Why Can a Sound Wave Not Travel Faster than the Average Molecule Speed?

[Amadeo]

I am having trouble understanding the following passage in my physics textbook, particularly the bolded sentence:

"The speed of sound in a gas is closely related to the rms speed of the molecules of that gas. In a sound wave, the disturbance is passed from one molecule to another by collisions. The wave cannot move any faster than the "average" speed of the molecules. In fact, the speed of sound must be somewhat less than this "average" molecular speed because not all molecules are moving in exactly the same direction as the wave. As examples, at room temperature, the rms speed of hydrogen and nitrogen molecules are 1920 m/s and 517 m/s respectively. The speeds of sound in these two gases at this temperature are 1350 m/s and 350 m/s respectively."

There is no explanation given.

I keep imagining a bunch of gas molecules, each with zero velocity, uniformly distributed throughout the volume of a container. Then, one of the walls of the container rapidly moves into the volume a small distance, setting the molecules in the volume immediate to it in motion in the direction of impact, thereby creating a wave through which the energy of the impact would be eventually transferred to the rest of the molecules in the container. Clearly, as this wave would move with some velocity, it would move with a velocity greater than that of the average velocity of the molecules, (which is zero).

Thank you for your assistance.

 

[jedishrfu]

Watch this video and consider that the sound wave can't move faster than the molecules.

Then consider that the RMS speed of the molecules is the basic limitation to the speed of the sound wave ie while some molecules move faster and some slower but on average they move at the RMS speed.

 

[mitochan]

I keep imagining a bunch of gas molecules, each with zero velocity, uniformly distributed throughout the volume of a container. Then, one of the walls of the container rapidly moves into the volume a small distance, setting the molecules in the volume immediate to it in motion in the direction of impact, thereby creating a wave through which the energy of the impact would be eventually transferred to the rest of the molecules in the container. Clearly, as this wave would move with some velocity, it would move with a velocity greater than that of the average velocity of the molecules, (which is zero).

I think the motion of wall would cause WIND, whose speed has no upper limit in non relativistic mechanics, that should be distinguished from SOUND.

 

[Herman Trivilino]

Then, one of the walls of the container rapidly moves into the volume a small distance, setting the molecules in the volume immediate to it in motion in the direction of impact, thereby creating a wave through which the energy of the impact would be eventually transferred to the rest of the molecules in the container. Clearly, as this wave would move with some velocity, it would move with a velocity greater than that of the average velocity of the molecules, (which is zero).

The moving wall sets the molecules in motion. You say so yourself. Then you say the molecules are not moving!

When the wall moves it sets up a wave that travels from one side of the box to the other. So eventually molecules will collide with the wall on the opposite side of the container. A certain amount of time has to elapse between the initial movement of the first wall and the collisions on opposite wall. Don't you see that the elapsed time can't be less than the time it would take a molecule to move from one wall to the other?

 

[Amadeo]

Watch this video and consider that the sound wave can't move faster than the molecules.

Then consider that the RMS speed of the molecules is the basic limitation to the speed of the sound wave ie while some molecules move faster and some slower but on average they move at the RMS speed.

I think the motion of wall would cause WIND, whose speed has no upper limit in non relativistic mechanics, that should be distinguished from SOUND.

The moving wall sets the molecules in motion. You say so yourself. Then you say the molecules are not moving!

When the wall moves it sets up a wave that travels from one side of the box to the other. So eventually molecules will collide with the wall on the opposite side of the container. A certain amount of time has to elapse between the initial movement of the first wall and the collisions on opposite wall. Don't you see that the elapsed time can't be less than the time it would take a molecule to move from one wall to the other?

Thank you for your responses.

jedishrfu, Thank you for your response. I watched the video and still do not understand why the speed of the sound wave must be slower than the average speed of the molecules in a container. In fact, the first animation in that video illustrated exactly what I mentioned in my first post: all the molecules in a volume uniformly distributed with zero velocity, then a mechanism which rapidly impacts the molecules on one side of the vessel, sends a sound wave through the vessel with a certain velocity, a velocity which is evidently greater than the average (zero) velocity of the vast majority of the molecules (which are stationary).

mitochan, Thank you for your response. I did not consider this difference and will look into it more to see how it bears on my question. However, I would mention that the video which jedishrfu posted illustrates sound in the way I described in my first post. But perhaps that video is inaccurate?

