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The differential equation of the spring mass system gives you a second order differential equation. Now, similarly wave equations have solutions that just like the spring system contain trigonometric functions (sine or cosine) as this function differentiated twice is itself negative.

Now my question is, how do we know this is the only solution to these equations? If in would think of another equation being itself after differentiated twice, it is $\exp(ix)$ which is a trigonometric function as well. So is the solution always a harmonic oscillation, and what is the proof that this is the only general solution?

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Well, in the case of a mass on a spring oscillating in 1-D, Picard's Theorem asserts that a unique solution exists for any given initial conditions. So, if your set of solutions can satisfy any given initial conditions, you know you've found all the solutions.

Now, there's a few subtleties here. Your assertion that all solutions are harmonic oscillations is incorrect; all solutions are linear combinations of harmonic oscillations. Once we allow ourselves to take linear combinations of solutions, only then can we match any initial conditions.

For more complicated, multidimensional problems, I don't know of any general uniqueness theorems for differential equations that would apply. However, we certainly know the solutions MUST be unique: in the real world, after all, a wave only moves in one way!

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  • $\begingroup$ Oke, that's an interesting theorem. I tried some things out with 2d and 3d wave differential equation and I got 2 possible solutions, one generating a wave field and the other a wave propagating into one direction only. That is strange! Should I combine these two linearly to obtain the unique solution? $\endgroup$ – user209347 Jul 22 '15 at 7:57
  • $\begingroup$ You should linar combinations in order to match your initial conditions/boundary conditions. $\endgroup$ – Jahan Claes Jul 22 '15 at 18:44

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