# Speed of an air molecule when transmitting sound of frequency f

#### Crashsite

Molecular Speed vs. Sonic Speed

Anybody knows any more advanced physics forums?
First, this is my first post here. Second, I've been wrestling with these same questions over in the, Electrotech Forum. Third, I am a math moron so I need to think of these things conceptually rather than mathematically (throw equations at me and you may as well be speaking Swahili). Fourth, I think you are asking the right question: Trying to nail down how the molecules themselves are acting but, I think that rather than going to a more advanced forum, perhaps simplifying things might be more beneficial?

If the question is how fast the molecules of air are moving, here's a web page that may be of interest.

But, let me quote a passage of interest here:

"There's a really neat mathematical equation based on a theorem called
the "equipartition theorem" which states that the energy of a gas system
(equal to 1/2*mv^2) is equal to the temperature of the gas (equal to 3/2*kT).
If we rewrite this equation to solve for velocity we get:

sqrt(3*T*k/m) = v

where T is the temperature in Kelvin, k is the Boltzman constant = 1.3805*10^-
23 J/K and m is the mass of the gas particle.

If we assume that the average mass of air (since it is a mixture of different
gases) is 28.9 g/mol (or each gas particle is around 4.799*10^-26), and room-
temperature is 27C or 300K, we find that the average velocity of a single air
particle is around 500 m/s or 1100 miles per hour
!"

The reaon I find this particularly interesting as it relates to the speed of sound is by thinking of what the average speed might be in a linear direction.

If an air disturbance is propelled by the collisions of the molecules and the molecules are moving at a nominal 1100 mph, some of the time the sound will be propagted at that speed. But, related to that selected direction, some will be at right angles to that direction and will propagate along that axis at zero mph. Other rates will depend on other angles and should average out to about the speed of propagation at 45 degrees.

That puts the average at about 770 mph along any given axis. To me that seems just a little too close to the nominal Mach 1, under standard conditions, of 761 mph to be a simple coincidence.

#### vin300

When A or/and f is zero, the molecule already has a speed. So its average speed is not 4Af? But IT IS 4Af.
Without A or f, there is no wave at all the molecule has no speed.
study of sound transmission requires "no wind", it disrupts wave

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#### vin300

One more clue: If the speed of the molecule increases as the f increases, then how come and the temperature of the air does not increase? One answer is that sound is not caused-transmited by all the molecules of the air which this sound traverses. Is this true though?
The temperature increases, it is A PRESSURE wave, it is transmitted by pressure differences of the medium it traverses.

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#### vin300

If the speed of sound is organizing the direction of the velocity of affected air molecule, then most or nearly all of their average speed could be in the direction of sound. Also it's an average speed, I don't know how much the instantaneous speed changes as a sound wave passes through a volume of air.
No, it's oscillation, I mentioned it in the previous post.When the wave velocity particle velocity r in the same dirn,it's rarefaction and when opposite,compression. The average speed in all directions is 0 so it does not change the average kinetic energy of the molecules.

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#### vin300

This is going far from the simple oscillatory equations to the kinetic theory. You just have tto superimpose the average initial velocity and the proposed velocity of particle oscillations to get the new velocity.

#### luckis11

According to their logic the average speed is 4Af and the the max speed is 2πAf.
The average speed is wrong, thus the max speed most probably wrong for the same reasons. Because when f tends to zero, the speed does not tend to zero.

Also, if we consider A to be the actual displacement of the molecule, then the factor A is correct. But we consider that when there is no sound, A is zero, thus the factor A is also wrong.

I guess these equations would be correct if the molecule was still when there is no sound.
The molecule is making a difinite number of oscillations (which is not definded by f) within a definite distance (which is not 2A) in each sec, when there is no sound.

Therefore, someone must find the correct equation. If I am wrong show me why.
My mistake could be that we can consider the speed of the molecule as zero when there is no sound, but why-how? Vin, don't say again that the average speed is zero "in all directions", because according to this logic, the speed in the above equations should also be zero.

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#### AIR&SPACE

Therefore, someone must find the correct equation. If I am wrong show me why.
My mistake could be that we can consider the speed of the molecule as zero when there is no sound, but why-how? Vin, don't say again that the average speed is zero "in all directions", because according to this logic, the speed in the above equations should also be zero.
I was just walking back from the library restrooms and noticed a book called, "A Textbook of Sound" and immediately thought back to this thread. I have just picked out some passages that I looked quickly at. Take it for what it's worth.

"A Textbook of Sound" 3rd Rv. A.B Wood cpyrt 1955. G.Bell and Sons Ltd.

Pg 248 - 249 "velocity of small amplitude waves" excerpts

$$\sqrt{\overline{u^2}}=482 metres/sec$$
= root mean square velocity ;

$$\bar{u}=447metres/sec$$
= mean velocity of molecules

The Kinetic Theory of Gases shows very simply that the pressure in a gas is given by
$$P=\frac{1}{3}Nm\overline{u^2}=\frac{1}{3}\rho\overline{u^2}$$

where $$\\N$$ is the number of molecules per c.c., $$\\m$$ the mass of a molecule, and $$\rho$$ the density of the gas.

This relation expresses the molecular velocity in terms of the pressure and density of the gas, thus
$$\overline{u^2}=\frac{3P}{\rho}$$

Since the energy of the motion of the molecules in a given (i.e. x-axis) we may write
speed of sound = $$\sqrt{\overline{u_{x}^2}}=\sqrt{\frac{P}{\rho}}$$

Which is Newton's "Isothermal Velocity" of wave-propogation in the gas. If we employ the mean molecular velocity $$\bar{u}$$ instead of $$\sqrt{\overline{u^2}}$$
we find
$$\sqrt{\overline{u^2}}=\bar{u}\sqrt{\frac{3\pi}{8}}$$

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Pg 275-276 "Velocity of Sounds of High Frequency -- in Gases" excerpt

H.O. Kneser, in a number of theoretical and experimental papers dealing with anomalous absorption and dispersion of sound, has derived the following expression for the velocity in terms of frequency, $$\\w=2\pi\\N$$, and molecular constants:
$$\\v=\sqrt{\frac{P}{\rho}}\left(1+R\frac{c_{v}+w^2\beta^2c_{va}}{c_{v}^2+w^2\beta^2c_{va}^2}\right)$$
where $$\\c_{v}$$ is the molecular heat at constant volume, $$\\c_{va}$$ is the specific heat of the translation degrees of freedom, $$\\R$$ is the universal gas constant, and $$\beta$$ is the mean life of the energy-quantum, that is the time involved in the quantum transformation -- translational-intramolecular-translational energy.

at low frequencies the velocity becomes $$\sqrt{\frac{P}{\rho}}\left(1+\frac{R}{c_v}\right)$$

whereas at high frequencies it becomes $$\sqrt{\frac{P}{\rho}}\left(1+\frac{R}{c_va}\right)$$
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Anyways it appears to me that you should be able to substitute around and solve for molecular velocity