Solving Wave Superposition: Amplitude & Phase

[Lavabug]

Homework Statement

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 ($\delta$,$2\delta$, ...$n\delta$)

The Attempt at a Solution

Using the trig identity cos(u+$\delta$), where $u=(kr-\omega t)$ 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:

$A^*cos\delta^* = A \sum cos\delta_n$

$A^*sin\delta^* = A \sum sin\delta_n$

Beyond that it gets ugly if I try to solve for A* or δ*, for example squaring both and adding gives me:

$A^* = \sqrt(A^2 ( \sum cos\delta_n)^2 + ( \sum sin\delta_n)^2))$

is there another way to do this?

 

[ehild]

Use the Euler form of the waves: B*e^iθ= A*∑e^inδ.

ehild

 

[Lavabug]

I also tried that but it leads me to the same set of equations, is my solution for A* correct? Cause I can't think of anything else to do with it.

 

[ehild]

Ae^iδn is element of a geometric sequence with quotient e^iδ and Ae^iδ as first element. The sum of N element is the resultant wave.

$$B e^{i \theta}=A e^{i\delta} \frac{e^{i \delta N}-1}{e^{i\delta}-1}$$

Factor out e^{iδ N/2} from the numerator and e^iδ/2 from the denominator:

$$B e^{i \theta}=A e^{i\delta (N+1)/2} \frac{e^{i \delta N/2}-e^{-i \delta N/2}}{e^{i\delta/2}-e^{-i\delta/2}}=A e^{i\delta (N+1)/2}\frac{\sin(N\delta/2)}{\sin(\delta/2)}$$

ehild