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This is inspired by this music.SE question, which I attempted to offer an answer for, but the more I think about it I am still quite unsure about it.

enter image description here

As physicists, we often try to describe features of experimental data using minimal models to capture the essential details of the system. I'm hoping to get some insight into what minimal physical mechanisms can be used to explain the basic features in this plot. The author of that question says that they plucked the E string of the guitar (82 Hz), and in the spectrogram, we see peaks at all of the integer multiples of this frequency. Here is what I think I can explain:

  1. The red line shows a heuristic fit for the peak amplitudes decreasing as $f^{-4}$, which makes sense: as shown in this physics.SE answer, the peak amplitudes of the Fourier components would be expected to decrease as $f^{-2}$, thus the power (amplitude squared) goes as $f^{-4}$. This is demonstrated by the red line, and gives a reasonable enough fit to the amplitudes for the third and higher harmonics.

  2. The broadening of spectral lines is a common occurrence due to any non-linear dispersive effects such as taking into account that the string tension is a function of amplitude and frequency.

  3. The minor peaks that appear between the larger peaks are likely resonances of the other strings of the guitar, for example just after the second harmonic the first peak appears to be the G (196 Hz) string.

Features I don't understand

  1. Should these peaks be Lorentzian or Gaussian? In trying to produce a similar toy graph I found that I needed both a Gaussian and Lorentzian part of each peak to get a roughly similar-looking plot. What physics would contribute to these two different channels? The peaks appear to be Gaussian (the Lorentzians are too sharp whereas these are rounded near the top), while the background may be some sort of Lorentzian tail, but now I am thinking perhaps it has a different explanation.

  2. How do we explain the "broad continuum" at low frequencies? Why does this broad continuum appear to decay as $1/f^2$ (see the black line)? My toy model with Lorentzian tails does not reproduce this behavior.

  3. Lastly the original question from the music.SE post, why are the first and second harmonics reduced in intensity? In my answer to that question, you can find my speculations. Now after thinking for a while I feel that I am most convinced by the idea that the string was plucked off-center, along with possible resonances that might enhance some of the higher peaks, e.g. the 2nd and 3rd.

As an example of what I mean by Lorentzian and Gaussian, this is what it looks like:

enter image description here

on the left I show both the Guassian and Lorentzian profiles, while on the right I show their sum. It's a bit hard to see, but the Lorenzians are sharply cusped near the maximum, while the Guassians give a nice round peak the way we see in the spectrogram. Clearly the $1/f^2$ behavior of the low amplitude background is not reproduced by the sum of Lorentzian tails (black line).

Bonus points if anyone has a nice quasi-particle analogy.

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Your interpretations all make two basic mistakes. You assume the recorded data was mathematically accurate, and the FFT algorithm used somehow produces "exact" results.

Some of the "broad spectrum" at low frequencies is most likely just environmental background noise. The signal to noise ratio compared with the peak amplitude is around 40 dB which is as good as you are likely to get unless you make the recording with professional-quality equipment and/or in an anechoic chamber.

The A/D conversion will also introduce quantization noise. Just because a "CD quality" signal is 16 bit data, that doesn't mean all 16 bits are accurate for every sample.

The FFT algorithm will be using a finite sized data window (probably with the number of points a power of 2 like 8192) and will be using a "window function" to eliminate the glitch caused by the fact that the frequencies in the recorded data are not exact multiples of the length of the data window. This smears out the width of the FFT peaks and fills the gaps between the peaks with non-zero data, even if the signal itself was synthesized from "perfect" sine waves.

The sound from a guitar decays over time, but the FFT algorithm assumes it does not and the finite length sample can be repeated indefinitely. Therefore making an FFT is a compromise: if the data window is short, the amount of decay is small but the resolution of the FFT is low, and if the data window is long, the FFT resolution is high but the decaying data does not contain "sharp" spectral lines because of the decaying signal amplitude.

There are probably a few more similar factors which I forgot to mention, but since we don't know exactly how the audio was recorded, it isn't possible to go beyond general ideas and into specific details to explain everything shown on the plot.

It does make sense to give a physical explanation of the 4dB/decade decay rate. If you consider the initial deformed shape of the string to be a triangle with the peak displacement at the plucking point, you can explain that by considering the FFT of the triangular displaced shape.

The 2dB/decade decay of the "noise" is probably just an artefact of the FFT algorithm. For a particular choice of data window length and windowing algorithm, the artefacts either side of a peak are at a constant level below the peak (typically around 40dB). The bandwidth of the artefacts around the peak is a fixed ratio of the peak frequency, but since the peak frequencies are a constant frequency difference apart, the artefacts overlap and add up more for higher frequencies than lower ones. The end result is the apparent "slope" of the noise being about half the slope of the peaks.

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  • $\begingroup$ Fascinating, I had not considered this, certainly, a proper analysis would require high-quality recording equipment, low background noise, and proper background subtraction. Any insight one whether we might expect Gaussian/Lorentzian behavior of the peaks or about why the fundamental peak appears to be significantly reduced? $\endgroup$ – Kai Jul 10 at 15:55
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    $\begingroup$ The damping ratio is different for different modes. (If there was no damping, there would be no sound produced by the string, or more accurately by the vibration of the entire guitar body for an acoustic guitar!) BTW if you want to produce better quality data for the frequency peaks (and damping ratios) Fourier analysis is the wrong tool. Methods based on autocorrelation functions don't need a windowing function, can measure both frequency and damping for each mode, and can track things like the change of modal frequency with amplitude for a real vibrating system. $\endgroup$ – alephzero Jul 10 at 16:03
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    $\begingroup$ For the "Gaussian/Lorentzian behavour of the peaks", you need to read a book on practical signal processing and topics like "system identification" in control theory or experimental modal analysis - which are very different subjects from "theoretical uses of Fourier transformations in physics". $\endgroup$ – alephzero Jul 10 at 16:06

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