# String vibration dimensions

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

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?