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Natural Science Forum / Physics / Acoustics / September 2004Physics, Sound, and Mass Transport
I am currently taking physics courses, at my advanced age of 65, at UC Santa Barbara, which has an absolutely first rate Physics department. One Ph.D. is all I need, so I am doing this just to keep my brain from rotting. At UCSB you can learn all about quantum mechanics and string theory, but forget about sound. The graduate students I talk to have at best heard a passing treatment of sound, usually presented as a trivial introduction to the study of waves (along with the obligatory violin string), as a solution of the one-dimensional scalar wave equation. Sound is totally ignored by "serious" physicists. Acoustics is still a respectable subject, but it is usually taught from an engineering point of view rather than as a topic in physics. A roadblock in the serious study of sound is that to really do it right, you have to deal with the statistical behavior of molecules. At UCSB I have taken an excellent course in statistical mechanics, but the course is focused on photon gases and superfluid behavior of liquid hydrogen - forget sound! When sound is taught at an introductory level, students typically do not have enough knowledge of statistics to teach it properly. I created my web site, www.physicsofsound, a serious physics treatment of sound, as my attempt to fill this gap. When you treat sound from the molecular point of view, aspects become evident that are not at all evident from the traditional point of view. I am only aware of one textbook, by Vincenti and Kruger, that treats sound from this point of view. (If my stuff is not easy to understand, this text is virtually impenetrable). One example, which recently came up in this newsgroup, involves the garden-variety solution for a harmonic sound wave, and mass transport. If you look at this solution at the molecular level, there is actually a net flow of molecules in the direction of wave propagation. How do I prove this? I consider a collection of molecules that are moving in mainly random directions. But if a sound wave is present there is a systematic component, a non-zero mean value, of the velocity. I compute the spatial flux of mass, momentum, and energy due to the velocity, both random and systematic components, of the molecules. (This is actually not all that complicated as long as you have a basic knowledge of statistics). If you evaluate the result for the usual sound wave solution, there is a non-zero time-average mass flow in the direction of propagation! In the real world, this doesn't happen. The error is in the solution - you need to include a small uniform velocity component. Now this is a small effect, and if you are a professional acoustical engineer or sound engineer you could probably care less. I have also never seen this mentioned in any of the many excellent books I have read. So why should you believe me? One thing I love about physics is that you don't have to believe any human authority - just check out the math for yourself. Hello Art, A good idea is the serious physics treatment of sound, as an attempt to fill the missing gap. Your link www.physicsofsound gives a "site cannot be shown" error. If I guess the address http://www.physicsofsound.com it is sent to: http://www.silcom.com/~aludwig/Physics/Main/Physics_of_sound.html There are many more nice entries: Art Ludwig's Sound Page http://www.silcom.com/~aludwig/ Table of contents http://www.silcom.com/~aludwig/contents.htm Crossover Design http://www.silcom.com/~aludwig/Sysdes/Crossove_Design.htm Construction of loudspeakers http://www.silcom.com/~aludwig/Loudspeaker_construction.html Maximum Length Sequences http://www.silcom.com/~aludwig/Signal_processing/Maximum_length_sequences.htm Glossary of Electro Acoustic Terms http://www.silcom.com/~aludwig/glossary.htm Kind regards Eberhard Sengpiel German forum for microphone recordings and sound studio techniques http://www.sengpielaudio.com > If you evaluate the > result for the usual sound wave solution, there is a non-zero time-average > mass flow in the direction of propagation! In the real world, this doesn't > happen. The error is in the solution - you need to include a small uniform > velocity component. Not sure what you mean here. You say that it is predicted from from your analysis yet it doesn't happen in the real world due to an error in the solution. What solution has the error in it? Any analysis that predicts a net mass flow will violate the conservation of mass in a situation like a diapragm in a tube. Does that not say that in that case the math, however elegant, fails to represent the physics? With physics, math can only do as well as the assumptions allow. A flawed assumption can lead to beautiful mathematical nonsense. Einstein demonstrated that