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what is the significance of the shape of a crest. For example, in the figure below, we can see that one graph of sound pressure shows sharp crests while the other shows smooth crests. How does this affect the sound quality or any other quality? To be clarify, the graph with sharp crests is from a steel plate oscillating while the one with smooth crests is from a aluminium plate. enter image description here

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A single frequency has a distinct mathematical shape function, sin(wt-kx). This is a smooth function. These are also solutions to the wave equations so any solution, e.g. phenomenon, can be described by a suitable expansion or superposition in terms of sine functions with different frequency, w, and/or wave number, k. The lower curve "looks" closer to pure sine wave but that doesn't mean it really is pure. In contrast the upper one has, as you mentioned, sharp jagged peaks and appears to have a higher frequency to these jagged peaks. This could be a couple of things and you haven't really provided enough data to determine what it is. While it may be safe to say there are some higher frequencies contributing to the plot the cause is not obvious.

It could be higher harmonics of the plate excited by how the plate was set in motion. Did you strike it with a hammer, or drive it with a piston placed on the plate?

It could be random noise, white Gaussian noise in the detector used to gather data.

It could be a sampling artifact although you claim to have the same frequency in both. It would help to know what that frequency is, e.g. the driving frequency or resonance frequency of each plate. If the former then identical plates made from different materials would have different harmonics and the Aluminum plate may be tuned to the driver while the steel is not tuned and hence exhibiting some combination of modes superimposed on the driving frequency.

It could be ringing of the plate against some other component of the system. There could by some part of the experiential set up that is not controlled.

There could be a defect in the steel, a crack or something causing rubbing that shows up as noise.

To know what it is it would help to perform an FFT on the time series data and look at the spectrum. The true harmonics of each plate are predictable using simple formulae. Noise would show up with a uniform distribution in frequency.

As for the effect of quality of sound. If the peaks are due to harmonics it would not necessarily be "dirty", it may be bright. Typically a pure tone sounds dull to the human ear. The more higher harmonics there are in the spectrum the "brighter" the sound is. You would get this from a guitar by plucking the string closer to the bridge. Whereas plucking closer to the center you get a spectrum more like the bottom plot. If it's noise then it will sound dirty or raspy. However I would not venture to correlate any of these terms just by looking at the time series data alone. If you did the same experiment with two aluminum plates struck differently you would get different results. The steel also seems to have a large starting amplitude which implies a percussive strike in addition to being driven.

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The lower plot would have almost all of its energy concentrated at one frequency and its sound would be a pure clean tone at that frequency. Whereas the upper plots would have higher frequencies in addition to the 'center' frequency and its sound would not be pure but would be slightly distorted from a pure tone. These harmonics could have a variety of undesirable effects: sound quality degradation, noise generation, heating loss, power loss, etc...

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    $\begingroup$ ...For some definitions of "pure" and "clean" and "distorted." There are some sounds (e.g., the sound of a bell, the sound of a plucked acoustic guitar string) which can have rich harmonic content, but which most people would not describe as "dirty" or "distorted." $\endgroup$ – Solomon Slow Sep 9 at 18:33
  • $\begingroup$ True--edited my answer. $\endgroup$ – user45664 Sep 9 at 21:58

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