# 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

## 1 Answer

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.

• So what I'm understanding from this (and let me know if I'm wrong) is that regardless of which mode/harmonic, the decay rate will stay the same and there's no relationship between it and the frequency of the oscillating string? Is this the only model applicable to the experiment I'm describing? Also, thank you so much for helping! @Puk – celeste Aug 27 '19 at 12:25
• If the damping force for your string is indeed strictly proportional to the (negative) velocity, then yes, the decay rate won't depend on the mode/frequency. However, I suspect that the dependence of the damping force on velocity is likely superlinear (one reason to believe this is that air drag tends to have a velocity squared dependence). Then the amplitude would decay faster for higher frequency (or "faster") modes. I don't know what kind of damping model is appropriate for your experiment. It may be worth looking into whether anyone has analyzed the nature of the damping force in a string. – Puk Aug 28 '19 at 0:01