e}^{i 2 \varphi}+\ldots+\mathrm{e}^{i(N-1) \varphi}\right]=\\ &=a\left(\mathrm{e}^{t \varphi N}-1\right) /\left(\mathrm{e}^{\mathrm{i} \varphi}-1\right) \end{aligned} \] Multiplying \(A^{*}\) by the complex conjugate value and extracting the square root, we obtain the real amplitude \[ A=a \sqrt{\frac{1-\cos N \varphi}{1-\cos \varphi}}=a \frac{\sin (N \varphi / 2)}{\sin (\varphi / 2)} \]