Exactly how do pressure waves work at the molecular level?

[Freixas]

I've searched high and low for an answer to my question and I can't find a thing.

Let's say I blow into the end of a tube. The opposite end is connected to a reed; when the air hits the reed, the reed vibrates and I can hear a sound. The harder I blow, the louder the reed sounds. I've tested this with a 15' tube while standing about 10' feet from the reed (the tube was partially coiled). The reed seems to respond immediately. If the pressure I create by blowing is transmitted down the tube at the speed of sound, then there is a 22ms latency in this system--that is, when I blow, the sound begins about 22ms later. If I blow less, the sound becomes quieter 22ms later.

Musicians can detect latencies of more than 12ms, but it varies by individual. Let's just say that I didn't notice any delays and so my test system suggests that the pressure changes at the starting end of the tube travel at close to (or faster than) the speed of sound.

Most web pages that cover this topic usually start talking about sound, where the air molecules vibrate back and forth a tiny distance and, on average, don't go anywhere. But in the case I'm looking at, if I blow air in at 0.2 m/s, then air needs to flow out the other end at around 0.2 m/s (with perhaps a slight loss due to friction) and this velocity change (not just the wave front) has to travel down the tube at roughly the speed of sound (or faster).

Because it's hard to study the situation when things are changing continuously, let's imagine that we can immediately go from a flow of 0 m/s to 0.1 m/s. At the instant I start, the molecules of air in the tube are moving, on average, at 0 m/s. The are about to encounter an incoming wave of molecules whose average velocity is 0.1 m/s.

I've tried to picture what happens by using a billiard ball model, with billiard balls sitting on a perfectly flat, frictionless surface. I imagine a row of balls spaced out evenly and lined up perfectly (this is a one-dimensional model of a 3 dimensional problem). An incoming ball hits the first ball in the line and transfers all its kinetic energy to that ball. If the incoming ball is moving at 0.1 m/s, then after the collision its velocity is 0 and the ball it hit is now moving at 0.1 m/s and headed to the next ball in line.

The ball that stopped after the collision has many more balls coming toward it at 0.1 m/s, so it will get moving again. But the wave front (the collisions at the head of the line) doesn't appear to be moving any faster than 0.1 m/s. What needs to happen is

• The wave front should be traveling faster than 0.1 m/s.
• Every molecule behind the wave front needs to be traveling 0.1 m/s.

The only derivation of the speed of sound that I've seen is at https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/17-2-speed-of-sound/. The derivation involves a lot of formulas; by the time I reach the end, I have absolutely no intuitive feel for what might be happening at the molecular level.

I'd be happy even with an explanation that showed how the wave front could move faster than the molecules flowing in without worrying about exactly how much faster this is. This still needs to explain how every molecule behind the wave front is moving at 0.1 m/s. In other words, the wave front, while moving (much) faster then 0.1 m/s, has to somehow speed up all the molecules it encounters to 0.1 m/s (on average).

 

[Arjan82]

This is a visualisation of sound waves, where the molecules on average travel at 0 speed:

For your case the cylinder should stay at the right most position. But I cannot find a visualisation of that.

 

[Freixas]

For your case the cylinder should stay at the right most position. But I cannot find a visualization of that.

That's been my problem, too. The visualization above is also a simplified to where it's a bit confusing. We can sort of intuit why the particles move to the right when the compression wave hits, but it's less obvious why they move the left when the rarefication wave hits. I understand why—I'm just saying this model doesn't provide any clues.

Also, it's probably not correct to say that, for my case, the piston should stay to the right. In my case, the piston should keep moving to the right, although maybe that's not correct either. In my case, the piston is a long ways outside the tube. In moves continuously in the direction of the tube without reaching the tube and we begin at time 0, when the first molecules moving at 0.1 m/s reach the tube entrance.

 

[hmmm27]

I'm not quite sure of the confusion...

If you take a solid steel bar, and push one end at $0.1m/s$ are you surprised when the other end moves (seemingly) immediately ?

 

[hutchphd]

.
Yes I find it more surprising for gas.

