Exactly how do pressure waves work at the molecular level?

[DaveC426913]

Am I wrong? Did I miss something?

Yes. The diagram is a simplification. You're taking it too literally. It was meant to answer one of your questions, not all of them.

 

[Freixas]

It's an average.

You do not have a leading particle always advancing at exactly 300.1 m/s all the time. You have a collection of particles moving in random directions at significantly more than 300 m/s. It is the peak of the density distribution that advances at 300.1 m/s.

Even having the peak density advancing at 300.1 m/s is a problem. As I mentioned, if the average incoming velocity were 50 m/s, then the peak density would advance at 350 m/s. I'm pretty sure this is not how it works.

In my [very strongly held] opinion, trying to understand this at the molecular level is the wrong thing to do. You know that it will work out because ideal gas behavior follows from molecular behavior and pressure waves follow from ideal gas behavior. Trying to jump directly from one abstraction layer to another two hops away is not optimal.

Yes, clearly my one-dimensional model is incorrect. I don't know why, but it is.

Another approach is to learn the formulas and derivations. You could then get the right answer without any intuitive understanding. I talked to a physics professor who said that his grad students had trouble resolving some of the basic "parodoxes" of special relativity. Yet they did their lab work correctly. How? They relied on the Lorentz transform.

I've though of modeling a few million particles, applying a lot more randomness and dimensions, but following the same basic principles as my simple case. I could write the code, but it's a pain. If I still got the wrong answer, it would say that that model is also incorrect.

If I were to model this

• Particles would have random positions but the average fluid density would be constant.
• Particles would have random velocities, but the average velocity in the tube would be 0 and coming into the tube would be +.1 m/s.
• Particles would have random speeds, but the average speed in all cases would be 300 m/s.
• All collisions would be perfectly elastic.
• All particles would have the same mass and size.
• It shouldn't matter if I use a monatomic gas or a diatomic gas, right? If so, I could simplify by assuming just 3 degrees of freedom (x, y, z).

Is there anything significant missing?

 

[jartsa]

This doesn't sound right. Now 0.1 m/s is not much speed, but it could be 50 m/s and now we're getting wavefronts traveling at significantly different speeds.

Am I wrong? Did I miss something?

Compressing gas increases its heat energy. The work done while compressing becomes kinetic energy of molecules, AKA heat energy. Or thermal energy, if heat energy is not quite right term to use.

The speed of sound depends on speed of molecules. So compression wave travels extra fast.

 

[hutchphd]

Even having the peak density advancing at 300.1 m/s is a problem. As I mentioned, if the average incoming velocity were 50 m/s, then the peak density would advance at 350 m/s. I'm pretty sure this is not how it works.

I believe the "thing" that is moving is the location of the 350m/s particle. If one does the analysis carefully I believe that location will move on average at 300m/s, because the local extra speed shifts the collision locations in the 1D gas. Need to work this out, though, (personally I rather like the model).

Edit: My supposition here is incorrect ( I worked it out) the speed of the disturbance is in fact 350m/s. Still OK though

 

[jbriggs444]

Even having the peak density advancing at 300.1 m/s is a problem. As I mentioned, if the average incoming velocity were 50 m/s, then the peak density would advance at 350 m/s. I'm pretty sure this is not how it works.

If you have a piston moving at 50 meters per second then (additional) density variations will indeed advance at 350 meters per second in the region of gas being displaced by the piston.

It is well known that wind velocity adds to sound velocity.

 

[russ_watters]

This doesn't sound right. Now 0.1 m/s is not much speed, but it could be 50 m/s and now we're getting wavefronts traveling at significantly different speeds.

Am I wrong? Did I miss something?

The diagram only tracks one set of interactions. If you're inducing a wave with a piston, the piston has to retract to complete the wave, otherwise you are just shoving the air out of the tube. If you make a wave, the average velocity is 0.1+(-0.1)=0 (and a cajillion other values that sum to 0).

However, you can do that if you want; with a reciprocating pump, a fan starting, a valve opening. In that case you simply end up with a positive flow rate(velocity).

 

[hutchphd]

I took the OP to ask about the wave front when one blows into the end of a tube, not the harmonic version. The result is, therefore, flow.

 

[russ_watters]

I took the OP to ask about the wave front when one blows into the end of a tube, not the harmonic version. The result is, therefore, flow.

You're right, I think somewhere along the line it got changed for illustrative purposes.

