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Wave superposition.

  • Thread starter Lavabug
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  • #1
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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?
 
Last edited:

Answers and Replies

  • #2
ehild
Homework Helper
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Use the Euler form of the waves: B*e= A*∑einδ.

ehild
 
  • #3
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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.
 
  • #4
ehild
Homework Helper
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Aeiδn is element of a geometric sequence with quotient e and Ae 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 eiδ N/2 from the numerator and eiδ/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
 
Last edited:

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