# Wave superposition.

## 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?

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ehild
Homework Helper
Use the Euler form of the waves: B*e= A*∑einδ.

ehild

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
Homework Helper
Aeiδn is element of a geometric sequence with quotient e and Ae 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 eiδ N/2 from the numerator and eiδ/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

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