most beautifully with special relativity. Bob  Signature"Things should be described as simply as possible, but no simpler." A. Einstein > One example, which recently came up in this newsgroup, involves the > garden-variety solution for a harmonic sound wave, and mass transport. If > you look at this solution at the molecular level, there is actually a net > flow of molecules in the direction of wave propagation. How do I prove this? Long ago, it was observed that a "Sonic Wind" occurs in the directions of travel of high amplitude sound waves. By the time of my MS thesis work (1952), this was a common observation. We found that the subsequent on-axis refractive effect could be minimized by a cross-winds from an ordinary fan. Looking at this in retrospect: It may be that this sonic wind enjoys a cumulative nature where the wind velocity results from the integral of the wind force along the entire propagation path; the fan disrupting this accumulation. Generally, then, assuming your conclusion based on statistical grounds is indeed true, a sonic wind always exists, but it takes either a high amplitude or a very long propagation path to demonstrate it. A pertinent experiment to perform is the following: Produce sound in a very long tube. Instrument a static pressure gage (water-column type the most common). A static pressure buildup should occur. Plan B is to perforate the source end of the tube wall. A, exit air flow should be observable. Acoustical pumping, anyone? Angelo Campanella Thanks Eberhard, my site is indeed www.physicsofsound.com I failed to communicate well in my last paragraph. My first mistake was to not be more specific about the "real world." I was talking about a system, such as a piston vibrating in a tube, where obviously no mass transport is possible. Allow me to try again: (1) postulate a system where mass transport is impossible; (2) the standard equation for a simple harmonic sound wave does involve mass transport; (3) therefore the standard solution is not the correct solution for this system. Now the standard solution is very nearly correct. The standard solution satisfies the wave equation, but it does not satisfy the condition of no mass transport. The only thing needed to achieve a valid solution is to add a small constant velocity. A constant velocity satisfies the wave equation, and the sum of the standard solution plus a constant velocity satisfies the wave equation. The constant velocity is adjusted such that the mass transport is zero, and viola there is the correct solution. I also should have specified that I was talking about a sound with amplitude in the linear region, not the case Angelo refers to. > Now the standard solution is very nearly correct. The standard solution > satisfies the wave equation, but it does not satisfy the condition of no > mass transport. The only thing needed to achieve a valid solution is to add > a small constant velocity. Yes, we usually call this the fudge factor, or in this case the fudge term. > A constant velocity satisfies the wave equation, > and the sum of the standard solution plus a constant velocity satisfies the > wave equation. The constant velocity is adjusted such that the mass > transport is zero, and viola there is the correct solution. Wow! Now, about the prediction of mass transport due to b0 in your equation (4) describing "Doppler distortion" in a tube. Can you make it vanish too? The problem with your your boundry condition solution is that in (4) the velocity of the air on the LHS is given relative to the instantaneous x measured from the rest position. This gives the volume velocity of the wave for any fixed x, but if x is moving with the piston face, then the LHS is always zero rather than the sinusoid you want it to be for your boundry condtion. Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein > One example, which recently came up in this newsgroup, involves the > garden-variety solution for a harmonic sound wave, and mass transport. [quoted text clipped - 16 lines] > love about physics is that you don't have to believe any human > authority - just check out the math for yourself. Art, if I understand this correctly, you are saying that there is no net mass flow in the direction of propagation in the real world, and that usual sound wave solution is incorrect because without the constant velocity term it predicts net mass flow that we know doesn't exist in the real world. Consequently,the correct sound wave solution requires a constant velocity term in order for the solution to yield a net zero mass flow in the direction of propagation, which conforms to the real world situation. > Art, if I understand this correctly, you are saying that there is no net > mass flow in the direction of propagation in the real world, and that usual [quoted text clipped - 3 lines] > term in order for the solution