For the OP Feynman's lecture on sound is worth looking at (just hit the highlights)

In particular he shows the speed of sound $$c_s=\sqrt \frac \gamma 3 v_{ave}$$

Where $v_{ave}$ is the average thermal gas molecule speed and $\gamma$ is ~1.4 for air

 

[Freixas]

I'm not quite sure of the confusion...

If you take a solid steel bar, and push one end at 0.1m/s are you surprised when the other end moves (seemingly) immediately ?

No force is transmitted "immediately". A force applied to one end of a steel bar is limited at least by the speed of light, although I believe it travels much more slowly.

Everything I'm interested in is in the parenthesized disclaimer "seemingly". In any case, the explanation for a steel bar is totally different from the explanation for a gas. But even for a steel bar, the question should not be whether I was surprised; it should be why you wouldn't be curious. How exactly does the force applied to one end of a steel bar get converted to kinetic motion for particles at the other end of the steel bar?

Forces travel differently in solids than in gases because the molecules are connected in one and not the other. I'm not saying I understand the process any better in solids than in gases; I'd be happy to learn the answer for just gases right now.

Here's a picture of what happens based on observations:

We have a stream of air flowing in. Let's assume we can increase the air flow in steps of 0.1 m/s. The diagram shows what happens as the air flow increases to 0.3 m/s and then drops back to 0.

After 1 ms, everything in the green box is moving at 0.1 m/s. The thin black vertical line near the opening is suggestive of how far the air the flowed into the tube has moved—it's not to scale. The pressure wave is traveling at the speed of sound, however, and has covered about 1/3 of a meter. The air behind the pressure wave is all traveling at 0.1 m/s.

In 3 ms (roughly), air flows out of the tube at 0.1 m/s. It continues to do this for 1 ms, after which it jumps to 0.2 m/s, etc.

If I try to picture this at the molecular level, I picture billiard balls. As I described in my OP, I can't see any way to make the pressure wave travel faster than the speed of the air coming in. Even if I could, I have to explain why the molecules behind the pressure wave are accelerated to 0.1 m/s, then 0.2 m/s, etc. Explanations for sound waves usually have the pressure wave moving at the speed of sound, but the particles just "vibrate" in place.

Since everything really happens as general trends in the somewhat random interaction of trillions of molecules, it seems it should be possible to explain what's going on at that level. I thought this would be a simple question, but I've asked it elsewhere as well to a deafening silence.

 

[Freixas]

.
Yes I find it more surprising for gas.

For the OP Feynman's lecture on sound is worth looking at (just hit the highlights)

In particular he shows the speed of sound $$c_s=\sqrt \frac \gamma 3 v_{ave}$$

Where $v_{ave}$ is the average thermal gas molecule speed and $\gamma$ is ~1.4 for air

Thanks. I'll take a look—maybe it will clarify things. As I said, I've seen a mathematical derivation (I included a link in the OP)—it makes nothing intuitive.

It's interesting the derivation I saw derives a different formula.

 

[DaveC426913]

If I try to picture this at the molecular level, I picture billiard balls. As I described in my OP, I can't see any way to make the pressure wave travel faster than the speed of the air coming in.

You are assuming the velocity of an undisturbed air molecule is zero. It isn't. It's between 300 and 500m/s. Air molecules don't just float there like in the animation. They are are moving all the time, constantly bouncing off one another, at 3-500m/s due to their kinetic energy. That is what gives a gas its volume. Otherwise, it would collapse to almost zero volume.

When a molecule gets bumped by the inrushing air, its velocity is only changed to somewhere from 299.9m/s to 300.1m/s.

In other words, it could be said that the individual molecules in the ebb/trough of the waves still have a velocity of a whopping 299.9m/s in the direction against the stream of air. They can easily backfill the rarefied volume.

 

[hutchphd]

Thanks. I'll take a look—maybe it will clarify things

Start at the end (sec 45-5) and expand from there. Hope it helps.

 

[Freixas]

You are assuming the velocity of an undisturbed air molecule is zero.

