# Speed of sound wave and particle velocity

1. Nov 7, 2013

### Gavinn

This may be fairly straightforward but it is a concept that I am really having problems understanding. A sound wave is a pressure disturbance that travels through a medium by means of particle to particle interaction, so why is the wave velocity so much faster than the velocity of the particles as they are displaced?

2. Nov 7, 2013

### Staff: Mentor

In gases, both are comparable.
In liquid and solid materials: the forces between the particles act much quicker than the motion of the individual particles. As soon as particle A starts to move, the force on the next particle B begins to change, accelerating this as well.

3. Nov 7, 2013

### AlephZero

It may help to know that the velocity of the individual particles depends on the amplitude of the sound wave, which is independent its speed (unless there are nonlinear effects like shock waves in air, or surface waves on water).

4. Nov 7, 2013

### Gavinn

I know that particle velocity is linked to pressure and the specific acoustic impedance of the medium (u=p/z) - what I don't understand is how the sound wave can travel so much faster than the particle movements that are giving rise to the pressure changes that make up that sound wave. I'm beginning to wonder if I have just fundamentally misunderstood the nature of the sound wave (and am giving myself a headache to boot).

5. Nov 8, 2013

### BruceW

I can see that it is a bit hard to imagine. But the particle only needs to move a small amount to exert a force on a nearby particle that will carry on the wave. So each particle does not have to move a large distance.

I'm just going to make up some numbers, as an example which I think will help. So, let's say that when the particle has moved by 1 nanometer, it exerts a 4 milliNewtons force on its nearest neighbour, which is 1 cm away from it. From this made-up example, you can see it is not so strange that the wave can travel faster than the particles.

6. Nov 8, 2013

### Gavinn

Thanks, that does help to an extent but I'm still left wondering how the speed of sound can be entirely independent of the particle velocity if it is the movement of the particles which passes on the pressure fluctuations that the sound wave consists of. I've had a look in a few textbooks and online but this query doesn't seem to come up (so I'm a bit concerned that I'm just overlooking something which should be really obvious), if anyone knows where I could find more information I'd be happy to hear about it.

7. Nov 8, 2013

### BruceW

If we assume that the particles do some kind of simple harmonic motion due to the wave, then yes, the amplitude of the particle's motion is independent of the speed of the wave (as AlephZero said). And the velocity of the particle due to the simple harmonic motion is also independent of the speed of the wave. To be clear, when I talk about the velocity of the particle, I'm talking about the RMS value of the velocity, since we are talking about simple harmonic motion, the average velocity would be zero.

hmm. Maybe thinking of the extreme example will help. OK, if the particle's RMS velocity is very very small (i.e. tends to zero), then does the velocity of the wave tend to zero? No! When the particle's RMS velocity is very small, then since it is doing simple harmonic motion, this means that the amplitude of its motion is also very small. It only has to move a tiny amount to be able to send the wave on to its neighbour.

So now maybe the question is why do we assume simple harmonic motion. Or (almost) equivalently, why do we assume linear waves? Well this is not necessarily true. But it is true for low energy excitations. And it makes our equations a lot easier :) So we often assume it is approximately true, to get an approximately correct answer. But non-linear wave analysis is pretty interesting too.

8. Dec 15, 2013

### darondny

Hello to everyone ,

I'm stuck also at this point with "sond velocity" vs "particle velocity" . Is verry clear that the particle doesn't have to move a lot , that being directly related to the energy of the sound wave .... But why , or better said .. isn't that true.. for sound waves in gases , sound velocity is the same with sound speed ?

I'm thinking for this kind of explication : in a long steel wire , we can have 2 kind of oscilations ,with a shear wave or a compresion wave .
Speed of sound in the seel is regarded to a compression wave ; the same thing is for watter .
When we consider a shear wave , the wave would be propagating with a much , much , slower rate . The direct example is a rock thrown in a lake or when you hit quickly with your hand a long wire suspended in air by both ends .
In those cases , the particle velocity will be.... small compared with the basic speed of sound in that medium ?!?

Do we have transverse waves in air ?
Mr. Gavinn , do you have some final conclusions?

9. Dec 15, 2013

### Staff: Mentor

Sound speed is the magnitude of the sound velocity (=a vector), both are not the same as the particle velocity/speed.
It is true that for gases, both speeds are often not completely different, as transmission of pressure differences happens mainly via moving particles.

No, both oscillations are independent types of sound in steel. In liquids (like water) and gases, there are no shear waves as there is no shear stress.

I don't understand what you try to compare. Surface waves in a lake are not even sound waves.

10. Dec 15, 2013

### sophiecentaur

If you consider a very simple model of sound transmission through a solid. Masses in a line, connected by springs. One mass only needs to be displaced by a small amount for the force on the connecting spring to connect with the next mass in the chain. Translating this into speeds, the speed of the mass is the distance it actually travels in a unit time. The speed of the wave / disturbance relates to the separation between the masses (the length of the spring) If the spring has a high k constant and is very long, then the speed that the information about the one mass being displaced is transferred to the next mass much faster than the original mass is actually moving.

