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From acoustics we know that sound is an elastic pressure wave. What our human senses perceive are "loudness" and "pitch" which are related to the amplitude and the frequency. While pitch can easily be expressed with the regular units of $Hz$ (e.g., the convention A440 means that the note $A_{4}$ has a frequency of $440 Hz$, and all others can be obtained through the Well Temperament) this is not the case for loudness. The most common units for loudness are decibels $dB$ which are defined through a logarithm on the pressure.

I find this to be counter-intuitive. I think that expressing sound waves in units of pressure $[P] = \left[\frac{F}{L^{2}}\right] =\left[\frac{E}{L^{3}}\right]$ would make more sense:

  1. There is a critical amplitude for which the valley reaches the negative atmospheric and you have a vacuum (if I had known this previously I would have asked my professor if this was the maximum value for the amplitude, or if wave becomes asymmetric beyond this point)

  2. The critical value changes with different atmospheric pressure, for example, it would be lower at higher altitude. From this follows that loud sounds are different at different altitudes.

  3. It would make it more compatible with the standard units for luminosity in electromagnetic waves. The lumen $lm$ is proportional to the radiative power, rather than some logarithm.

Why is the use of decibels so widespread when these points make it clear that using pascals or psi would be better? Am I missing something?

Thank you very much.

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    $\begingroup$ For the decibel scale the reasons are mostly historical and for good visualisation, I think. However, many of our human senses, including hearing, are logarithmic. Therefore a logarithmic scale has an almost linear relationship to the perceived intensity of a stimulus. Keyword: Weber-Fechner law. $\endgroup$ – scaphys Apr 22 at 20:34
  • $\begingroup$ @scaphys I whink I had already read something about the logarithmic perception of stimuli, but it still feels somewhat unsatisfactory. Human experience is one thing, but we cannot let it hinder our understanding of physics. $\endgroup$ – S V Apr 22 at 20:39
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    $\begingroup$ @SalvadorVillarreal It's not at all hindering our understanding since we have a very precise definition of soumd intensity, while loudness is a completely subjective phenomenon. $\endgroup$ – Gabriel Golfetti Apr 22 at 20:43
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    $\begingroup$ Well, a logarithmic scale doesn't really hinder our understanding. The logarithm is bijective on the positive reals, so it doesn't really matter that much. Also, mathematics in general is just a human-made thing with more or less arbitrary definitions. The models work, and that's enough. Some other lifeform might perceive and model the universe totally differently, resulting in the same physical predictions. I don't think there is anything wrong with that. $\endgroup$ – scaphys Apr 22 at 20:45
  • $\begingroup$ It wouldn't make much sense to use units of pressure. The pressure difference fluctuates between negative and positive values, and its average value is zero. $\endgroup$ – Ben Crowell Apr 22 at 22:41
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The basic reason is that sound pressures span a WIDE range of orders of magnitude, despite that it may not seem like such.

The faintest sound you can hear(*), at least if your ears are good, is taken by standard to be 20 μPa. And this sets the level for 0 dB. That's micro pascals, by the way, or $10^{-6}$ pascal. The loudest sound you can hear is one where the pressure equals the actual pressure of the atmosphere - again, by standard, taken as 101.325 kPa, exactly (so the actual atm. pressure at your own location may be different). You don't want to hear this sound, though: not only would it destroy your ears, but it'd be effectively like getting hit by a bomb blast - quite easily lethal! (You can, of course, make a pressure wave in air that's even stronger - that's what a bomb does very close in - but you can't really call that "sound" anymore. Actually, even this may be "pushing the envelope" as to whether it's sound.)

Suppose though we tried to use instead the micropascal as our choice unit. Then that level is 101 325 000 000 μPa, against 20 μPa. That's kind of awkward to write. Moreover, using scientific notation, $1.013\ 25 \times 10^{11}\ \mathrm{μPa}$, is ugly, and switching to larger/smaller pressure units by changing the prefix makes the comparisons less clear (there's a sort of guideline that you, generally speaking, should stick to a single unit of a particular size for a given physical quantity throughout a context, even if it may result in large numbers and/or imprecise ones perhaps requiring scientific notation along with more reasonable ones, for just this reason). But writing 0 dB and 194 dB, respectively for lowest and highest sound, is much cleaner. One unit and one reasonably compact number.