Mister T, Thank you for your response. Perhaps my description was not clear. To see what I intended to convey, please see the first animation in the video posted by jedishrfu. Indeed, after the first pulse is made in the stationary molecules, the average velocity of all of the molecules in the container will be greater than zero. However, if we imagine that the size of the container approaches infinity, the average velocity of the molecules inside of the container will approach 0, yet, the speed of the sound wave pulse will still remain unchanged (and will be larger than that average speed).

 

[davenn]

Don't you see that the elapsed time can't be less than the time it would take a molecule to move from one wall to the other?

the molecule(s) don't move from one wall to the other. they just oscillate about their position as the wave passesd ( is passed/propagated by them)

http://www.sciencebuddies.org/Files/3303/5/CE_img048.gif

 

[jedishrfu]

Imagine a relay team of runners running back and forth at the same speed passing a baton forward from the start to the finish line.

Can the baton ever travel faster than the runners?

As @davenn points out the molecules are oscillating around their position. As they move right they hit another molecule that causes them to move left. The sound wave comes out of this motion.

 

[Amadeo]

Imagine a relay team of runners running back and forth at the same speed passing a baton forward from the start to the finish line.

Can the baton ever travel faster than the runners?

The baton could never travel faster than the runners.

But, I am having trouble seeing how the analogy applies. For, in the case of the baton, an object is transferred by carriers (runners) which do not change their speed when they are in the process of transferring the object. In the case of sound, at least according to the video, the carriers (molecules) do change their speed when they are in the process of transferring energy.

 

[anorlunda]

However, if we imagine that the size of the container approaches infinity, the average velocity of the molecules inside of the container will approach 0, yet, the speed of the sound wave pulse will still remain unchanged (and will be larger than that average speed).

That's your problem. You are including molecules in the average that are not involved in the sound. The statement about speed of sound versus particle speed applies to each individual molecule, no averaging of anything.

That sentence you quoted in #1 (where's the link?) should not have included the word average.

 

[mitochan]

@mitochan, I did not consider this difference and will look into it more to see how it bears on my question.

Say speed of the wall is v. The molecules hit by the wall have speed 2v. Many of them do inertial motion forever. Some others hit another molecules and stop. Newly hit molecules move with speed 2v. and so on. These are billiard like motion not sound motion. v is not limited under sound speed of medium.

You see in the video you refer hit molecules back to original position. Above scenario includes no coming back.

 

[sophiecentaur]

I keep imagining a bunch of gas molecules, each with zero velocity, uniformly distributed throughout the volume of a container.

You have asked this same question several times, despite some very informative posts so, in an attempt to resolve the problem:
I think this statement of yours is the root of your problem because, in a way, it's a reasonable situation to start with but it just doesn't represent the way a gas behaves. Your model represents a gas at 0K but, ok let's take it from there. Each molecule that's next to the moving wall will move off at the velocity of the wall until it hits another and transfers its Kinetic Energy. If they are all the same mass, all the energy will be transferred, one step at a time, till it reaches the 'microphone' at the other end. It's as if there was just one row of molecules going the whole distance. The speed of propagation will be the same as the speed of the input wall. The speed of the pulse IS the speed of the molecules. So the very simplistic approach actually makes a bit of sense to start with BUT. . . . (try to follow my ramblings to the end)

That model falls down for any subsequent motion of the wall (especially when the wall moves backwards because the molecules in that 'first row' will have nothing to make them follow the wall. As with Newton's cradle, the molecules will be left stationary where the wall left them. They will only follow the wall is there is pressure in the gas. The necessary pressure has to be due to the temperature. You can only get propagation of a wave if the temperature (relating to the molecular speed) can make the molecules move as fast as the wall and to follow it. (When the wall moves too fast, you generate a shock wave - a third level subject!)

Further along the line, you have molecules that are being hit from both directions, according to 1. The thermal agitation and 2. The additional sound wave pressure. They can only transfer their momentum to the next in line after they have traveled over the distance between. We are now in a random situation (thermal) so it is the average speed that will determine how fast a pulse will propagate.