So maybe to complete the thought about speed of sound in a flowing fluid: if it is 300m/s and you introduce a flow in a tube at 0.1m/s, the speed of sound with respect to the ground is now 300.1m/s.

 

[Freixas]

I took the OP to ask about the wave front when one blows into the end of a tube, not the harmonic version. The result is, therefore, flow.

Yes, that's correct. Blowing into a tube could be considered as a very low frequency wave but with no zero crossings (I mean, one could suck air, but I'm trying to analyze a musical instrument in which one only blows air in).

So maybe to complete the thought about speed of sound in a flowing fluid: if it is 300m/s and you introduce a flow in a tube at 0.1m/s, the speed of sound with respect to the ground is now 300.1m/s.

It is well known that wind velocity adds to sound velocity.

What I think needs to happen is illustrated in reply #6 (sorry, I don't know how to create links to specific replies). The drawing shows the pressure wave traveling at the speed of sound regardless of the incoming average velocity.

If airflow speeds affect the speed of the pressure wave, the drawing would be wrong. However, if that were true, the effect would also distort sounds. The higher the frequency, the sooner the distortion would occur.

Consider a concert A (440 Hz) where the airflow speeds (amplitude) range from -100 to 100 m/s. The wavefront would move at 243 to 443 m/s. The wavefront peaks would overtake the troughs in a fraction of a second.

So I'm still looking into this. I've found descriptions of the characteristics of perfect gases and my one-dimensional model seems to follow all the rules (except that it's one-dimensional). I'm also reading articles on the kinetic theory of gases, but I'm just starting. The term "root mean squared" gets bandied about a lot and might offer a key to this conundrum. Or not.

Most of the things I see deal with sound, in which the flow reverses. What I'm looking at is like taking the compression part of the sound wave, stretching it out and examining it at the molecular level using kinetics.

 

[russ_watters]

Consider a concert A (440 Hz) where the airflow speeds (amplitude) range from -100 to 100 m/s. The wavefront would move at 243 to 443 m/s. The wavefront peaks would overtake the troughs in a fraction of a second.

That's not what amplitude does. Amplitude affects how much/far/fast an individual particle will displace as the wave passes; it doesn't mean that different parts of the wave travel at different speeds.

 

[Freixas]

That's not what amplitude does. Amplitude affects how much/far/fast an individual particle will displace as the wave passes; it doesn't mean that different parts of the wave travel at different speeds.

It always looks so simple with sound waves. The individual molecules aren't moving very far. They get bounced first one way and then the other for a net velocity of 0.

Everything seems hunky-dory until I stretch out, say, the compression phase of the sound wave, stretch it into seconds and then try to analyze it. Then it looks like the velocity of the flow is being added to the average speed of the molecules. That can't be right, but I just don't know where the error is.

I'm doing some studying to figure out the error. If someone has the answer, feel free to pipe in. The answer might look something like this: https://courses.physics.ucsd.edu/2016/Spring/physics4e/kintheory.pdf, which sadly seems to mention, but not analyze kinetically, the speed of sound.

 

[russ_watters]

It always looks so simple with sound waves. The individual molecules aren't moving very far. They get bounced first one way and then the other for a net velocity of 0.

Everything seems hunky-dory until I stretch out, say, the compression phase of the sound wave, stretch it into seconds and then try to analyze it. Then it looks like the velocity of the flow is being added to the average speed of the molecules. That can't be right, but I just don't know where the error is.

I'm doing some studying to figure out the error. If someone has the answer, feel free to pipe in. The answer might look something like this: https://courses.physics.ucsd.edu/2016/Spring/physics4e/kintheory.pdf, which sadly seems to mention, but not analyze kinetically, the speed of sound.

As others have said, it's pretty hard, on a molecular level, with air because real air molecules bounce around so randomly.

If the system is a bunch of masses spaced out on a spring, it is more obvious; the particles do indeed move around different speeds and attempt to overtake each other, but in the wave the individual particle speeds and locations are constantly varying. The particles are not permanent parts of the wave. The speed up, get closer and then slow down and move apart.

For a transverse wave like a bull whip, the amplitude is perpendicular to the direction of motion of the wave. So its much more obvious that amplitude is independent of the speed of the wave.

 

[Freixas]

OK, here's the best I've got.

My one-dimensional model lacks some behaviors that are visible if more dimensions are added. I don't know if it's true for 2 dimensions, but let's say:

• I use 3 dimensions
• I model an ideal gas
• I begin with all particles having a speed of $V_i$, but with equal probabilities of moving in any direction

Therefore:

• The average speed is $V_i$
• The average velocity is 0

If I simulate this for a while, the probability distribution of speeds changes for reasons beyond my pay grade (Maxwell-Boltzmann distributions?). The average speed is no longer $V_i$, but $V_{rms}$ (the root mean square of all the individual speeds).

