# How air molecules still carry acoustic messages when multiple sources present?

• arashmh
In summary, sound waves are pressure waves that propagate through a continuum of air molecules. The speed of sound is determined by a ratio of principal molar heat capacities and the individual molecules' velocities do not directly affect the movement of the ear drum. The ear is able to hear sound because of its sensitivity to organized superimposed velocities rather than the random motion of individual molecules. The concept of linear superposition helps to explain how multiple sound sources can be perceived simultaneously.

#### arashmh

We know that air molecules are subjected to random motion in different directions. A sound wave is a push toward these molecules to make a wave front to make them expand or contract in groups so that we can hear a voice message in our ears.

The question is that what happens when different sources are propagating sound waves at the same time. Every molecule that feels the waves stars moving in a different direction depending on the direction of the source. We have to plus this to the random motion of air molecules as a gas. How this makes a wave that carries the information of multiple sources ? I know all the science about wave addition but from the view point of single molecules moving, it's not that much clear. Can anybody explain this ?

The mean free path is small enough to make air behave like a fluid, at which point you treat it as a continuous medium and go on from there. It doesn't really make sense but somehow it works. Especially for room temp air where the speed of sound is on the order of the average molecular speeds, so yes it is confusing.

We'll see if someone can come out with a good explanation.

You should think of sound as pressure waves and not individual air molecules moving around. For small pressure variations the propagation of such waves can be modeled as a "linear system" adhering to the superposition principle which means that waves propagate independently of each other.

I've often wondered how to treat sound propagation on the level of molecules.

[The specific question I keep coming back to is how to show by direct consideration of molecular motion, that the speed of sound is √(γ/3)xcrms, in which γ is the ratio of principal molar heat capacities. After all, the individual molecules do carry the sound as small superimposed velocities, so a direct kinematic/dynamic approach based on the molecules' underlying random movement (rather than the standard macroscopic approach) would seem possible. I think this amounts to treating sound propagation as a transport phenomenon like transverse momentum transport (viscocity).]

But, returning to your point, the key is surely that ears and microphones 'average out' the random motion of the molecules; they can hardly respond, even mechanically, to individual hits, but they will 'integrate up' momentum transfers due to organised superimposed velocities which constitute sound. If you accept this, then I don't see why multiple sources present any more of a problem than a single source. As you say, you know all the stuff about wave addition.

You could, if you like, regard a molecule's velocity as vrandom + v1 + v2 in which v1 and v2 are due to sound sources. v1 and v2 will be almost the same (I suspect) for many neighbouring molecules. When we average over this neighbourhood group, the random velocities will drop out, leaving that due to the sound.

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I see what you mean Philip, but how can u explain this integral momentum transfer when the average speed of molecules in room temperature is close to that of the sound waves ?

It's easy to show from macroscopic ('air layer') theory that the peak velocity, u, of oscillation of the air layers due to the passage of a loudish sound is far smaller than the speed, v, at which sound travels, and therefore far smaller than the rms speed of the molecules.

It's my contention that we can regard all the molecules in a 'layer' of air between two nearby wavefronts as having a the same small superimposed oscillatory velocity. Hits of these molecules on the ear drum will all have this superimposed velocity, so their combined effect will be to move the ear drum – at audio frequency! Random velocity fluctuations (though far greater in magnitude) will not move the eardrum.

Philip, thanks for sharing ur reasoning. but i don't get "Random velocity fluctuations (though far greater in magnitude) will not move the eardrum."! why ?

Curl said:
The mean free path is small enough to make air behave like a fluid, at which point you treat it as a continuous medium and go on from there. It doesn't really make sense but somehow it works. Especially for room temp air where the speed of sound is on the order of the average molecular speeds, so yes it is confusing.

We'll see if someone can come out with a good explanation.

arashmh said:
I see what you mean Philip, but how can u explain this integral momentum transfer when the average speed of molecules in room temperature is close to that of the sound waves ?

Going off of what Curl started, the mean free path between molecules is small, meaning that the fluid is effectively a continuum. This means that even though the fluid is composed of many particles moving and colliding randomly, the bulk fluid moves with a seemingly uniform velocity. In other words, the fluid appears to be motionless as a whole (or moving uniformly in one direction or any number of flow fields) regardless of the fact that individual molecules are moving wherever they want at great speed. If you took a volume of air, the average of all the velocities of the particles would be the overall velocity of the fluid in that volume.

With that in mind, sound waves act on a much more macroscopic scale than individual particle motion. The propagate through the continuum, not through individual molecules because it relies on the pressure produced by collisions of many, many molecules.

Aside from that, except in special cases, sound waves are linear and therefore obey linear superposition.

The sensitivity of the ear is as high as it practically can be for hearing sound; if it were just a little more sensitive you WOULD hear the constant noise of random molecular banging onto the ear drum. The ear operates just above that threshold.

Going off of what Curl started, the mean free path between molecules is small, meaning that the fluid is effectively a continuum. This means that even though the fluid is composed of many particles moving and colliding randomly, the bulk fluid moves with a seemingly uniform velocity. In other words, the fluid appears to be motionless as a whole (or moving uniformly in one direction or any number of flow fields) regardless of the fact that individual molecules are moving wherever they want at great speed. If you took a volume of air, the average of all the velocities of the particles would be the overall velocity of the fluid in that volume.

With that in mind, sound waves act on a much more macroscopic scale than individual particle motion. The propagate through the continuum, not through individual molecules because it relies on the pressure produced by collisions of many, many molecules.

Aside from that, except in special cases, sound waves are linear and therefore obey linear superposition.

Philip, thanks, ur way of reasoning is interesting. Do u think that we can simulate this sound propagation with multiple sources on molecular scale ? I just want to c the connection between molecular scale and scales beyond

boneh3ad thanks for sharing ur idea

The molecular impacts (forgetting about the superimposed sound) and the eardrum is a essentially a brownian movement problem. The eardrum's mass is much greater than that of a dust-speck, so we wouldn't expect to see it move under a microscope. But I'm sure that what Bahamagreen says is right, that with just a little more ear sensitivity we would hear the movement (as white noise?). [I remember working out on once that the faintest audible 'quantum' of sound had roughly the same energy as that of three or four visible photons - the smallest number you can detect with the eye. A wonderful example of nature's economy.]

I'm sure it should be possible to simulate sound propagation on a molecular scale, because it's via the movement of molecules, and collisions of molecules, that the sound must travel. It's obviously much easier to make calculations about sound propagation by considering macroscopic 'layers' containing huge numbers of molecules, but the basic process must be molecule to molecule. Unfortunately I'm not clever enough to do the modelling.

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Philip Wood said:
After all, the individual molecules do carry the sound as small superimposed velocities.

At 25°C, the average dry air molecule will change its speed and its direction of movement some five billion times a second. It is impossible for any individual molecule to carry any sort of meaningful signal. Instead, sound waves involve collections of molecules whose mean direction of movement and speed carry the signal. With some 1025 molecules per cubic meter at NTP, there are more than enough collections to go around.