The theory tells us that the density / pressure in the gas does not matter; it's just this average speed that counts. Lower the density enough and we are in your initial situation, with a free path over the whole way.

 

[sophiecentaur]

Just a further comment. The speed that a regular vibrating sound source (Loudspeaker or squeaky door) involves vibrations in which the motion is a small fraction of the speed of sound. This means the air molecules have no problem keeping up with the diaphragm when it moves 'backwards'. It can be regarded as a linear interface to a linear medium.

 

[cjl]

The baton could never travel faster than the runners.

But, I am having trouble seeing how the analogy applies. For, in the case of the baton, an object is transferred by carriers (runners) which do not change their speed when they are in the process of transferring the object. In the case of sound, at least according to the video, the carriers (molecules) do change their speed when they are in the process of transferring energy.

In the case of sound waves, they really don't change their speed. Technically, sure, there's a small increase, but the magnitude of that increase is basically negligible.

 

[sophiecentaur]

Technically, sure, there's a small increase,

The mean velocity will be zero, though.
The distribution of the (thermal) velocities of molecules will involve some molecules traveling slower than the wall, even for a low amplitude sound wave. It seems to me that this constitutes a non-linearity - but I don't think it can be.

 

[cjl]

Sure, but that number will be vanishingly small for any reasonable wall speed and gas temperature. Besides, when a molecule traveling that slowly interacts with the wall, it will come off at a much higher speed purely due to heat transfer, regardless of the bulk wall velocity.

 

[sophiecentaur]

Sure, but that number will be vanishingly small for any reasonable wall speed and gas temperature. Besides, when a molecule traveling that slowly interacts with the wall, it will come off at a much higher speed purely due to heat transfer, regardless of the bulk wall velocity.

But does ordered motion mean heat transfer? Isn't that to confuse microscopic and macroscopic? With a 'very loud' sound from a loudspeaker peak displacement of say 2cm at 10kHz, that would mean a peak wall speed of around 100m/s. Not too far below the speed of sound. There would be a significant proportional of thermal speeds less than that. Could that be regarded as a linear process? Perhaps we are nearer, in everyday life to the limit of linearity of sound than I always assumed.
That loudspeaker idea is a bit extreme but there could be many mechanical examples with that sort of amplitude (turbines?). Edit: But I guess one could say that we wouldn't be expecting hifi in that case?

 

[cjl]

2cm at 10kHz is an astonishing amount of displacement. A 4 inch (~100mm) diameter speaker with 2cm of excursion at 10kHz will be putting out nearly 190dB SPL at 1m. Even jet aircraft are quieter than this by ~50dB. For a more realistic value, at 10kHz, you only need a 1 inch diameter driver at 0.1mm of excursion to achieve 119dB, which is probably as loud as you'd ever want a 10kHz sound to be. In general, we are quite far from the linearity limits of sound in everyday life.

As for your first question, I guess I don't understand what you're asking? My heat transfer point was that, assuming the wall temperature is not close to absolute zero, a slow-moving gas molecule that encounters the wall will very likely leave the wall with a high velocity due purely to the wall temperature itself, even if the wall is not moving.

 

[sophiecentaur]

slow-moving gas molecule that encounters the wall will very likely leave the wall with a high velocity due purely to the wall temperature itself, even if the wall is not moving.

If there is thermal equilibrium then there will be no net gain or loss. Fast molecules can move away slower than they arrived etc..

But I guess my example was perhaps way out of scale - even allowing for the skirt of the Maxwell Boltzman velocity distribution. But your SPL figures at 1m distance would increase a lot, just in front of the speaker cone so what happens right in front of the cone? Once the non linear products have been formed, would they not propagate away?

However, the conditions are much more extreme than the assumptions made about sound in general. It's up in the Shock Wave region.

 

[snorkack]

If you put a small amount of molecules next to the wall in motion at a speed far greater than the average speed of molecules away from the wall, then as they move towards the gas, the energy will be progressively distributed to more and more molecules at slower and slower speed. It behaves as a shock front, not as a wave.
It is only when the additional energy is small enough compared to the average kinetic energy of molecules that the sound wave can propagate as a wave and leave undisturbed gas behind.
The transition from shock wave to sound wave is a gradual phenomenon, and very common one which happens as bangs propagate away from source.