Actually, no, I was assuming the average velocity of air molecules in the tube at the start was 0.

When a molecule gets bumped by the inrushing air, its velocity is only changed to somewhere from 299.9m/s to 300.1m/s.

Agreed. And, on average, the velocity of a billion or so of these molecules is now 0.1 m/s in the direction of the tube's axis.

They can easily backfill the rarefied volume.

Yes, agreed, I have no problem with the concept of how many random molecule collisions cause molecules to travel from areas of compression to areas of rarefication.

I still have the same two questions:

• From a molecular point of view, why does the pressure wave travel much faster than the average speed of the molecules that caused it? I don't need the formula or the derivation of the formula or an exact number—just a sense for how it happens. Everything that happens in a gas is due to transfer of kinetic energy through collisions—or, at least, that's how I think it works.
• Again, from a molecular point of view, why the do molecules trailing the pressure wave get accelerated up to (in the first step) 0.1 m/s? Keep in mind the pressure wave's speed is the same in all cases, no matter what the speed is of the air left behind. (And, yes, in some of my steps the air following the pressure wave is decelerated.)

The one-dimensional billiard ball model was my attempt at condensing billions of molecules into averages. Each billiard ball represents the average velocity of billions of molecules. The spacing between the billiard balls also represents the average spacing of those molecules. My theory is that what happens to the billiard balls and the average of what happens to the molecules is the same. Clearly, this can't be right since I can't create a pressure wave that travels faster than the incoming molecules.

 

[hutchphd]

From a molecular point of view, why does the pressure wave travel much faster than the average speed of the molecules that caused it?

It does not! Read Feynman. Thats the point...

 

[hutchphd]

The collisions all take place at the speed of sound. The wave travels at that speed. But the drift velocity of the net motion mimics the drift velocity of the incident wind. (0.1 m/s)

 

[DaveC426913]

From a molecular point of view, why does the pressure wave travel much faster than the average speed of the molecules that caused it?

Because that molecule is actually moving at >= 300m/s. If it gets a bump of .1ms, now it can travel at 300.1m/s to reach the next molecule - way faster than .1m/s. It's only adding .1m/s to the already existing 300m/s. And that is the speed at which one molecule can transmit its kinetic energy to the next.

Again, from a molecular point of view, why the do molecules trailing the pressure wave get accelerated up to (in the first step) 0.1 m/s?

They don't. They get accelerated from 300m/s to 300.1m/s.Think about a bar of steel. If a wave could only be transmitted as fast as the input, then moving one end up at .1m/s would mean that wave could only propagate the length of the bar at .1m/s.

In fact, we know that atoms in a bar of steel can transmit their energy at the speed of sound in steel.
You move the bar, and that signal is propagated at 6000m/s.

In air, it's similar. The air can transmit a signal at >= 300m/s (i.e. the speed of sound in air).

 

[Freixas]

I just got a bunch of comments. Before I say anything else, let me say I appreciate everyone's time in helping me work through this.

It does not! Read Feynman. Thats the point..

This is in response to my comment "From a molecular point of view, why does the pressure wave travel much faster than the average speed of the molecules that caused it?"

Hmm... I thought I phrased this carefully. The average speed of the molecules is 0.1 m/s (as an example). The speed of the pressure wave is something like 343 m/s. So perhaps I'm being dense, but it looks like my original comment is correct.

I did wade through part of Feynman's post. He seems to drop the molecular view pretty early on: "It is clear that we are going to describe the gas behavior on a scale large compared with the mean free path, and so the properties of the gas will not be described in terms of the individual molecules."

So perhaps there is no way to ever conceptualize the behavior at the level I want. I'll keep wading, though.

The collisions all take place at the speed of sound. The wave travels at that speed. But the drift velocity of the net motion mimics the drift velocity of the initial wind. (0.1 m/s)

The Feynman post doesn't include the term "drift" anywhere, but I can look it up. My guess is that you mean the average velocity.

The collisions all take place at the speed of sound? I'm just not sure what you mean. I thought the molecules were all zipping about with different velocities, so collisions could occur at all sorts of speeds, but maybe you meant something else.

