A question about wave interference.

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The discussion centers on wave interference and how to analyze it using multiple solutions of the wave equation. Two main methods are proposed: calculating the analytic sum of the wave functions to assess interference based on amplitude and wavelength, and directly summing the solutions while considering phase differences. There is uncertainty about whether the phase difference between eigenmodes accurately reflects the real phase difference in the field distributions. It is suggested that all solutions must be combined to derive a unique solution, implying inherent interference. The phase is ultimately determined by boundary conditions.
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Hello All!
My question is about the interference if the waves.

For example, I have SEVERAL solutions of the wave equation in some area (several eigenvalues and corresponding eigenfunctions). I want to see if the would be an interference between these solutions.

What should I do?

I see the 2 ways:

1) use analytic sum of this functions.
by this I mean to calculate the interference of the two plain waves of different amlitudes and wavelength. How can I do this?

2) sum directly two my solutions.
BUT a question arises: can I suppose that the difference in phase between my two solutions of the wave equation is what I need?
Can I state that the phase difference between the 2 eigenmodes of the equation is the real phase difference between these two field distributions?
 
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Seems to me you have to add all your solutions to get the unique, general solution. Adding them inherently means they interfere. Not sure I see any difference between your 1) and 2). The phase should be determined by boundary conditions.
 
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