Why sine wave is the natural wave, why not something else?
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That’s not quite true. For example, if you study the KdV equation which describes shallow water wave, the natural wave is the *soliton*. Same goes for light in a fiber optic. These can be related to asymptotically sinusoidal wave satisfying the Schrodinger equation, through the process call inverse scattering.
In general, the natural wave we want to study are solitons. Over time, solitons keep its shape as it moves, while being sturdy and highly resistant against disturbance. This allows us to think of them as concrete object.
When it comes to physical field, it’s common for those field to satisfy a law that is both time and space invariant. After all, we expect the law of physics to be unchanged over time and space, so as long as there are no external forces we expect these fields to be space and time invariant. In that case, complex exponential on imaginary axis has a very specific property: it diagonalizes the translation, in other word, translation only causes a multiplication by a constant factor. If the equation is also linear, which is a common situation when you’re looking at *perturbed* equation (that is, equation describe tiny changes from the equilibrium), then these equation are linear and hence has superposition, so complex exponential on imaginary axis satisfies all the other properties of solitons automatically. This makes them the most natural wave. Sine wave is equivalent to these wave, but in real number.
If the law is not space invariant but time invariant, due to an effect that is localized in space, one can look at the far-field approximation: what do the waves look like far enough, where they behave almost as if there are no external forces, and the above analysis still work approximately.
The difficulty started to come in when you no longer have perturbed equations and have to face non-linear equation. Then you would have to look at solitons that are very different from sine waves.
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