$V_{rms}$ can be calculated using $$V_{rms} = \sqrt { \frac {3RT}{M}}$$ where $R$ is the gas constant, $T$ is the absolute temperature and $M$ is the molar mass.

The speed of sound can be calculated from the root mean square speed using $$V_{sound} = \sqrt { \frac { \gamma } {3} } V_{rms}$$where $\gamma$ is heat capacity ratio (again, not my pay grade).

Into this pile of gas particles, I send in particles whose average speed is also $V_i$ but whose average velocity is, say, 0.1 m/s. Before I send them in, I let them stew for a while for the Maxwell-Boltzmann distribution to settle in. I'm guessing that this changes the average speed, but not the average velocity, which remains 0.1 m/s.

The speed of the wavefront travels at the speed of sound. In both cases (gas in tube, gas coming into tube), none of the values used for calculating $V_{rms}$ (and $V_{sound}$) seem to much care about the average velocity. I'm not 100% certain, because the velocity difference might change the temperature $T$, but I don't think so.

So then I would have to conclude that the speed of any wavefront is the speed of sound which is only dependent on things that don't care about the velocity difference.

I have to do a little hand waving to explain how the particles behind the wavefront now have a relative velocity of 0.1 m/s. The best I can say is that the wavefront is the transfer of the relative velocity through collisions moving through the system at $V_{sound}$.

It's not the clearest explanation, but at least I know where the holes in my understanding lie. Assuming, that is, that the explanation is otherwise correct.

 

[hutchphd]

I have time for only a few answers:

If I simulate this for a while, the probability distribution of speeds changes for reasons beyond my pay grade (Maxwell-Boltzmann distributions?).

As noted previously (@vanhees71 ?) an ideal gas in a perfectly elastic box will never equilibrate because particles essentially do not interact. If started with a single speed distribution that will remain. Interaction with real world non-perfect Temperate walls gives them a temperature and velocity distribution. Not difficult to understand I think (?)

Before I send them in, I let them stew for a while for the Maxwell-Boltzmann distribution to settle in. I'm guessing that this changes the average speed, but not the average velocity, which remains 0.1 m/s.

My problem with your original 1D attempt was that you were introducing gas at zero temperature...obviously the result will be strange. This requirement just says use gas at the same temperature as your sample or the result will be complicated with lots of extraneous results.

I still want to play with the !D model if I get my brain up past idle speed...programming anything is becoming a real chore.

 

[Freixas]

As noted previously (@vanhees71 ?) an ideal gas in a perfectly elastic box will never equilibrate because particles essentially do not interact. If started with a single speed distribution that will remain.

Odd as I was just looking at a bunch of simulations which seemed to say otherwise. Here's one: https://www.falstad.com/gas/ You can set the model to start with equal speeds for all particles and watch them fall into the Maxwell-Boltzmann distribution. It's rather neat.

I still want to play with the 1D model if I get my brain up past idle speed...programming anything is becoming a real chore.

The 1D model has some interesting properties that are actually easy to picture without needing any programming.

The setup:

• All particles are identical.
• In the tube: particles evenly spaced alternating between either -300 and + 300 m/s.
• Outside the tube: particles evenly spaced alternating between either -299.9 and 300.1 m/s.
• Assume + means toward the right. The tube is at the right. The incoming particles are to the left.

In this setup, I don't believe the gas is at 0 temperature. With a little bit of contemplation, it should be easy to see that:

• In the tube, the average speed is 300 and average velocity is 0.
• Outside the tube, the average speed is 300 and the average velocity is +0.1.
• Any elastic collisions will produce particles with one of the four velocities.
• Particles with the same velocity will always remain an equal distance apart (they are traveling at the same speed).
• Between any two particles with the same velocity will be just one particle with a differing velocity (this might be a little harder to picture).
• Because the elastic collisions exchange speed and reverse direction, particles of the same speed can be treated as though they "ghost" through each other.

We could now paint strips of film with evenly spaced dots. Each strip represents one velocity and we could slide the strips along at the corresponding velocity to track the effective position and velocities of the particles. We know the "real" particles are just bouncing back and forth, but since they are identical, one is as good as another. We just need to know where a particle is located and its velocity.