Logarithmic scales - and actually, the unit "decibel" isn't just for sound, it is a generic logarithmic measure that can be used for just about anything - are quite standard for dealing with quantities that span many orders of magnitude, even ones that have nothing to do with sound: for example, electronic signals in signal processing circuitry like that used in radio communication, are often measured using dB as well, relative to some other reference level appropriate for them (and also that is not a pressure, but typically a power [rate of energy delivery], or voltage, or something of that sort). Astronomy even uses this to measure the brightness of stars, whether actual or as seen in the sky at distance, quantities which also have similarly-great variation, but there, their unit system is a terrible mess and their logarithmic scale is a weird one that not only does not use base ten but also rates things as higher in "magnitude" when they are fainter, not brighter. I really think they should use dB like everyone else, with suitable reference points, but fat chance to tilt at so much entrenched tradition.

(*) People with highly sensitive ears, such as in some manifestations of autistic spectrum disorders and other disorders with sensory processing modification, seem to report the ability to hear sounds much fainter, like a tiny bug crawling on the wall nearby - but I've never heard of a measurement study done thereupon. 20 μPa is the threshold for an average human, not the lowest possible sound such that no existent human can hear lower. Also, if you have tinnitus (as I do), chances are quite good you can't even hear this low.

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The perception of loudness is logarithmically related to the amplitude of the oscillation and not linearly (nor quadratically). We use the dB scale because it is also logarithmically related to the amplitude of the oscillation.

This logarithmic relation also holds for pitch. Consider an A and an E on the piano. The frequency of the A is 440 Hz and that of the E is 3/2 * 440 = 660 Hz (to very good approximation when 'well tempered'.) The frequency difference is 220 Hz. The A and the E one octave higher, are perceived to be the same distance ('a fifth') apart, even though their frequency difference is now 2*660 - 2*440 = 440 Hz.

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    $\begingroup$ In fact, "loudness" has its own unit, the phon. The phon is equal to the decibel for a 1kHz pure sine wave , but at other frequencies is corrected for the average sensitivity of human hearing. There is a third unit, the sone, which is calibrated directly by human perception - i.e. a sound amplitude of $2x$ sones always sounds "twice as loud" as an amplitude of $x$ sones. The point about measurement systems is that they need to be useful, and for psychoacoustic experiments, direct or even logarithmic pressure measurements are not as useful as units based on human perception. $\endgroup$ – alephzero Apr 22 at 21:52
  • $\begingroup$ The first statement: "The perception of loudness is logarithmically related to the amplitude of the oscillation and not linearly" makes it seem as if it has high accuracy. How was this assessed? Which papers back it up? How was the data handled statistically? How reliable/reproducible were the results? I get that human perception could roughly be approximated as logarithmic, but I insist in the analogy I made with light. Why don't we measure luminosity in logarithmic units when we get most sensory information from vision? (30% of the brain is dedicated to vision but only 3% for hearing) $\endgroup$ – S V Apr 22 at 22:24
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    $\begingroup$ We also use a logarithmic scale for brightness, see en.wikipedia.org/wiki/Apparent_magnitude. $\endgroup$ – Georg Apr 23 at 6:01
  • $\begingroup$ As to your quesions "How was this assessed? Which papers back it up? How was the data handled statistically? How reliable/reproducible were the results?", just look at the many references given on Wikipedia, en.wikipedia.org/wiki/Decibel#Perception $\endgroup$ – Georg Apr 23 at 6:12
  • $\begingroup$ Ok, I will concede that point. I did not know such a unit existed. Thank you, @Georg. $\endgroup$ – S V Apr 23 at 16:13
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dB is chosen because it relates more closely to human ear's perception of loadness.

Roughly, although not exactly equal across the spectrum of the frequencies we hear, our ear perceives loudness of a sound when its intensity is increased by tenfold, only twice.

Also a logarithmic scale lends itself well to the huge spectrum of loudness that our ears can perceive.

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