Is there a relationship between the frequency of a standing wave on an oscillating string and the half-life of its amplitude as it decays? I'm asking in the context of an experimental exercise where we used a speaker to create a standing wave on a string with constant tension, then turned off the speaker and allowed the amplitudes to decay over time. I'm trying to see if increasing the frequency in order of the harmonics appearing affects how long it takes the amplitude to reach half of the initial amplitude.
I am a truly amateur physics student and am assuming an ideal string that decays exponentially, and I definitely do not know very complicated approaches to this problem through math. I really just want to know if I am chasing something that either doesn't make sense or is only approachable through college level physics/math. I would greatly appreciate any help, and I apologize if this is too rudimentary for this forum (first time using it).
 A: The simplest model for the vibrations in a damped string is the damped wave equation:
$$ \frac{\partial^2y}{\partial t^2}y +2b\frac{\partial y}{\partial t}=c^2\frac{\partial^2y}{\partial x^2} $$
where $y$ is the wave amplitude, $t$ is time, $x$ is the position coordinate along the string, $c$ is the speed of the waves in the string (which depends on the string material and the tension) and $2b$ is a damping consant. You don't need to understand exactly what this equation means and how to solve it, but it models the damping force acting on a unit length of each piece of the string as being proportional to its speed, and $2b$ is the proportionality constant. For a string of length $L$ fixed at the end points $x = 0$ and $x = L$, and a sufficiently small $b$, the standing wave solutions are
$$y_m(x,t) = e^{-bt} \cos\left(\sqrt{\omega_m^2 - b^2}t+\phi_m\right)\sin\left(\frac{\omega_m}{c}x\right) $$
where $m$ numbers the different modes ($m=1$ being the fundamental), $\phi_m$ is some constant phase and
$$\omega_m = m\frac{c\pi}{L}. $$
$e^{-bt}$ is the factor governing how the amplitude decays over time, and the decay is exponential as you guessed. However, in this simple model the time constant of the decay is $b^{-1}$ (the half-life is $b^{-1}\ln2$) which is the same for all modes. This may be more or less true for an experiment such as yours, but especially at very high frequencies I suspect this may no longer be the case.
I would expect in most strings the damping force may have a stronger dependence on velocity than what this simple model assumes (such as an additional velocity squared dependence due to air drag), resulting in a greater damping force and a faster decay in amplitude at high frequencies. Perhaps someone will chime in with a more sophisticated model for a damped string, or with experimental results.
