Determine the amplitude and phase of the luminous disturbance produced by the superposition of N waves of the same amplitude and phases which increase in an arithmetic progression ([itex]\delta[/itex],[itex]2\delta[/itex], ...[itex]n\delta[/itex])
The Attempt at a Solution
Using the trig identity cos(u+[itex]\delta[/itex]), where [itex] u=(kr-\omega t) [/itex] I rewrite the resulting wave(with asterisks), which is a linear combination of n waves with different phases. Associating the coefficients I get the following 2 equalities:
[itex]A^*cos\delta^* = A \sum cos\delta_n[/itex]
[itex]A^*sin\delta^* = A \sum sin\delta_n[/itex]
Beyond that it gets ugly if I try to solve for A* or δ*, for example squaring both and adding gives me:
[itex]A^* = \sqrt(A^2 ( \sum cos\delta_n)^2 + ( \sum sin\delta_n)^2)) [/itex]
is there another way to do this?