Speed of an air molecule when transmitting sound of frequency f

[luckis11]

The speed of sound in the air is constant no matter what its frequency f is. Fine. But what is the speed of an air molecule which transmits that sound of frequency f? The higher the frequency, the higher the speed of the molecule, right? What is the equation which gives this velocity? (and a couple of arithmetical examples).

 

[negitron]

The net velocity is zero.

 

[diazona]

The speed of sound in the air is constant no matter what its frequency f is. Fine. But what is the speed of an air molecule which transmits that sound of frequency f? The higher the frequency, the higher the speed of the molecule, right?

Not really. Sound waves are pressure/density waves, and it takes a really large number of molecules to have a meaningful pressure or density. So the speeds of individual air molecules carrying sound waves are pretty much irrelevant; only the average properties of those large numbers of molecules matter.

 

[luckis11]

If you don't know the answer you'd better not fill the thread with nonsense.

 

[luckis11]

According to some equations I have seen, the speed of the molecule is not constant. So I want its average velocity of which its Δx and Δt are defined by two successive collisions of the molecule with its neighbour molecules.

See how the sound waves are produced by the collisions of each molecule with its neighbour molecules:http://www.surendranath.org/Applets/Waves/Lwave01/Lwave01Applet.html
Each molecule corresponds to one moving dot of that simulation.
Set the "frequency" botton at a higher frequency, and you will see that the average speed I am asking for, increases as the frequency increases. However that simulation might not be accurate for this case, because the speed of each wavefront seems to increase as the frequency increases, whereas in the case of sound waves in the air is the same no matter what the frequency is. But what is the cause of a higher frequency? It must be that the molecules are running faster, or else what causes it? Besides, the higher the frequency, the higher the energy and pressure of each wave (although this might not proove the higher speed of the molecules).
That simulation is also confusing, as the problem has the extra complication that the average speed of ALL the molecules of the air is defined by the temperature, which implies that their speed does not increase as the frequency increases (since the temperature remains the same irrespective of the frequency) , but this conclusion might be wrong because not all molecules transmit the sound wave. And we are referring to the molecules which by definition, their motions and collisions cause the particular sound wave. The more molecules transmit it the higher the intensity?

So, I want the equation which gives that average velocity, in relation to the frequency.

Do not quote my posts, as I usually edit them correcting my mistakes. Correct my misconceptions if any, without quoting, unless if necessary.

 

[Nick Bruno]

id suggest lookin thru a heat transfer book. may find something interesting there...
at least with the MFP or mean free path?.. just a suggestion.

 

[luckis11]

The molecule goes back and forth f times in 1 sec, traversing at each of these f times a distance equal to 2A (where A the amplitude), as it traverses twice the amplitude A if every period T, and f=1/T.
So (in order to find the average speed in relation to the f), I only need to find the equation which gives the A in relation to the f.

correct?

To add to the confusion, it seems they have given two different meanings for the term amplitute: I am refererring to the meaning of the distance, the displacement of the molecule between two succesive collisions.

 

[AIR&SPACE]

They're right. A sound wave is a pressure wave. The particles bounce into each other and then come back to their original place. Since the net displacement is zero, then the average velocity is zero.

http://paws.kettering.edu/~drussell/Demos/waves/wavemotion.html

Longitudinal wave is what you're looking for.

 

[negitron]

There is no definite relationship betwen amplitude and period (or frequency).

 

[AIR&SPACE]

The molecule goes back and forth f times in 1 sec, traversing at each of these f times a distance equal to 2A (where A the amplitude), as it traverses twice the amplitude A if every period T, and f=1/T.
So (in order to find the average speed in relation to the f), I only need to find the equation which gives the A in relation to the f.

correct?

Sorry, started my last post before this was up.

Let me see if I understand you. You want the magnitude of the energy of the molecule.

I say that because if we assume ideal gas, the collisions take zero time to occur. There is no acceleration/deceleration period. If we could graph the velocities it would be non-differentiable. It would look like abs(x) @x=0. And I don't believe there is any negligible energy loss between interactions. There should be radiation energy-loss occurring but that's the negligible loss occurring.

So since the energy transfer is equal, the particle is given a velocity (based on it's mass) instantaneously. This velocity should stay the same until the next interaction.

Now then, tell me if I've got this next part right. YOU want to know if there is a correlation between the energy of the particle between interactions and the frequency of the sound. Correct?