 

[vanhees71]

The speed of sound for an ideal fluid is given by the adiabatic compressibility of this fluid, i.e.,
$$v_{\text{sound}}=(\partial_{\rho} p)_{s}.$$
For an ideal gas you get
$$v_{\text{sound}}=\sqrt{\gamma k_{\text{B}} T/m},$$
where $\gamma=c_p/c_v$ is the adiabate coefficient. The mean kinetic energy of a molecule (mass $m$) is
$$\frac{m}{2} \langle \vec{v}^2 \rangle=\frac{3}{2} k_{\text{B}} T,$$
and thus the rms "thermal" speed
$$v_{\text{RMS}}=3 k_{\text{B}} T/m.$$
So the speed of sound is of the same order as the average thermal speed of molecules since $\gamma \simeq 5/3, 7/5, 4/3$ for monatomic, two-atomic, $\geq 3$-atomic molecules.

This makes physical sense, because after all sound is a pressure wave in a fluid and pressure is due to the collisions of the molecules. Thus changes of pressure are propagator roughly at the average speed of the molecules.

 

[cjl]

If there is thermal equilibrium then there will be no net gain or loss. Fast molecules can move away slower than they arrived etc..

Yes, but any molecule moving slowly enough that the molecule speed is of comparable order to the wall speed will be way, way down the tail of the velocity distribution (on the slow side), so it is overwhelmingly likely to gain a lot of velocity when it interacts with the wall purely because of the thermal interaction (even ignoring the wall velocity)

But I guess my example was perhaps way out of scale - even allowing for the skirt of the Maxwell Boltzman velocity distribution. But your SPL figures at 1m distance would increase a lot, just in front of the speaker cone so what happens right in front of the cone? Once the non linear products have been formed, would they not propagate away?

Of course they would propagate away, but my point was that your example with 100m/s cone velocity is way out of the linear region for air anyways. At 190dB like you would get there, the pressure wave amplitude is so large that you're approaching a complete vacuum in the troughs, and you can no longer make the small amplitude assumptions that are typically made with soundwaves. At a very loud (but more reasonable) level, the cone velocity of the speaker will be on the order of a couple meters per second or lower, so it will always be <1% of the molecular velocity.

However, the conditions are much more extreme than the assumptions made about sound in general. It's up in the Shock Wave region.

For your example? Yes, exactly.

 

[snorkack]

While a gas at absolute zero is a metastable condition, note that He has a very broad gaseous range. Down to 3,2 K at 1 bar, and less than that at lower pressure.
You could watch, e. g. a fan running at a constant speed which is far below sonic at room temperature, and see what happens to aerodynamics and acoustics as the sound speed drops to supersonic on cooling.

 

[cjl]

That's not as possible as you might think. Due to helium's very low molecular mass and monatomic nature, it has a very high sound speed. As a result, even at 4K, the speed of sound in helium (assuming it still behaves as an ideal gas, which I wouldn't necessarily trust here) should be right around 100m/s. I don't know of any gas that would have a sound speed slow enough to observe the kinds of effects you're talking about. Tungsten hexafluoride should have a sound speed of around 95m/s at normal(ish) conditions, but it has a boiling point of 17C, so you can't cool it much. Maybe you could have a low pressure WF6 gas and get it a bit colder that way?

 

[Amadeo]

Note: this response is made after reading the messages up to sophiecentaur's Saturday post. I don't have time to review the other messages right now. So, please excuse if this message contains things potentially made irrelevant by those subsequent posts. Also, my book is Haliday Physics 7th Ch 19.4

I think I have resolved the question, with the help of considerations raised by multiple respondents. The question of the average speed was an issue.

I will describe a few different physical situations which will illuminate the different ways in which the term "average speed" is to be interpreted, and why it is that the average speed, properly interpreted, represents the limit of the speed of wave propagation.

In each of the following situations. Imagine that we have a long sealed cylinder filled with a gas of uniform pressure and temperature. At the end of the cylinder is a movable piston.

