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I appreciated your analysis very much. I do take some exception to the idea of "a sharp corner" moving along the string.
In my experience, the material "stiffness" of the metallic string on a guitar or violin precludes a "sharp corner", and the string does behave very much like a sine wave--essentially what your linear example predicts. In that regard, and from the viewpoint of a background in electronics, I have observed movement on a metal guitar string excited in such a way as to be vibrating in several harmonically related modes at once, and viewed axially, that its movement describes "Lissajou Patterns"--very similar to what we used to view on an oscilloscope and use to count frequencies.
If on a string of non-metallic material, as on a classical guitar or perhaps a ukelele, the wave form moving along the string does in fact have a "sharp corner", does that not imply a non-sinusoidal wave form similar to an asymetrical sawtooth or ramped wave form? When these forms are consistent and enduring, as from a signal generator or from a perfect violin, they should not be too difficult to analyse mathematically--though I'm not sure what might be learned that the ear does not already tell us.
Thank you for an interesting bit of research.