# 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).

• Note that exponential decay is normally expressed in terms of its time constant, which is the time that it takes for 63% of the decay to take place (e.g., $1-e^{-1}$). Due to this, the standard solutions to this problem are not formulated as a function of half-life. Are you open to the idea of getting an answer in terms of time constant? – David White Aug 27 '19 at 2:50
• @DavidWhite Actually yes, I'm glad you mentioned this because I started looking into it today as an better measure of comparing the decay, so I'm hopefully going in the right direction (?) and I think I understand it enough to find an answer involving it. – celeste Aug 27 '19 at 3:01
• Welcome to Physics SE :) Your curiosity and description will make it a great place to stay here. – Stefan Bischof Aug 27 '19 at 5:05

$$\frac{\partial^2y}{\partial t^2}y +2b\frac{\partial y}{\partial t}=c^2\frac{\partial^2y}{\partial x^2}$$ 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.