Because that molecule is actually moving at >= 300m/s. If it gets a bump of .1ms, now it can travel at 300.1m/s to reach the next molecule - way faster than .1m/s. you're only adding .1m/s to the already existing 300m/s. And that 300/ms is the speed at which one molecule can transmit its kinetic energy to the next.

I cannot picture a particle moving at 0.1 m/s catching up to a particle traveling at 300.1 m/s to transfer all its kinetic energy to the latter and create a particle traveling at 300.1 m/s.

But let me try your system. I'm going to have to simplify to have any hope of an intuitive understanding:

• Let's keep all motion 1-dimensional.
• Let's begin with all particles evenly spaced.
• Each particle in the tube is moving at either -300 or +300 m/s.
• The particles in the tube alternate velocity, so if one is going at -300, then next is going at +300.
• The incoming particles are set up the same way, but with velocities from -299.9 to +300.1 m/s.

Let's say the first incoming particle is moving at +300.1 and the first particle in the tube is moving at -300. We place these two particles at the tube entrance to begin.

This leaves us with one particle traveling at +0.1 at the entrance. All other particles in the tube then collide with a net velocity of 0. All other incoming particles collide with an net velocity of +0.1. I'm back to my original billiard ball model. I don't see how randomizing the velocities will help or even randomizing the velocity's direction as long as the averages remain at 0 for the tube and 0.1 for the incoming air.

Oh, well... I'll go ponder Feynman some more and maybe lightning will strike.

 

[russ_watters]

Let's say the first incoming particle is moving at +300.1 and the first particle in the tube is moving at -300. We place these two particles at the tube entrance to begin.

This leaves us with one particle traveling at +0.1 at the entrance.

How did you get that one particle at +0.1? These are elastic collisions, so after the collision the incoming particle is traveling at -300 and the first particle in the tube is traveling at +300.1.

 

[Freixas]

How did you get one particle traveling at +0.1 ? Assuming elastic collision and same mass, the -300 will change to +300.1

Ignorance? Walk me through this: +300.1 meets -300. Are you saying that after collisions the +300.1 becomes -300 and the -300 becomes +300.1?

If so, I can try to step through my simplified version again.

 

[russ_watters]

Ignorance? Walk me through this: +300.1 meets -300. Are you saying that after collisions the +300.1 becomes -300 and the -300 becomes +300.1?

If so, I can try to step through my simplified version again.

Consider if the particle hits a brick wall and the collision is elastic. What velocity does +300.1 rebound at?

 

[Freixas]

Consider if the particle hits a brick wall and the collision is elastic. What velocity does +300.1 rebound at?

No, let's not use a brick wall. Let's get two elastic cars (equal mass). First, the head straight towards each other at 300 m/s. My assumption was that the net result would be tow cars at 0 m/s, but I realize now that that doesn't conserve kinetic energy (I peeked at Wikipedia while you were typing). They swap directions and maintain speed.

The next case is that one of the cars is moving at 300.1 m/s and the other at 300 m/s. According to Wikipedia, the velocities and direction swap.

As I said, let me work through the one-dimensional example again.

 

[russ_watters]

No, let's not use a brick wall. Let's get two elastic cars (equal mass). First, the head straight towards each other at 300 m/s. My assumption was that the net result would be tow cars at 0 m/s, but I realize now that that doesn't conserve kinetic energy (I peeked at Wikipedia while you were typing). They swap directions and maintain speed.

Elastic cars are tough because cars are designed to be inelastic, but ok...

 

[jbriggs444]

average speed

There is a difference between average speed (300 m/s, give or take - directionless) and average velocity (0.1 m/s, give or take - vector-valued).

 

[hutchphd]

But let me try your system. I'm going to have to simplify to have any hope of an intuitive understanding:

In your model the gas you introduce will also be at temperature so your model is no good. The gas you supply (blow in) is at room Temperature so it is also zooming at +/-300m/s plus 0.1m/s.
Let's do this instead:
Suppose you increase the "rebound" speed of the left end molecule of the 1D gas (by 0.1m for one rebound). That speed anomaly will be transferred at each subsequent particle collision along the chain until it reaches the right end. The speed of the transfer will be 300m/s and the excess speed of the "other" end particle will be 0.1 m/s . I think this works to explain the physics?

