Wave speed dependent on medium not source?

[Essence]

Homework Statement

So in my textbook it says

"The speed of the wave depends on properties of the medium, not on the motion of source or observer. An explosion causes pressure variations in the air around it. This "deformation" propagates outward as a sound wave at a speed dependent only on properties of the medium-air."

Homework Equations

This doesn't make sense to me using the idea of conservation of momentum. I am assuming I can make an analogy between air and billiard balls. Pretend I hit the first billiard ball with a lot of force. I expect it to accelerate to a certain high speed. Once it hits the next billiard ball (let's pretend they are the same mass) the second billiard ball will take that same speed and so on. If I ignore the collision process I can say that the location of the object holding the momentum is moving at speed v1 which is faster than some other hypothetical if I had hit the first billiard ball with less force.

I could elaborate with different possibilities of interpretation, but it's probably better if I wait.

The Attempt at a Solution

 

[DrClaude]

Springs are a better analogy to understand the propagation of waves. Depending on the stiffness of the springs, it will be easier or harder to compress the first spring, and then the propagation of the compression/expansion zones will depend essentially on how a spring can compress its neighbors and relax.

 

[BvU]

Dear E, welcome to PF !

You put down a challenging analogy. I think it would work somewhat like that if you throw a stone in a sandbox with quite some speed: the sand flies out and stops where it lands. The essence of waves is that there is some restoring force (hence Claudes springs). Throw the stone in a pond and the water returns to its equilibrium level. The disturbance spreads out like a wave. Circular and with a speed that doesn't depend on the shape, weight of the stone, nor on its speed at landing.

The billiard ball analogy doesn't hold very well for sound because, unlike your billiard balls, air molecules move around at high speed. What we call pressure is the force per area of the collisions of these molecules. A wave is a disturbance in the randomness of the motion that evens out very quickly. That is the restoring force and the strength of the restoring force doesn't depend on the source of the disturbance.

 

[Essence]

Ok so would it be fair to model this as a bunch of springs lined up in a row such that when I hit the first one it compresses the first end towards the second and then follows by expanding the second away from the first into the next spring. The way that I think this might be a fair model is that if the air molecules were so close together that the collisions between the air molecules was the majority of the process (I suppose I would have to justify this logic more later, if the model is ok). I am still not sure if this will fit the expected result and I haven't yet found a way to fit the high speed of air into anything. I can think the disorganization of air could affect the chain, but I'm leaving that as a deviation from the ideal rather than a cause for it. Hope I haven't missed anything.

Note: the way I put the string model into two steps of the first approaching the second and the second moving away from the first I believe isn't what really happens, but I put it into two steps in order to gain access to a visual.

Also I apologize if what I wrote may have reiterated what was mentioned; I want to make sure that my intake was what was expected from the text I read.

 

[BvU]

Been pondering over this hard-ball analogy. After all, you are definitely right that momentum conservation holds. But I'm also pretty shure that even supersonic bangs propagate with the speed of sound and not with the speed of the airplane.

Come to think of that, an airplane would impart an enormous amount of momentum to an air molecule: the air molecule would fly forward after a collision with speed approaching the speed of light ? Not likely...

Fortunately all can be harmonized by realizing the initial momentum isn't redistributed by single one-on-one central collisions with non-moving air molecules, but by many collisions. The disruption in the evenness and symmetry of the molecular velocity distribution is restored by processes that aren't influenced by the amplitude of the initiating disturbance.

Notice how I try to word it carefully. My compliments for your not taking everything that's thrown at you for granted !

The speed of propagation depends on the force that restores equilibrium. That's very clear with solids . There they use the spring analogy with great success. For gases the derivation of the speed of sound is awkward in my view -- not suitable for introductory texts. I suppose that's why they just state things without rigorously underpinning. Didactically reasonable, because they want to move on to wave physics, not to much more involved thermostatistics.

 

[Essence]

Ok this is on my phone.
http://www.britannica.com/EBchecked/topic/541339/shock-wave

Seems to suggest that the deformation of pressure (shock wave) travels faster than sound, which would mean the book is just wrong (or meant to be interpreted some other way). Perhaps I'm totally missing something about what is being talked about but that seems about right. From a "large" distance away maybe pressure differences go at the speed of sound. I hope I'm not misinterpreting something completely or misconstruing the definitions.

Also somewhat related, but wouldn't the momentum of the air particles "die down" eventually and sound travel slower than the speed of sound?
I suppose if what you said earlier is true than I have nothing to worry about and I should just wait for a higher level course.