Another point: the speed of sound is independent of the displacement of the medium. The displacement is proportional to the speed of the particles. So the two speeds do not have to be related. When the particle speeds start to approach the speed of sound, a shock wave is generated and the situation changes.

11. Dec 15, 2013

### darondny

Ok... but could you explain , please , if you don't mind , in a rationalized manner why , the sound in air doesn't have a constant particle speed ? Or .. how many m/s for particle speed we will have in the case : 30Hz/20 degrees C/ X (Pa) ?

P.S. :
sophiecentaur , those mases with springs .. does not have the same speed regarding we have X meters / Y seconds , or X-Z(m)/Y-Z(s) ?

I'm asking those questions because I can not implement a logical explication in my head and is really bugging me .
Thanks for all of you ;)

Last edited: Dec 15, 2013
12. Dec 15, 2013

### sophiecentaur

The particles in a gas do not have a uniform speed. There is a wide distribution so the model you want to apply just doesn't work,

13. Dec 15, 2013

### klimatos

Statistical mechanics gives the speed of sound under conditions of equilibrium as:

vs = γ1/2σ

Here vs is the speed of sound in meters per second, gamma is the ratio of the specific heat of the gas at constant pressure to the specific heat at constant volume, and sigma is the root-mean-square axial molecular speed in meters per second. If we accept a value of 1.40 for gamma for dry air, then:

vs = 1.18 σ

At NTP, σ = 280 meters per second for dry air, which gives us vs = 331 meters per second for the speed of sound at NTP in dry air.

The mean impulse molecular speed for dry air at NTP is 351 meters per second. This is the mean speed of air molecules passing through an imaginary plane measured normal to and toward the plane. It is somewhat faster than the speed of sound.

I suspect that the speed of sound is actually the mode of the impulse molecular speed distribution.

14. Dec 16, 2013

### AlephZero

I accept the statistical mechanics based answers, but IMO when talking about waves traveling in a medium, "particle velocity" means the mean velocity of the particles in a small region. That is amplitude dependent, and of course much smaller than the velocity of individual particles.

If you want to relate the "particle velocity" to the velocity of transducers like loudspeakers or microphones, the mean particle velocity in the gas is certainly the relevant quantity.

15. Dec 16, 2013

### darondny

Last night , so I can sleep , came in my mind an old idea of mine : since the particles whom are involved in the 'business for propagation of a wave travel in a rectilinear motion without changing the mean position (+-X um ; no mass transportation by the wave) we will agree that the particle will have 2 positions of rest in a moment of time.
Further away , since the particle posess mass , the stopping will be gradually slower , respectively an acceleration in the very next second reaching a top speed (like "klimatos" said , bigger than the speed of sound in air) .

Now we can conclude , I think so , with this : since the particle have a guidance in a sine manner , it's average speed over a big period of time , like 2 seconds , will be ...... :D

16. Dec 16, 2013

### sophiecentaur

The speed of the particles relates to their Kinetic Energy and the temperature is defined as the average KE (mv2 of the particles. At a given temperature, the more massive molecules will be travelling slower (mean speeds). As the sound wave travels through the air, the mixture of molecules will be moving backwards and forwards (on average), pushed by the tiny pressure differences between peaks and troughs of the sound waves. These differences in pressure are due to different concentrations of molecules. The rate at which the pressure communicates itself will depend on the speed (mean) of the molecules involved.- i.e. speed of sound depends on speed of molecules. You find that the speed of sound in a gas is dependent upon temperature and not pressure (which may not be too intuitive).

17. Dec 16, 2013

### ZapperZ

Staff Emeritus
I'm re-read this, and the thread, and unless I missed something, most of the responses seem to either misunderstood the question, or simply missed the mistake.

Here's what I understood from this question: Say sound travels in air. It has a speed. The OP claims that the speed of sound in air is "higher" than the speed of the air particles themselves! So he/she wants to know why.

That's my understanding of this question.

If that is your understanding as well, then there is an obvious mistake in the premise. The speed of sound in air is NOT higher than the average speed of the air "particles". The speed of sound in air, at STP, is ~340 m/s. Now, while there isn't one single "air particles", we can look at gasses such as N2 and O2 and look at the average speed of these molecules at STP. Even rounding off, these numbers are around 500 m/s.

So the speed of the sound wave is NOT higher than the average speed of the individual particles of the medium (air). The question cannot be answered because it is based on a false premise.

Zz.

18. Dec 16, 2013

### klimatos

Your premise is certainly valid if you are referring to the "true path" molecular speeds. However, IMO these speeds are largely irrelevant to the speed of sound. I believe that the speed of sound is proportional to the component molecular speed measured along the straight line connecting the emitter to the sensor, since the sensor only measures impulses normal to its sensing surface.

My work has been solely with the atmosphere, so I cannot speak for other media. For dry air, under conditions of equilibrium, the mean molecular axial impulse speed is about 1.06 times the speed of sound. This is certainly suggestive regarding the premise that sound impulses are carried by the normal movements of air molecules.