 

[DaveC426913]

 

[Freixas]

Thanks everyone!

Using the helpful prod on elastic collisions and the notion of speed (directionless) vs. velocity (vector), all this seems to fall into place. The speed of sound in dry air at 20° (presumably at 1 atmosphere) is 343 m/s. So all the air in my example has an average speed of 343 m/s with the air in the tube having an average velocity of 0 and the air coming in having an average velocity of 0.1 m/s into the tube, then the solution seems intuitive (easy to imagine without needing pencil and paper or other device).

But Wikipedia says: "The speed of sound in an ideal gas depends only on its temperature and composition. The speed has a weak dependence on frequency and pressure in ordinary air, deviating slightly from ideal behavior." Let's ignore frequency and pressure for the moment.

I set up my one-dimensional model. Using my new concepts, I see the wave front advancing at a velocity of, not 343 m/s, but 343.1 m/s. Oops!

My one-dimensional model test case had the particles in the tube traveling at either +343 m/s or -343 m/s. The ones streaming in had velocities of either -342.9 m/s or 343.1 m/s. So the wave front basically swaps a -343 m/s particle with a +343.1 m/s particle.

I had better draw this. In the tube at the wave front, we see this

+343.1 → ← -343 becomes -343 ← → +343.1​

The particles streaming in look like:

+343.1 → ← -342.9 +343.1 → ← -342.9 +343.1 → ← -342.9 +343.1 → ← -342.9​

so all the rightward moving particles are at 343.1. At the wavefront, we can expand the picture to look like this:

+343.1 → ← -343 +343.1 → ← -343 +343.1 → ← -343 +343.1 → ← -343 +343.1 →​

Hmm... The average speed is 0.1 m/s. The average velocity is 343.05 m/s, but the wavefront is at 343.1 m/s. You can't have the wavefront influenced by the average speed or else you get all sorts of artifacts.

(DaveC426913, your drawing arrived while I was typing all this. It helped me find the problem in the above diagram)

Ok, the error is that I have particles colliding in synch even though they are moving at different speeds. Using DaveC426913's diagram, it is much clearer that the wavefront advances at +343 m/s (or whatever the average speed is) and the particles left behind move at an average speed of 0.1.

The other thing is Wikipedia's claim that this depends on temperature and composition. I seem to have only taken temperature into account. I'm assuming composition includes fluid density. If we pack the particles closer or further in my little model, it makes no difference (a change in temperature, however, seems to have a direct influence).

This is quite a step up from where I was when I started!

 

[jbriggs444]

The speed of sound in dry air at 20° (presumably at 1 atmosphere) is 343 m/s. So all the air in my example has an average speed of 343 m/s

The speed of sound is the speed at which bulk density variations propagate. It is related to the speed of the air molecules but it is not identical to that speed.

 

[Arjan82]

The other thing is Wikipedia's claim that this depends on temperature and composition. I seem to have only taken temperature into account. I'm assuming composition includes fluid density. If we pack the particles closer or further in my little model, it makes no difference (a change in temperature, however, seems to have a direct influence).

I think what they mean by composition is really the type of molecules. If I remember correctly from classes a long long time ago molecules also have internal vibration modes, i.e. vibration of the different atoms within a molecule, and molecules can also rotate and thus may have different angular momentum (These are considered internal degrees of freedom). In this case the elastic collisions can also convert energy into these vibrations or into rotation which means the velocity after collision is different, this matters.

