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 ([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?

 

[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.

[tex]B e^{i \theta}=A e^{i\delta} \frac{e^{i \delta N}-1}{e^{i\delta}-1}[/tex]

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

[tex]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)}[/tex]

ehild

 

[paranormal]

I would approach this problem by first understanding the underlying principles of wave superposition and how it affects amplitude and phase. I would then use mathematical equations and principles, such as the trig identity mentioned in the attempt at a solution, to solve for the amplitude and phase of the resulting wave.

One approach could be to use the principle of vector addition, where the amplitude and phase of the resulting wave can be represented as the sum of the amplitudes and phases of the individual waves. This can be visualized using a vector diagram, where the length and direction of the resulting vector represents the amplitude and phase, respectively.

Another approach could be to use complex numbers and Euler's formula to represent the individual waves and their superposition. This would allow for a simpler and more elegant solution, as the amplitude and phase can be represented as the magnitude and argument of the resulting complex number.

Ultimately, the solution will depend on the specific context and application of the problem, and it may require a combination of different mathematical techniques. As a scientist, it is important to critically analyze the problem and choose the most appropriate approach to solve it accurately and efficiently.