 

[DrClaude]

Seems to suggest that the deformation of pressure (shock wave) travels faster than sound, which would mean the book is just wrong (or meant to be interpreted some other way).

Sound waves are infinitesimal pressure perturbations and travel through the medium at the speed of sound, which is completely dependent on the properties of the medium. Finite amplitude pressure fronts are shock waves, and travel necessarily at a speed that is greater than the speed of sound. I guess (since that I don't know the context) that the textbook was concentrating on the first case.

Also somewhat related, but wouldn't the momentum of the air particles "die down" eventually and sound travel slower than the speed of sound?

The perturbation gets weaker (the sound dies down), but its travel speed does not change.

 

[BvU]

Ok this is on my phone.
http://www.britannica.com/EBchecked/topic/541339/shock-wave

Seems to suggest that the deformation of pressure (shock wave) travels faster than sound, which would mean the book is just wrong (or meant to be interpreted some other way). Perhaps I'm totally missing something about what is being talked about but that seems about right. From a "large" distance away maybe pressure differences go at the speed of sound. I hope I'm not misinterpreting something completely or misconstruing the definitions.

Also somewhat related, but wouldn't the momentum of the air particles "die down" eventually and sound travel slower than the speed of sound?
I suppose if what you said earlier is true than I have nothing to worry about and I should just wait for a higher level course.

I am impressed ! My image was that you justifyably questioned something in your textbook that was stated without reasonable underpinning and I tried to steer towards the usual wave physics with which I expect the book continues -- didn't verify that: what kind of book is it ?

But yes, shock waves exist (and I know very little about them: they aren't your usual bread and butter physics). As your link documents. For a normal curriculum treatment is generally limited to phenomena where the wave equation is valid. Difficult enough already ... and a very rich subject with many, many applications.

 

[Essence]

Sound waves are infinitesimal pressure perturbations and travel through the medium at the speed of sound, which is completely dependent on the properties of the medium. Finite amplitude pressure fronts are shock waves, and travel necessarily at a speed that is greater than the speed of sound. I guess (since that I don't know the context) that the textbook was concentrating on the first case.The perturbation gets weaker (the sound dies down), but its travel speed does not change.

I don't really expect you to have one but just in case, is there some sort of site that you know of that describes this in a ground-up fashion. I happen to know that for springs oscillation time is constant regardless of amplitude (and I believe I can still prove this out); I imagine that this fact may relate to the constant speed of sound but I'm having trouble getting from point A to point B. I have conceptually considered the idea of pushing one spring into another one and so forth, but this seems to (without rigorous mathematics) lead to a conclusion that is deviated from the expected. I suppose that if the added pressure due to the forced movement of the air is practically infinitesimal relative to the pressure of the atmosphere the expression for sound movement may be dominated by predictable atmospheric reactions rather than the actual amplitude of the disturbance. The atmospheric reaction that I am thinking of is just other air molecules filling in the vacuum caused by the disturbance, and so forth. I don't know if this leads to the right answer, but judging from the comments I think it might. I will think about this more and try to formalize. I hope I'm heading down the right track. Thanks.

Edit:

I have come across

http://www.grc.nasa.gov/WWW/K-12/airplane/snddrv.html

sifting through it now... it seems that the equations are generally stated without prior build-up

 

[Essence]

I am impressed ! My image was that you justifyably questioned something in your textbook that was stated without reasonable underpinning and I tried to steer towards the usual wave physics with which I expect the book continues -- didn't verify that: what kind of book is it ?

But yes, shock waves exist (and I know very little about them: they aren't your usual bread and butter physics). As your link documents. For a normal curriculum treatment is generally limited to phenomena where the wave equation is valid. Difficult enough already ... and a very rich subject with many, many applications.

This is actually more for my own curiosity. I am technically opening up the book for next year's Physics class called "Modern Physics" by Randy Harris. It mentions this early in chapter 2 related to the idea of ether (for light); I believe it sets up the idea of waves moving independently of the source for the purpose of suggesting an experiment that could disprove the existence of ether. For that experiment to hold it assumes things like ether having a uniform velocity over its entirety, which experimentally would make sense (et al.). For the most part my question is unrelated to the material in the book.

Thanks for the compliment BTW.

 

[DrClaude]

I don't really expect you to have one but just in case, is there some sort of site that you know of that describes this in a ground-up fashion.

You can have a look at Hyperphysics: Sound speed in gases, but I don't know if the explanation will satisfy you. To get more details, you need to look in a book on fluid mechanics. A good one is Fluid Mechanics by Kundu and Cohen, see ch. 16 Compressible Flow.