 

[Lok]

Usually the average speed of gas molecules is close to the speed of sound in that medium.

Even though there is no apparent movement while sound gets transmitted, there is an increase in speed followed by a decease inside the wavefront. It is necessary or no information ( momentum, energy, sound) will be transmitted.

The individual molecules travel faster with the increase of freqency but only locally, the speed of sound (of the wavefront) remains the same for the whole sound range.

 

[luckis11]

There is no definite relationship betwen amplitude and period (or frequency).

The amplitude A (the distance in which the molecule is moving), does not depend at all on the frequency and thus does not depend at all on the wavelength λ? And reversly, λ does not depend at all on A? How can this be? Then what causes the different frequencies-wavenengths? Only the speed of the molecules no matter how much the A is? And the A can be however small or large without this affecting how much the λ is?

c=λf, thus λ depends on f. Thus if A does not depend on f, it does not depend on λ either.

 

[russ_watters]

The molecule goes back and forth f times in 1 sec, traversing at each of these f times a distance equal to 2A (where A the amplitude), as it traverses twice the amplitude A if every period T, and f=1/T.
So (in order to find the average speed in relation to the f), I only need to find the equation which gives the A in relation to the f.

correct?

No, not correct. You are confusing amplitude and wavelength. If what you were suggesting were true, then every sound wave of a certain frequency would have the same amplitude. Turning up the volume on your stereo will show you that that isn't true. And if the speaker driver doesn't have a cover, you can actually see its amplitude change as you turn up the volume. Perhaps more to the point, the speed of that speaker driver is well below the speed of sound. If it wasn't, it would destroy the wave it was trying to create.

Amplitude in a sound wave is not measured in units of distance, it is measured in units of pressure.

Your initial question was what is the speed of an air molecule when transmitting sound and that question was answered correctly as the speed of sound. But you also said this:

According to some equations I have seen, the speed of the molecule is not constant.

What equations? Where? What you are trying to describe and saying you have seen - if it existed - would already have given you the answer you were looking for!

Now from your last post:

Then what causes the different frequencies-wavenengths? Only the speed of the molecules no matter how much the A is?

What causes frequency? The frequency of a sound wave is the frequency of oscillation of the driver. It's cause has nothing to do with the speed of the molecules or the their amplitude.

Longitudinal waves are tricky this way, so maybe it would help to read up on them some more: http://en.wikipedia.org/wiki/Longitudinal_wave

 

[luckis11]

At page 539 of the "university physics by Hugh Young", at drawing No 19-5, regarding longitudinal waves, it is stating clearly that the amplitude A is the distance in which "the particle" is moving from its equilibrium position till one of the two edges of its displacement. I.e. it travels back and forth within a distance equal to 2A. And in that drawing 19-5, the speaker is also vibrating within a distance equal to 2A. And this 2A, IS NOT the wavelength λ, which is the distance between two successive wavefronts. In that drawing 19-5, the distance λ is greater than the distance 2A. Also, the definition of amplitude in a "harmonic oscillation" is also defined as the above dispacement of the moving object, thus the moving object also corresponds to an air molecule. Each molecule is regarded as not moving with constand speed but accorsing to the harmonic equation equations. I have a small doubt that they might refer to the motion of each wavefront, although all the above clues suggest that they are referring to each molecule. Indeed, even the vibration of the atomic nucleuses within each molecule, are regarded as moving according to harmonic oscillation.

I now see that the molecule traverses a distance equal to 4A (and not 2A), f times in each sec. Thus its average speed is 4Af/sec (sorry, f has already the sec in its denominator thus it's 4Af). Why is this wrong?

I doubt that the speed of sound equals to the speed of each air molecule which transmits this sound, for reasons some of which I mentioned. For example, in that simulation, the speed of the circles increases as f increases. But I repeat that this simulation might be misleading, as the speed of the first wavefront (which is the speed of sound) increases as the f increases, whereas the speed of sound remains the same as the f increases.

You are correct that frequency does not depend on the maximum distance that the speaker is traversing. Thus the distance of the speaker in that drawing 19-5 refers to another vibration of the speaker. So it seems that amplitude has two different definitions. So I am referring to the definition I mentioned, which however, does not identify with the wavelength λ. But wtf, the wave equations have the same symbol A for both of these two definitions? Hm, the two A's might identify. If they identify, then the A does not depend on f, and thus it does not depend on λ either, so...?

