My interest in mathematical art is at an all time low, but this thread is not bad, especially with recent addition of video data for oscillating latex.
I checked some references, and now believe my initial positing is partly rushed / confused / wrong. For a first pass, we can just start with the frequency result for a linear string of mass
$m$ and length
$L$
$f \propto \sqrt{\frac{T}{m L}} = \sqrt{\frac{k}{m}} . $
From this equation we can see that
$f$ is only constant if tension force obeys a simple Hooke Law
$T = k L .$
For small changes in
$L$ a linear approximation will usually apply, but larger changes in
$L$ could explore the curvature of the Stress vs. Strain curve, which does depend on the material. In the above, when I say "anharmonic deformation", I mean that tension
$T$ expands in a power series of length
$L$:
$T = k_1 L + k_2 L^2 + k_3 L^3 + . . . $
The coefficient
$k_2$ does not have to be large to cause some effect. For example, In Large Amplitude Motion of a String Fig. 1 purports to depict linear data, but it looks to me like it could have non-negligable quadratic components in a series expansion in powers of
$L$. As the article says, whenever
$k_i \neq 0$ for
$i > 1$, it is not possible to derive the wave equation.
Anharmonic motion could be solved by dividing the string into many equal masses and iterating Newton's equations using numerical methods, with the additional requirement that the force function between masses is not a simple Hooke's law as is usually assumed. The same approach generalizes to two or three dimensions.
Where I've seen a great many harmonic calculations, I rarely ever see calculations including anharmonicity, so again, could be an interesting / worthwhile direction.
On the subject of anharmonic wave equations, it's also interesting to take note of anharmonicity and quantum vibrations. Harmonic oscillation is often a low-amplitude expectation for vibrational motion, especially in the adiatbatic approximation ( Born Oppenheimer ). But force or potential expansions in nature usually contain higher order terms, easier to notice and measure with laser technology.
One commonly found example in laboratory quantum mechanics is molecular Iodine. Peaks in the vibrational frequency spectrum ( the eigenvalues ) are expected to have uniform first differences if harmonic, but even linear spacing is easy enough to find in laboratory data. This is the basis for the famous Birge-Sponer method.