It may be that this finding also clears up the OP's confusion.

19. Dec 17, 2013

### darondny

I still can not understand how you , "klimatos" , have reached this conclusion "1.06 times the speed of sound" .

If we say mean , we must to be sure that the "mean" value is over a big enough period of time . In a periodic movement we have the mean speed... zero! Is a back and forth motion .

And , I do not understand also how we can have an "impulse" in a sine wave .

If we analize a sine wave we can see that , on a big amount of the sine wave/over time , the acceleration is steady , and then is decreasing . So , if we take off the steady state , even the decreasing slope , we have a climbing on the graph with a much bigger rate than .....

Next question : the acceleration in that sine wave is proportional with the SPL ? Which is the formula for that ?

20. Dec 17, 2013

### sophiecentaur

If you want an alternative source of this sort of information, try this Hyperphysics link.

21. Dec 17, 2013

### ZapperZ

Staff Emeritus
I'm not exactly sure what this has anything to do with what the OP asked. I am trying to clarify if he/she simply misunderstood the "particle" speed with the speed of sound. The way it was described, he/she got it completely backwards where he/she seemed to indicate that the speed of sound is HIGHER than the speed of the air molecules.

In any case, the OP hasn't responded to the latest queries, and we may be huffing and puffing here for nothing.

Zz.

22. Dec 17, 2013

### klimatos

I think the major source of confusion is that you are looking at the movement of waves in a continuous medium (the classical view) and I am looking at the movement of molecules in largely empty space (the kinetic gas/statistical mechanical view).

The molecular speed that is 1.06 times the speed of sound is the mean axial speed normal to and toward the sensing surface. Molecules whose axial arm speed is normal to and away from the sensing surface play no immediate role in sound transmission by air molecules.

This mean axial arm speed is the mean speed at the instant of impact of that select sub-population of air molecules that impacted on the sensing surface during the period of measurement. The "impulse" that you questioned, is the mean impulse transferred by these impacts.

Statistical mechanics shows that both the mass and the axial arm velocity of this select sub-population is different from the mass and the axial arm velocity of the general population of air molecules. This is because faster moving molecules have a greater probability of impacting on a surface than their simple numbers suggest. That is, a molecule moving with a speed of 2v has twice the probability of impacting upon a surface than one with the speed of only v--if they are equal in numbers.

In a mixture of gases with different molecular masses, such as air, these faster molecules are more likely to be lighter in mass. Hence, the mean impact mass is slightly lighter than the general population mass. This difference is very slight for dry air, and even slighter for water vapor; but it is real.

I am not a physicist, my field is the earth sciences--particularly the atmospheric sciences. I have been told by physicists that neither statistical mechanics nor kinetic gas theory are particularly popular with physics students today. I think this a pity. I have found them to be fertile fields for insights into atmospheric behavior.

23. Dec 17, 2013

### nasu

What do you mean by "axial arm speed"?

24. Dec 17, 2013

### klimatos

π
Under conditions of equilibrium, the distribution of velocity probabilities along any axis of a standard orthogonal tri-axial reference system is the same. That is, the mean velocity along the + arm of the x-axis is identical to the mean velocity along the – arm and is indeed identical to the mean velocity along any arm of any axis—no matter how the reference system is oriented. More interesting, it is identical to the mean velocity along all arms averaged together. That speed (vp) is the mean axial arm speed of a general population of gas molecules. I use the “p” subscript to denote that the directional component of the velocity is “poly” directional.

This gives rise to the classical concept that the pressure of a gas (in the absence of unbalanced forces) is everywhere the same within that volume of gas. No matter where you postulate a surface, the impact rate of molecules impacting upon that surface (½nvp) will be identical and the mean impulse per impact (2mivi) will also be identical. Hence, the pressure (mean impact rate times mean impulse transferred per impact) will everywhere be the same.

However, when I used that term in my post, I was clearly referring to the mean axial arm impulse speed (vi). This is the mean axial arm speed of the interactive molecular population, not the mean axial arm speed of the general population (vp). The two have different parameters. The mean axial arm impulse speed is the axial arm speed of that select sub-population of molecules that are passing through an imaginary plane or impacting upon a surface. The speed is measured at the instant of passage or impact. The mean molecular axial arm impulse (vi) speed is always higher than the mean molecular axial arm speed (vp). The relationship between the two is always vi = π/2 vp (that's a pi symbol, not an n). The directional component of the mean axial arm impulse velocity is always normal to and toward the object of interest, be it an imaginary plane, a sensing surface, a raindrop, or anything else you can think of.

Last edited: Dec 17, 2013
25. Dec 22, 2013

### darondny

"If a force is not of constant magnitude over time, the impulse is the integral of the magnitude of the resultant force (F) with respect to time: J = (integral) Fdt ."-- wiki

And is not an impact because is an elastic movement , is not like a hammer . In a sine wave we do not have a sudden change in amplitude , everybody knows that .

"Maximum rate of change of sine wave occurs at zero crossing point(..) 2 x Î  x f x V volts per second " -- slew rate .