# Why do vibrations of a string create inharmonic noise past a certain amplitude of excitation?

When a real world string is plucked hard, like a guitar string, it enters a very brief period of noise generation after the release of the plectrum. Analysis of the output appears as pure chaotic noise. Then the harmonic content follows and you hear a musical note evolve.

What relationship between the increased tension or kinetic energy from high amplitude vibrations and the transition to noise generation would exist? How would you predict at which point the noise generation begins and where it transitions from harmonic to noise and back?

• Are you wanting how the material changes behavior as the tension changes? Aug 25, 2020 at 4:27
• So let me give you the same hint in a different fashion: check out Hooke's Law. Aug 25, 2020 at 4:46

When you really dig into a string with a pick and pull the string far off-center with it, the string is not in a sine-wave shape or anything like it: it is two lengths of straight string with the pick tip at the point where they meet. Those segments, along with the trace of the string in its unperturbed location, form a triangle.

Then, when the pick starts sliding off the string, a short burst of pick noise is generated, followed immediately by a lot of high-frequency vibrations that travel away from the pick release point towards opposite ends of the string as the sharp kink in the string caused by the pick "snaps back". Those high frequency waves are not even multiples of the string fundamental and hence sound harsh and strident ("anharmonic"), and they propagate back and forth along the entire length of the now-released string until the components of that noise which aren't harmonics of the fundamental get suppressed by cancellations and die out.

By this point the string has begun vibrating at its fundamental, the initial burst of high-frequency, random crackle has been quenched, and what remains superimposed on the fundamental are all the higher harmonics that were left over.

High-speed videos of bass players slamming a roundwound string with a pick show this effect clearly.

• +1 for the pick noise "riding" over the natural harmonics, but I'd also add that some of the differential equation solutions that give nice results for standing waves are based on small-angle approximations with regards to initial deflections. If you pull a string hard enough then the "clean" solutions aren't valid and the system doesn't respond like a tuned resonator. That is, a very hard pluck doesn't just add noise to the system, it changes the response of the system. Aug 25, 2020 at 13:16
• agree; how large would those pick excitations have to be to invalidate the small-angle approximations? Aug 25, 2020 at 15:58
• You get to 1% error at 8 degrees for the $cos{\theta} \approx 1$ approximation at about 8 degrees and for the $sin{\theta} \approx \theta$ approximation at about 14 degrees. Assuming 3 inches from the bridge to the pickup, those would be vertical distances of 0.42 and 0.73 inches, respectively (10.6 and 18.5 mm at 76.2 mm from the bridge). In looking for videos online to show this, though it looks like you would likely get distortion from fret buzz first. Aug 25, 2020 at 17:58
• I don't know how either. Aug 27, 2020 at 4:29
• @mike, do you plan to publish your results? Aug 27, 2020 at 6:03