We can conclude that:

• The wavefront (the location of the first particle traveling at +300.1) moves at +300.1.
• There is a reverse wavefront moving at -300.
• These wavefront divide the 1D world into 3 regions.
• Region 1 (right of the rightmost front): average speed 300, average velocity 0 (as at the start).
• Region 2 (between the two fronts): average speed 300.05, average velocity +0.05.
• Region 3 (left of the leftmost front): average speed 300, average velocity +0.1 (as at the start).

It's a curious model. Besides being 1 dimensional and imposing a very regular starting condition, it seems to follow all the laws of an ideal gas without really behaving as one would expect.

• The wavefront advances faster than the average speed of either set of starting particles (+300.1 instead of +300).
• The particles behind the front are traveling at +300.05, not +300.1.
• There is a reverse wavefront that shouldn't exist.

The single dimensionality seems to prevent the Maxwell-Boltzmann distribution that you might get with 2 and certainly with 3 dimensions.

 

[vanhees71]

The Java applet alone is useless, because it's not explained what's simulated. As far as I can see they start with a Boltzmann equation and then let it run. Whether or not they have interactions between the particles or how they treat the walls I can't say.

 

[A.T.]

The 1D model has some interesting properties that are actually easy to picture without needing any programming.

The setup:

• All particles are identical.
• In the tube: particles evenly spaced alternating between either -300 and + 300 m/s.
• Outside the tube: particles evenly spaced alternating between either -299.9 and 300.1 m/s.

I assume in this 1D-line model the -300 m/s particle only collides with the +300.1 m/s particle. But in reality, you would have many different types of collisions. The -300 m/s particle could also hit a -299.9 m/s particle. Arranging the particles introduces a bias, compared to the chaotic reality.

You could try a simpler model, that introduces a disturbance without changing the speed of the tube particles:

Consider just a closed tube, filled with -300 and + 300 m/s particles. To introduce a wave you shift one of the end walls, while it's not interacting with a particle. So the next wall collision happens at a different place, but keeps the particle speed at 300 m/s. This disturbance in density then propagates at 300 m/s through the tube.

 

[hutchphd]

The 1D model has some interesting properties that are actually easy to picture without needing any programming.

Yes I sat down to work it out and realized its just a fishnet of lines when plotted t vs x. Considering

onsider just a closed tube, filled with -300 and + 300 m/s particles. To introduce a wave you shift one of the end walls, while it's not interacting with a particle. So the next wall collision happens at a different place, but keeps the particle speed at 300 m/s. This disturbance in density then propagates at 300 m/s through the tube.

obviously if one moves the wall and then later moves it back a pulse is created that moves at exactly the average particle speed. All is copacetic, yes?
If one were to try to add temperature to this 1D model the appropriate distribution of velocities would be Gaussian centered at zero, because the nonzero average speed in Boltzmann arises from the dimensionality (the volume of phase space to be pedantic). That's also the "3"that shows up in the Feynman equation I pointed to earlier.

So that was fun. Thanks for the question.

 

[Freixas]

So that was fun. Thanks for the question.

I'm glad it was entertaining!

 

[trainman2001]

There is a real world problem created by the latency as a result of the speed of sound. Train brakes work by the engineer reducing air pressure in the train line (a pipe that connects every car in a train). The triple valve in each car senses this pressure drop and allows air of about the same proportionality to enter the brake cylinders at each wheel truck and applies the brakes. The system is ingenious and failsafe since a break in the train line automatically applies all the brakes in case of a break-in-two (wish my model trains had that).

The problem arises where you have mile long (or longer) freight trains. The pressure wave is traveling approximately at the speed of sound so it can be five seconds before the tail of the train knows that the brakes should go on. At 60 mph, the rear of the train has moved 440 feet while the front has already begun stopping. Placing a slave engine in the middle of the train that receives its commands electronically and then it reduces the air in the train line, acts from its point backwards to the end effectively reduces latency by half. It's still not great.

To solve this problem, the latest innovation is to electrically apply the brakes in each car. The triple valve remains, but what activates it changes. With electronic application, the signal propagates almost instantly (we all know it's not "instant") and all the brakes apply simultaneously. This greatly reduced stopping distance since you no longer have a train still moving in the rear pushing on cars in front that are attempting to stop. It's a very expensive proposition to retrofit all the railroad stock in America to the new system and it will take a long time to do so, but it's necessary. This change would greatly reduce the amount of jack-knifing when there is a derailment since the rear of the train won't be doing all that pushing.