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We're all familiar with the typical diagrams of standing waves of a string, as in this image from Wikipedia:

Standing waves

The thing that bothers me is that they ignore the reality that the string is vibrating in three dimensions. If we say X is the longitudinal direction and Y is the up-down vibration, what about Z (vibration towards/away from our view)?

Consider a simple pure sine case of Y vibration at 2n (2nd harmonic, upper right animation in the gif) and Z in 1n. Besides I imagine a superposition of the harmonics, there is also the fact that the Z vibration will affect the length and tension of the string and thus the Y vibration (and vice-versa).

In a real world string of course, we'd never see a standing wave only in Y, but in a radial (meaning Y-Z vectorial) direction at any given moment. I can also imagine this vector rotating at some rate as well (precessing?).

A further complication seems that the longitudinal tension of the string would not actually be uniform but could have its own "tension wave" in X.

I'm wondering what effect all this has on the frequency content of the string (harmonics and inharmonics or side-bands).

Or am I way over-complicating reality here?

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First point: The wave equation is a linear approximation. This does not mean other than the amplitude has to be so small, that it is valid by neclecting any term nonlinear in the amplitude.

Second point: The linear system has independent modes as fundamental solutions. In the linear approximation, the fundamental modes don't interact, meaning their superpositions oscillate indepently.

For a simple 3d elastic chain there is one longitudinal mode and two transversal modes.

For periodic elastic chains with 1,2,3 different masses and spings between them, there are many more fundamental modes, seen as phonon modes in solid state physics.

It may be very difficult, to pepare an extended mechanical system to oscillate in a pure mode, but one my always abstract from small perturbations of other modes.

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  • $\begingroup$ Thanks for your answer. It's a little beyond my reach ATM, but I'm currently dusting off my linear algebra & PDE skills. Can you point me towards equations that do a better job of describing these types of vibrations than a simple Fourier series? $\endgroup$ Jan 24 at 16:12

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