Density itself is not directly influencing the speed of sound. Read also this part on wiki:
https://en.wikipedia.org/wiki/Speed_of_sound#Details

In the end the speed of sound, $c$, for an ideal gass is:

$$
c = \sqrt{\frac{\gamma \cdot k \cdot T}{m}}
$$

It is a function of:

• $T$: Temperature: some measure of the average kinetic energy of the particles
• $k$: The Boltzmann constant, relating the temperature to the average kinetic energy
• $m$: The mass of a molecule, equal momentum for a particle with higher mass means lower velocity
• $\gamma$: And the adiabatic index, this is actually taking into account the internal degrees of freedom of a gas. According to the lecturenotes on gasdynamics I still have: "for air considered to be a di-atomic gas with 5 degrees of freedom (i.e. $n= 5$$) \gamma = \frac{n+2}{n} = 7/5 = 1.4$)" which is indeed the adiabatic index for air.

By the way, this formula for $\gamma = \frac {n+2}{n}$ apparently follows from the kinetic theory of gasses. Which is actually trying to model the properties of a gas by colliding particles.

 

[Freixas]

I think what they mean by composition is really the type of molecules.

Thanks for the clarification. I had seen the notes you mentioned, but I didn't know how to translate them into my billiard ball model. I think I see now. Terms $T$, $k$, and $m$ are what you need to calculate the average speed of the molecules using things we can measure at the macro scale.

$\gamma$, the adiabatic index, is harder to understand. I looked into it a bit. It sounds like it is another factor needed to compute average speed of the particles because the collisions may not be purely elastic; some energy could be transferred to rotational motion (I'm imagining it like putting spin on a billiard ball), for example, and not contribute to the average speed.

So, if we somehow (magically) know the average speed of the particles, we don't need to look at anything else—we know the speed of sound in that medium. Wikipedia's statement about temperature and composition just means that, lacking magic, we need to know those things in order to determine the average speed.

Thanks, everyone! I didn't think I'd get an answer I could follow.

 

[hutchphd]

Which is of course the same result derived in Feynman (https://www.physicsforums.com/posts/6398388/bookmark) .
In equilibrium (meaning Temperature is defined) each degree of freedom (possible motion of a system) is apportioned on average $\frac {kT} 2$ of energy. It is the definition of Temperature. So physics supplies the magic...

 

[vanhees71]

It's often forgotten to mention a very important assumption when quoting the equipartition theorem. The complete statement is: Each phase-space degree of freedom that appears quadratically in the Hamiltonian provides and average energy $k_{\text{B}} T/2$ when in thermal equilibrium with a heat bath at (absolute) temperature $T$.

 

[Freixas]

View attachment 270168

Sorry, I'm back. I was thinking of how to explain this to someone else (always a good test) and things aren't working out.

The diagram above mixes average relative velocities (.1 m/s) with absolute speeds (300 m/s). So the blue line, if we were to remain consistent, would be a particle traveling at 300.1 m/s. We can still follow the diagram from this point, though.

The diagram states that the .1m/s is propagated at 300 m/s. Given what I learned about elastic collisions, the particle hit by the incoming 300.1 m/s particle will now move right at 300.1 m/s. Every 300 m/s particle it hits from then on will move at 300.1 m/s. So the wave front is propagated down the tube at 300.1 m/s.

To infer this, I don't need to calculate exactly when each collision occurs. I just need to track the leading particle, which is always going to move at 300.1 m/s. It's not the same particle all along the way, but that doesn't matter.

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?

 

[jbriggs444]

Sorry, I'm back. I was thinking of how to explain this to someone else (always a good test) and things aren't working out.

The diagram above mixes average relative velocities (.1 m/s) with absolute speeds (300 m/s). So the blue line, if we were to remain consistent, would be a particle traveling at 300.1 m/s. We can still follow the diagram from this point, though.

The diagram states that the .1m/s is propagated at 300 m/s. Given what I learned about elastic collisions, the particle hit by the incoming 300.1 m/s particle will now move right at 300.1 m/s. Every 300 m/s particle it hits from then on will move at 300.1 m/s. So the wave front is propagated down the tube at 300.1 m/s.

To infer this, I don't need to calculate exactly when each collision occurs. I just need to track the leading particle, which is always going to move at 300.1 m/s. It's not the same particle all along the way, but that doesn't matter.

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?

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.

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.

 

[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 Boltzman 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.