The more loud the sound, the greater the distance the speaker is traversing. This also does not affect the speed of the molecules? And no matter how loud the sound, its speed remains constant. If it doesn't affect the speed of each molecule, then how does the pressure and energy of the wave increase? Because (and only because) the number of the molecules which transmit the sound increases? Or only because the number of molecules concentrated in each wavefront increases?

Here's an important clue: Before we switch on the speaker, the air molecules were colliding with the average speed defined by the air temperature. So after we switch on the speaker, we have collisions of the molecules of the speaker with the molecules of the air. So we have a significant change of the speed of the air molecules which transmit and cause the sound, to the direction of the sound, as each air molecule exchanges velocity with its neighbour molecule? Or this change is insignificant? But the simulation at the link I gave above, shows that the waves were produced exactly because there was a change in the speed of the molecules, because of their collision with the speaker, as they were still before that.

An equation which refers to harmonic oscillation says that the maximum speed (of the moving particle) is ωΑ=2πfA. This indicates that the speed of each molecule increases as the frequency f increases, but is this conclusion correct, and also correct regarding the average speed I am asking for?

Anyways, it seems that 4Af is correct, but the question is, does that A decrease as f increases? If yes, it decreases that much so that the speed of the molecule does not increase as the f increases? But then, A does depend on λ and f, so the loundness of the sound defines λ and f? Or the two A's do not identify... here we go again. ARE A AND λ RELATED OR NOT AND IF YES HOW? In the wave equations they are related, but there are extra variables in there (x, y, and/or t) which I must get rid off.

 

[vin300]

The molecule goes back and forth f times in 1 sec, traversing at each of these f times a distance equal to 2A (where A the amplitude), as it traverses twice the amplitude A if every period T, and f=1/T.
So (in order to find the average speed in relation to the f), I only need to find the equation which gives the A in relation to the f.

correct?

To add to the confusion, it seems they have given two different meanings for the term amplitute: I am refererring to the meaning of the distance, the displacement of the molecule between two succesive collisions.

Amplitude is the maximum distance a particle in the wave travels from its mean position and does not affect any of wavelength, frequency or velocity. The only difference it makes is in the intensity given by
I=2(pie)^2A^2df^2v
The frequency is the number of times the particle oscillates between two extremes of its amplitude so it need not make a difference in the amplitude.Wavelength is the distance between two particles in phase.
If according to these considerations you are able to represent a transvese wave as longitudinal or vice-versa it makes no difference except that the particle vibrations in a longitudinal wave are exactly in and opposite to the direction of wave propogation while in a transverse wave are both in the direction of the wave and perpendicular.

 

[luckis11]

Have you got a link showing that A and λ do not depend at all on each other?

 

[vin300]

Have you got a link showing that A and λ do not depend at all on each other?

First speak about the equation relating A and lambda

 

[vin300]

c=λf, thus λ depends on f. Thus if A does not depend on f, it does not depend on λ either.

Plain right. If the amplitude is zero, then there is no wave because it is a basic property of any wave.
You mentioned it is 4A(and not 2A). This is because the particle travels from mean to the positive extremity, reverse, mean to the negative extremity and reverse back to the mean, each constituting 1/4 of one complete oscillation, because after this, the particle travels from mean to the pos...(cycle repeated). after vibrating a length of 4A and changing its starting mean position to a distance equal to lambda, the particle vibrates in the same phase again(same manner, same velocity).
Next,they are basically vibrations, so they do not have a constant velocity but follow the oscillatory displacement equation:
x=Asin(omega*time + phase constant)

 

[luckis11]

Yes, but there is the extra variable x in the last equation you mentioned.

I need the proof that A does not depend on f, or if it does, how.

Anybody knows any more advanced physics forums?

 

[vin300]

Yes, but there is the extra variable x in the last equation you mentioned.

The extra variable is nothing but particle displacement.
The velocity is omega*sqrrt.(A^2-x^2)
After all that explanation tell me what tells you there is a link between A and f?
The average velocity is the average of the maximum velocity which is at the centre, A*omega and the minimum velocity 0 at the Amp.,A*omega/2 which is at x=sqrrt.(3)/2A and not x=1/2A as it is not uniform but oscillatory motion.
You got the relation between velocity and frequency
How do I get you a relation between A and f if it does not at all depend on f?