to yield a net zero mass flow in the > direction of propagation, which conforms to the real world situation. That is exactly correct. For the people who are surprised that the usual solution involves mass transfer, I will repeat a description I give elsewhere. What I call the "usual solution" has a pressure variation p=p0sin(wt-kx) perfectly in phase with a mean molecular velocity in the x-direction u=u0sin(wt-kz), where p0 and u0 are related by the impedance of air. Both velocity and pressure are perfectly symmetrical functions. At a wave crest the pressure is maximum, and the velocity is maximum in the +x direction. At a wave null, both pressure and velocity are the exact negative of the values at the crest. Now what is kind of hidden in this description is the fact that pressure is proportional to temperature and the molecular density. Specifically, the molecular density is higher at the crest than at the nulls. So you have a high density moving forward at a velocity u0 at the crest, and a low density moving backward at a velocity negative u0 at the null. Thus there is a net mass in the forward direction. >> Art, if I understand this correctly, you are saying that there is no >> net mass flow in the direction of propagation in the real world, and [quoted text clipped - 25 lines] > velocity negative u0 at the null. Thus there is a net mass in the > forward direction. From the courses that I took years past, I thought it was standard practice to add a constant to any solution of a differential equation and to then determine the value of the additive constant using the boundary conditions of the problem. Is this what either should have or could been done with the standard solution, or is there a more fundamental underlying issue? [snips] > From the courses that I took years past, I thought it was standard practice > to add a constant to any solution of a differential equation and to then > determine the value of the additive constant using the boundary conditions > of the problem. Is this what either should have or could been done with > the standard solution, or is there a more fundamental underlying issue? Yes, that's my experience as well regarding the solution of differential equations, and that is simply what I have done in this case. But I have never seen this done or even mentioned in the case of sound plane waves or the piston in a tube problem. As you know, when this constant term popped up in my numerical solution for a piston in a tube, I pointed out that it corresponded exactly to the value I had previously computed based on the zero mass transfer condition. There is nothing in the numerical solution that directly imposes a zero mass transfer condition, so it is a nice correspondence. By the way, in my diatribe concerning the lack of interest regarding sound among physicists, I was referring to sound in air. Physicists are very involved with sound in crystals, and the interactions of phonons and electrons, but that is another "matter" entirely. > [snips] >> [quoted text clipped - 16 lines] > imposes a zero mass transfer condition, so it is a nice > correspondence. That's outright scary. I wonder how many other solutions to problems involving satellite tragectories and nuclear reactions are missing the additive constants! [snip] >density. Specifically, the molecular density is higher at the crest than at >the nulls. So you have a high density moving forward at a velocity u0 at the >crest, and a low density moving backward at a velocity negative u0 at the >null. Thus there is a net mass in the forward direction. That's a second order effect. You're multiplying the acoustic velocity by the density change. You might consider formally upping your solution to second order via Whitham's rule, then seeing if that makes the mass flow go away. Ken Plotkin > [snip] > >density. Specifically, the molecular density is higher at the crest than at [quoted text clipped - 6 lines] > your solution to second order via Whitham's rule, then seeing if that > makes the mass flow go away. First off, it is a second order effect. Second, I don't see anything approximate in the evaluation I did (the analytic evaluation, not the heuristic argument above). I took the "standard solution," which is an exact mathematical expression, computed the exact density variation, and did an analytical integration over time to compute the average. I don't know what you mean by upping my solution to second order, and I don't know what Whitman's rule is. Could you please elaborate? Art Ludwig >First off, it is a second order effect. Second, I don't see anything >approximate in the evaluation I did (the analytic evaluation, not the [quoted text clipped - 3 lines] >what you mean by upping my solution to second order, and I don't know what >Whitman's rule is. Could you please elaborate? Your solution of the wave equation