# Is there a limit to loudness?

Is there any reason to believe that any measure of loudness (e.g. sound pressure) might have an upper boundary, similar to upper limit (c) of the speed of mass?

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## 4 Answers

Let's talk about the sound waves in the air first. Physically they are longitudinal waves of pressure. The bunch of air in one place will get compressed (in comparison with equilibrium state) and after that will expand, compressing the adjacent air and so on the wave propagates. These (single frequency) waves are essentially described by three numbers: the speed of propagation (called speed of sound; it depends on the type of the material and the temperature), the frequency of oscillation (determining the pitch), and the amplitude of oscillation.

It is the amplitude of oscillations which determines the loudness. So you are essentially asking whether there is any limit on amplitude of compression. Well, of course. At high enough pressures, the air would freeze even at normal temperatures. So this is the limit for air. Similar thing would happen with liquids: at certain pressures they would condense into solids. You could continue with phases of matter in this way and applying higher and higher pressures you would eventually end up with a black hole. That would be an ultimate limit.

But of course, in reality the limit is set by our engineering capabilities and I doubt it's possible to create sound waves that would be able to freeze air.

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You've completely missed the effects of adiabatic compressional heating. You won't liquify the air by compression, its temperature rises with pressure, so it will remain gaseous, and the soundwave transitions into a shockwave. So it becomes an issue of symantics, is a shockwave a highamplitude soundwave, or is it something different? Of course the local soundspeed scales as the squareroot of temperature, so as the compression ratio of the shock increases its velocity increases as well. – Omega Centauri Jan 5 '11 at 17:10
Of course you can liquify it, but it's true that it will heat somewhat (depending on precise process used). I had intended to discuss also temperature dependence and other phenomena but apparently I forgot to mention it for some reason. – Marek Jan 5 '11 at 18:31

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I really doubt it will liquify. Adiabatic heating is stronger than you imagine. For a perfect gas, T is proportional to density to the 2/3rds power, air has higher heat capacity, so at gamma =7/5ths it only increases at the .4 power of density. That still means a 300x increase in density implies a 10x increase in temperature. Can we get liquid phase on that curve, or will we end up much hotter than the triple point? These pressures are readily reachable via high explosives. – Omega Centauri Jan 6 '11 at 5:31
@Omega: I guess you meant critical point? Sure if the temperature got bigger than that there would no longer be a distinction between gas and liquid but the gas/liquid should still condense to a solid. But I have to admit, it's not clear to me whether at those pressures and temperatures other processes (like nuclear and eventually GR) wouldn't take over. – Marek Jan 6 '11 at 10:34

Yes - there is a sound pressure limit for undistorted sound. Over that limit we have a shock wave. It depends on the environmental pressure, but there is a theoretical limit to loudness which you can find here.

The limit is basically equal to the pressure.

Theoretical limit for undistorted sound at 1 atmosphere environmental pressure 101,325 Pa ~194.094 dB

The lower limit of audibility is therefore defined as 0 dB, but the upper limit is not as clearly defined. While 1 atm (191 dB) is the largest pressure variation an undistorted sound wave can have in Earth's atmosphere, larger sound waves can be present in other atmospheres, or on Earth in the form of shock waves.

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Well, the short answer is: there is, when the hydrodynamic approximation (that fluid is composed of small "fluid particles" i which real particles move in the reference frame of the "fluid particle" like in stationary fluid) breaks.

The upper bound can be approximated with wave amplitude equal to ambient pressure, so that the pressure is going down to 0 in wave minimas (this plus minus corresponds to cavitation); yet this corresponds to loudness of $\infty$dB.

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+1. @mbq what is "weave pressure"? – user346 Jan 5 '11 at 13:49
@space_cadet wave amplitude, which is pressure... But I'll fix it. – mbq♦ Jan 5 '11 at 15:03
lol. you meant "wave". doh. sorry, it was very early in the morning where I live! – user346 Jan 5 '11 at 21:13
"yet this corresponds to loudness of ∞dB."  No, it's 194 dB SPL. – endolith May 8 at 3:59

The only practical limit to sound-pressure might be when the medium were to be compressed into a black hole. Although I do not know about the sound-propagating features of black holes. Long before that the medium would come apart, f.e. into plasma.

After all, you compare two things: an upper limit of a speed with a power. How much power can you put into a particle to speed it up with c in mind?

PS: I'd rather not guess what kind of sound-generator would be necessary. One might ask the band "Disaster Area" (https://secure.wikimedia.org/wikipedia/en/wiki/List_of_minor_The_Hitchhiker%27s_Guide_to_the_Galaxy_characters#Hotblack_Desiato)

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You'll reach a vacuum in the negative direction long before you reach a black hole in the positive direction. – endolith May 8 at 4:00
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