 

[kanato]

Yes, but there is the extra variable x in the last equation you mentioned.

I need the proof that A does not depend on f, or if it does, how.

Anybody knows any more advanced physics forums?

If you think about it, there are two degrees of freedom when it comes to a single sinusoidal sound wave; a sound's tone and its volume. Sit down at a piano. Select any key. That key plays notes with a constant tone (frequency). You can press as hard or as soft as you want. The volume changes, but the tone does not. You can vary the amplitude (volume) of a wave without changing its frequency (tone). Or you can change keys, and change the frequency of sound you produce, but you can generate the same amplitude of wave with either piano key. So there's empirical proof that frequency and amplitude are independent.

 

[luckis11]

I told you what is the conclusion I am interested of. It is the title of this thread. I want to conclude (or to reject) that the higher the frequency, the higher the average speed of an air molecule that its collisions cause this sound, whereas the speed of that sound remains the same no matter how high the frequency is. You are saying that this is obvious. Man, it is not obvious, otherwise why so many people say that the speed of sound is equal to the speed of the molecule?

The average speed of the molecule is 4Af. Therefore if A and f are not dependent at all, then this conclusion is correct. But I fear whether A gets smaller as f increases, so that the speed of the air molecule does not increase as the frequency increases.

I don't see any clue that suggests that A and f depend on each other, otherwise I would have an indication of how to solve the problem. But I also can not be certain that they do not depend on each other. If I do not see it prooven clearly through the relative equations, I cannot be sure. In that simulation I mentioned, A and f are shown as independent, but that simulation has the mistake that the speed of the firts wavefront (the speed of sound), increases as f increases. I think I read somewhere that this indeed happens in the case of the sound in solids, but in the air its speed does not increase as f increases.

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?

Something is wrong in all this. When A or/and f is zero, the molecule already has a speed. So its average speed is not 4Af? But IT IS 4Af.

 

[rcgldr]

The speed of air molecules is much faster than the speed of sound.

For the specific example of dry air at 20°C, the speed of sound in air is 343 m/s, while the rms speed of air molecules is 502 m/s using a mean mass of air molecules of 29 amu :

http://hyperphysics.phy-astr.gsu.edu/Hbase/sound/souspe3.html

more links:

http://www.ems.psu.edu/~bannon/moledyn.html

http://en.wikipedia.org/wiki/Acoustic_velocity

Getting back to the original question, what I don't know is if the average kinetic energy of the air (temperature) is affected by sound waves. Do sound waves just organize the otherwise random movements of air molecules without any change in average kinetic energy (temperature)? I'm also not sure how shock (super sonic) waves, which have the low pressure part of the wave clipped at zero pressure (negative pressure doesn't exists in the real world), affect the air.

 

[luckis11]

502m/sec is their resultant velocity (when not transmiting sound). Thus its component velocity at each of the three axises should be (502m/sec)/3^(1/2)=289.8m/sec. Correct?

Their speed at the direction of sound is only one of the 3 component velocities, at only one of the 3 axises xyz. Correct? (This is also weird if it implies that the speed of sound is greater than the component velocity of the molecule at the direction of sound. But I guess it does not imply that, but that after sound starts, the 502 velocity increases, so the 502 refers to before the sound starts?)

And that component velocity should be 4Af.
THIS IS IMPOSSIBLE if A and f are independent.
It means that when it does not transmit any sound, f=0, thus its speed is zero at one of the three axises. Correct?

ANY PHYSICISTS IN THIS FORUM? THERE MUST BE SOME!

 

[rcgldr]

This is also weird if it implies that the speed of sound is greater than the component velocity of the molecule in the direction of sound.

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.

 

[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.

http://www.Newton.dep.anl.gov/askasci/chem03/chem03448.htm

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

 

[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.

 

[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.

 

[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.

 

[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
$$\sqrt{\overline{u^2}}=\bar{u}\sqrt{\frac{3\pi}{8}}$$

---------------------------------------------------------------------------------
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 $$\sqrt{\frac{P}{\rho}}\left(1+\frac{R}{c_va}\right)$$
----------------------------------------------------------------------------------------


Anyways it appears to me that you should be able to substitute around and solve for molecular velocity