is exact. But the wave equation is itself a linearization of the Euler equations, which means it's the first term of a Taylor series in wave amplitude. If you make a second order quantity out of the first order solution, that's only part of the story. The rest of it is the second order terms that were dropped in the linearization. Whitham's rule is described by G. B. Whitham, "The Flow Pattern of a Supersonic Projectile," J. Fluid Mechanics, I, pp290-318, 1956. He applied it to linearized supersonic flow, which is exactly equivalent to linear acoustics. (Just a matter of the frame of reference being fixed to the supersonic vehicle or the air.) He takes the linear acoustic solution, and presumes it to give the correct amplitude to first order. Because the waves disturb the medium and change the local propagation speed, the location of the waves is correct to zeroth order. The wave speed is then corrected to first order via the linear solution, thus bootstrapping things into a second order (some would say consistent first order) result. This method is used to analyze weak nonlinear propagation. I suspect that if you applied it to your exact solution for the piston, you'd resolve the mean flow issue without having to invoke an arbitrary (IMHO) constant of integration. If you e-mail me (take the nospam- out of my address) I can point you to some other stuff on this. Ken Plotkin > If you e-mail me (take the nospam- out of my address) I can point you > to some other stuff on this. I wouldn't mind being pointed there myself, and I suspect several readers here feel the same way. Could it be done here, in public? best regards Peter >I wouldn't mind being pointed there myself, and I suspect several readers >here feel the same way. >Could it be done here, in public? Whitham's method is cool because once you have a linear solution it's straightforward to bootstrap it up to second order. The more traditional way of handling weak nonlinearities is via the Burgers equation. ASA sells a nonlinear acoustics book by Bob Beyer. David Blackstock and Mark Hamilton (who are the top of the heap in nonlinear acoustics) and Whitham have written books, but those are expensive, and may be hard to find. On the derivation of the wave equation as a formal perturbation expansion, the only things that come to mind (and I'm sure they're not the only ones, or even original) are a couple of things I've written, but I'd rather not cite them here. Ken Plotkin [snips] > >an analytical integration over time to compute the average. I don't know > >what you mean by upping my solution to second order, and I don't know what [quoted text clipped - 6 lines] > the story. The rest of it is the second order terms that were dropped > in the linearization. Ken: Thanks for the clarification. I thought you were disagreeing with my contention that there is mass flow with the specific solution to the linear equation I refered to. I understand now that you agree with that, but you suggest that the solution could be corrected more naturally by including non-linear terms in the wave equation. I definitely would like to look into the solution you refer to. I might add that there certainly are situations where there really is mass transport - e.g. when the wind is blowing. So the non-linear equations cannot exclude this posibility. Therefore it seems to me that "no mass transport" must always be imposed as a boundary condition. However the form of the solution could be (and almost certainly will be) different when the non-linear terms are included, as you suggest. Art Ludwig Some (perhaps) interesting historical references to this discussion 1) Devik O, Dahl H, Acoustical output of air sound senders, J. Acoust Soc. Am. 10(1938) July, 50 - 62. 2) Sleator W W, Proofs of the equation U = (E/rho)^(1/2) for the velocity of sound J. Acoust Soc. Am. 17(1949)2, 51 - 62. 3) Beyer R T, Nonlinear acoustics, AJP 41(1973)Sept. 1060 - 1067. 4) Stapper M, A Simplified approach to mechanics of acoustical wave propagation applicable to problems of radiation pressure. Part I: On the theory of longitudial wave propagation, Acustica 39(1978) 105 - 110. 5) Stapper M, A Simplified approach to mechanics of acoustical wave propagation applicable to problems of radiation pressure. Part II: The application to Rayleigh and Langevin radiation pressure, Acustica 39(1978) 111 - 116. 6) Thurston R N, Defininition of a Linear Medium for one-dimensional longitudial motion, J. Acoust Soc. Am.45(1969)6, 1329 - 1341. 7) McCormack F J, Craven D E, Kinetic theory of sound propagation in gaseous mixtures 1. Two-fluid 5-moment, 13-moment, and Navier-Stockes Theries, J. Acoust Soc. Am. 55(1974)4, 775 - 782. 8) Wyrzykowski R, The dependence of the speed of sound on space coordinates, Acustica 79(1993), 128 - 134. Kari Pesonen |
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