1.) The gas in the cylinder is uniformly distributed and each molecule has zero velocity (as in the first illustration in the video). Then, the piston makes a rapid movement inward, and impacts some of the molecules, setting them in motion. In this situation, it is evident that the average speed of the molecules set in motion by the piston will limit the speed at which the pulse will be able to travel through the medium. Assuming that the energy is transmitted in the fashion of a Newton Cradle, the other molecules in the cylinder have no effect on the motion of the pulse whatsoever, and this speed of transmission will be maintained throughout the process.

2.) A gas fills the cylinder in a uniform distribution and at certain non-zero pressure and temperature. Now, imagine that a certain volume of the gas is instantly removed from one end of the cylinder, leaving, temporarily, an empty volume. The gas in the rest of the cylinder will move into fill the empty volume. The speed at which the molecules will fill this empty volume is limited by the average speed of the molecules remaining in the cylinder. Incidentally, a low-pressure front will move though the remaining gas in a direction opposite to the direction in which the gas moves to fill the evacuated volume. The speed of this pulse also will be limited by the average speed of all the remaining molecules in the cylinder.

3.) A sound wave of constant amplitude is established in the cylinder by moving the piston back and forth repeatedly. This gas carrying this sound wave will be characterized by areas of relatively high pressure and areas of relatively low pressure. These areas of high and low pressure need not be of different temperatures however, we can imagine that they contain different numbers of molecules, and that the molecules in the low pressure areas will have the same average speed as those in the high pressure areas. So, all of the molecules in such a cylinder will have equal average speed. This average speed will limit the speed at which the wave will be able to move through the gas in the cylinder, for this is the highest speed at which the position of molecules could be altered to realize the changes in the molecular density of different portions of the volume of space in which they move which characterize this wave motion.

Now, everything seems to be in order. However, we have one more physical situation which wee must discuss, a physical situation which may seem to invalidate some of our above conclusions:

4.) A pulse of the piston establishes a wave front in the gas of the cylinder. Another part of the cylinder contains gas which has not yet received any energy from the wave moving through the gas. Imagine that the gas not yet affected by the wave in one part of the cylinder has a lower temperature than the temperature of the gas which is transmitting the wave in another part of the cylinder. Experiment reveals that when sound travels from a medium of high to low temperature, the speed of sound decreases. However, based on our observations in case 1, it would seem that, theoretically, the transmission of the sound wave should not be affected at all by the average speed (average temperature) of the molecules which are yet to receive additional energy from the wave. If it must be said of a wave pulse moving through a gas of otherwise very low or even zero velocity, that the speed of transmission is limited by the average speed of the molecules in the pulse, and is not affected at all by the speed of the molecules yet to be impacted by the molecules transmitting the pulse, why do we observe that the speed of sound changes when sound travels from one medium to another of different temperature?

The answer is that the model of the physical process of sound wave transmission described in case 1, (and illustrated in the video) must be incorrect. If we imagine that the molecules in a gas do not move and collide with one another in only one direction when subjected to an impact by a piston, or speaker, but, rather, move and collide in all directions, the increase of energy (and speed) of the molecules corresponds to an increase in the pressure exerted by those molecules in all directions.

Imagine that the front of a sound wave comes into contact with a gas of a lower average energy/speed. The energy of the high energy molecules will be transmitted to those of lower energy with a speed equal to the average velocity of the high energy molecules, but only momentarily. For, the molecules receiving this energy are at a lower energy than the molecules delivering additional energy, and, therefore, the molecules which receive this additional energy will only be able to transmit the disturbance to their neighboring molecules at a reduced speed. As this process continues, the additional energy introduced by the initial pulse, will be dissipated. The disturbance in molecular density (sound pulse) will eventually be carried by molecules which are of the average energy/speed that prevailed before this sound pulse was transmitted into the lower energy gas. So, the speed of transmission of the disturbance will eventually reach the speed of sound which corresponds to the gas at the lower average speed/energy (temperature). This, of course, is assuming that the volume of the lower temperature gas is large, and, thus, the addition of energy into it by the pulse is not enough to appreciably alter its average energy distribution.

The model illustrated by the picture of motionless particles transmitting energy in the fashion of a Newton cradle is a preliminary heuristic device, not intended to offer a full basis by which the experimentally observed behavior of sound is to be comprehended. Thus, treating it as such will likely lead to disjunctions between the conceptual model with experimental observation, as it did in my case.