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Natural Science Forum / Physics / Acoustics / September 2004A Proof For No Doppler In a Tube
I will show that in a tube any piston velocity function will be exactly transcribed (i.e., without distortion of any kind) to a propegating wave. (0) Given: http://www.silcom.com/~aludwig/Physics/Piston_collisions.htm Then: (1) The piston can, by superposition of step functions, create any piecewise constant propegating velocity wave with the same shape in time as that of the piston velocity. (2) It can thus create a propegating Harr wavelet at any magnitude, scaling and translation or any superpositon of them. (3) A proof exists here: http://amath.colorado.edu/courses/4720/2000Spr/Labs/Haar/haar.html that from (2) it can produce any smooth propegating velocity function whatsoever and from (1) it will be exactly the same as the motion of the piston. Q.E.D Thus, however Doppler mixing is happening, it is _not_ happening at the piston/air interface. Bob  Signature"Things should be described as simply as possible, but no simpler." A. Einstein Bob Cain has posted a "proof" that Doppler distortion doesn't exist for a piston vibrating in a tube. A standard test of audio equipment is its square-wave response, so let's test Bobs proof with a 500 Hz square-wave. Our piston velocity profile is a constant positive velocity for 1 millisecond, followed by a constant negative velocity of equal magnitude for 1ms. The piston moves between x=0 and x=1cm. At t=0 the piston launches the leading edge of a pulse from its position at x=0, and the pulse starts moving down the tube at the velocity of sound, 344 m/s. At t=1ms the piston is at x=1cm and launches the trailing edge of the pulse. Freeze-frame at this instant. The leading edge has traveled down the tube to the point x=34.4 cm. The trailing edge is at x=1cm. The width of the positive part of the pulse is therefore 33.4 cm. The pulse will continue to travel down the tube at the speed of sound without changing its contour, so this width will remain constant. At t=2ms the piston is back at x=0, and launches another leading edge. Freeze-frame again. The original leading edge is now at x=68.8 cm, the trailing edge at 35.4 cm, and the new leading edge at x=0. The wavelength of one complete cycle of our square wave is 68.8cm (and that would be the same it the piston moved 1mm or 1cm). But the length of the positive part of the pulse is 33.4 cm and the length of the negative part is 35.4 cm. Our test microphone down the tube records this sound wave moving past the mike at the speed of sound. We record a time signal that our oscilloscope shows as a positive pulse .9709 ms long, followed by a negative pulse 1.0291 ms long. We send this signal to our spectrum analyzer: oops! Its not the correct spectrum for a perfect square-wave! We turn down the volume so the piston velocity profile is still a 500Hz square-wave, but it is only moving .01mm. Aha, now the oscilloscope shows almost equal widths of the positive and negative part of the pulse and we get a nearly perfect square-wave spectrum. As we crank the volume up however the spectrum changes. If the frequency spectrum changes as a function of the volume level, that's distortion in my book. Final test report: System exhibits severe Doppler distortion. P.S. I realize that someone is going to say that my example involves a piston traveling at a constant velocity, and of course you get Doppler distortion in that case. So this P.S. is to point out that you can make exactly the same argument for a triangle wave. The vertices of the triangles take the place of leading an trailing edges. It's also true for a sine wave, or any other wave; the landmarks and timing are just harder to describe. Thanks, Art. Yours is a challenge to step (1) of my proof, which says that velocity step superposition causes pulses of the piston to be transcribed to the air as piecewise constant waves having the same time between edges as the piston and the same piecewise constant magnitude. I will study it to see if I can refute it or support it and get back one way or the other. I very much appreciate your addressing it. Bob > Bob Cain has posted a "proof" that Doppler distortion doesn't exist for a > piston vibrating in a tube. A standard test of audio equipment is its > square-wave response, so let's test Bobs proof with a 500 Hz square-wave. > Our piston velocity profile is a constant positive velocity for 1 > millisecond, followed by a constant negative velocity of equal magnitude for > 1ms. The piston moves between x=0 and x=1cm. Signature"Things should be described as simply as possible, but no simpler." A. Einstein I withdraw the proof. It is flawed. In considering your challenge I realized that my proof begins with an unrealistic assumption which allows no conclusions to be drawn, much less a proof of anything. The flaw is that I began with: http://www.silcom.com/~aludwig/Physics/Piston_collisions.htm having a value of zero for delta-t. For any medium with non-zero mass density that requires infinite force to achieve a step change in velocity. Obviously, nothing that follows from the assumption of a step change in velocity can be valid for a system that disallows that. Back to the drawing board. Thanks, Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein > The flaw is that I began with: > > http://www.silcom.com/~aludwig/Physics/Piston_collisions.htm > > having a value of zero for delta-t. Make that zero for delta-x. Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein > Our test microphone down the tube records this sound wave moving past the > mike at the speed of sound. We record a time signal that our oscilloscope > shows as a positive pulse .9709 ms long, followed by a negative pulse 1.0291 > ms long. We send this signal to our spectrum analyzer: oops! Its not the I'm not following the qualities your example. Is this a measurement you actually made, or is it a hypothesized scenario? > correct spectrum for a perfect square-wave! We turn down the volume so the > piston velocity profile is still a 500Hz square-wave, but it is only moving [quoted text clipped - 3 lines] > If the frequency spectrum changes as a function of the volume level, that's > distortion in my book. True. This is called "finite amplitude distortion" (FAD). If the pressure amplitude is high enough, the pressure peaks travel faster than the pressure valleys. The final wave is always a saw tooth wave, regardless of the input. A sine wave is often used, as with it, the picture of peaks advancing and sharpening while catching up to the valleys is clearest. > Final test report: System exhibits severe Doppler distortion. The Doppler effect is a different phenomenon. I suppose one could superpose a high frequency on a low frequency wave as discussed before. An explanation would be that while the the piston advances according to the low frequency wave, it will be accelerating a series of several small waves, advancing their phase and thus frequency. But I have to say that the doppler effect and the FAD are two separate entities, one linear (Doppler), the other nonlinear (FAD). The term "nonlinear" in the FAD case is appropriate since for a captive parcel of gas, the P-V curve is not a straight line, rather it is an hyperbola. Angelo Campanella Hi Angelo: > > Our test microphone [...snip] > I'm not following the qualities your example. Is this a measurement you > actually made, or is it a hypothesized scenario? It is a mental experiment. [more snips]> > > > If the frequency spectrum changes as a function of the volume level, that's > > distortion in my book. > True. This is called "finite amplitude distortion" (FAD). If the > pressure amplitude is high enough, the pressure peaks travel faster than > the pressure valleys. The final wave is always a saw tooth wave, > regardless of the input. A sine wave is often used, as with it, the > picture of peaks advancing and sharpening while catching up to the > valleys is clearest. My "experiment" does not involve FAD. Once the wave is launched, the peaks and valleys maintain exactly the same spatial relationship as they continue down the tube. I'm talking about an effect that occurs when the air is totally linear, and the effect is due simply to the position of the piston when the wave is generated > > Final test report: System exhibits severe Doppler distortion. > > The Doppler effect is a different phenomenon. I suppose one could > superpose a high frequency on a low frequency wave as discussed before. Personally I hate to get bogged down in semantics. If the signal coming out of the speaker is different than the electrical signal going in, I call it distortion. My "experiment" (I hoped) is a way for people who are not physicists or mathematicians to understand what is going on. The real proof of Doppler distortion, or whatever you want to call it, is given on my web site http://www.silcom.com/~aludwig/Physics/Exact_piston/dopdist.htm There I derive an exact boundary-value solution to the problem. An exact boundary-value solution is the gold standard in physics. Even if you don't get into the details of my solution, I start out by demonstrating that an undistorted sine wave is not a valid solution, despite the fact that the air is perfectly linear, and the piston is moving in a perfect sinusoidal motion. This by itself proves the existence of distortion. The remainder of the derivation simply quantifies the distortion. Art Ludwig P.S. I sincerely hope that this is my last post, and even if provoked (I don't mean you, Angelo), I will probably not reply. The only exception I would make is that I would participate in further discussion of my boundary-value solution. > P.S. I sincerely hope that this is my last post, and even if provoked (I > don't mean you, Angelo), I will probably not reply. The only exception I > would make is that I would participate in further discussion of my > boundary-value solution. Art, I find that your equation (5) has no solutions other than when v=0 or w=0. Otherwise, no set of a and b coeficients can yield the pure sinusoid on the left. Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein > Art, I find that your equation (5) has no solutions other > than when v=0 or w=0. Otherwise, no set of a and b > coeficients can yield the pure sinusoid on the left. > > Bob Man I love it! An actual real honest technical question completely free of invective. (That may sound like I'm being sarcastic, but I am quite sincere). OK. If v=0 then all the coefficients are zero. That is the solution for that case. If v>0 then there is set of coefficients that solves the equation. In fact I sent you my Matlab program that generates solutions to this equation. You don't need to believe my program. You can program your own version of my equation (5) and plug in my solution values to verify them. Art >>Art, I find that your equation (5) has no solutions other >>than when v=0 or w=0. Otherwise, no set of a and b [quoted text clipped - 5 lines] > invective. (That may sound like I'm being sarcastic, but I am quite > sincere). OK. Heavy sigh. If anyone would but look at the the threads I started on this topic other than the ones with the inflamatory subjects, it should be more than apparent that that is all I ever wanted here. I really don't understand the protocol here at all. I sincerely hope that from here on out it can stay at a level appropriate to learned professionals. One shouldn't need to know enough in order to come here that he doesnt't need to come here at all. Running a gauntlet of insult is not helpful to anyone. OK, enough of all that, lets roll. Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein > Art, I find that your equation (5) has no solutions other > than when v=0 or w=0. Otherwise, no set of a and b > coeficients can yield the pure sinusoid on the left. I received a private communication requesting clarification of my boundary value solution. The purpose of my equation (5) is to obtain the coefficients that solve the boundary value problem. Once those coefficients are obtained, then they are used in my equation (4) to give the solution for the wave in the tube at any time and at any place in front of the piston. Anyone with a spreadsheet can program my equation (5) and verify that the set of coefficients below is a solution. In Matlab this solution is accurate to 2 parts in 10^14. Adding more terms would increase the accuracy, but this is close enough for government work. v=10 meters/sec, c=344 meters/sec, omega=2*pi*100 b0 = -0.14534883720930 n an bn 1.00000000000000 9.99683124329033 0.00000000000000 2.00000000000000 0.00000000000000 -0.14526696535919 3.00000000000000 -0.00316643279050 0.00000000000000 4.00000000000000 0.00000000000000 0.00008180181848 5.00000000000000 0.00000232171944 0.00000000000000 6.00000000000000 0.00000000000000 -0.00000006996041 7.00000000000000 -0.00000000219738 0.00000000000000 8.00000000000000 0.00000000000000 0.00000000007115 9.00000000000000 0.00000000000236 0.00000000000000 10.00000000000000 0.00000000000000 -0.00000000000008 > I received a private communication requesting clarification of my boundary > value solution. The purpose of my equation (5) is to obtain the coefficients > that solve the boundary value problem. Once those coefficients are obtained, > then they are used in my equation (4) to give the solution for the wave in > the tube at any time and at any place in front of the piston. Art, I have verified with my own Matlab calculation that, with the coeficients you gave here and v=10, a pure sin of magnitude 10 and any frequency can be _extremely_ well approimated by (4). That does certainly surprise (and impress) me but I like surprises like that. The particular value of those coeficients depends on the magnitude v of the sin being approximated but are independant of frequency. What I am understanding you to say, then, is that for any sinusoidal oscillation of the piston, an approximation to it can be found algorithmically which uses (5) as the basis function and further that the coeficients of that basis determine the spectrum of the propegated wave according to (4). The coeficients depend only on the magnitude of the sinusoid and not its frequency. Correct? With those coeficients (4) does not converge to (5) for x=0 as required. Can you reconcile that? The b0 component adds a DC offset to (4) which vanishes when calculating (5) with your coeficients. You have said elsewhere that purely alternating oscilation of a piston in a tube produces a DC air current. That seems to me to be a matter generator but we shall see. > In Matlab this solution is accurate > to 2 parts in 10^14. Adding more terms would increase the accuracy, but this > is close enough for government work. > > v=10 meters/sec, c=344 meters/sec, omega=2*pi*100 Ok, you've asserted equations that predicts exactly what distortion components should be seen for any sinusoidal driving function as well as a prediction that the spectral components of the measurement should scale exactly with frequency so long as magnitude is held constant. That is a _big_ step over what has gone before because it yields definite predictions that a well designed experiment, free of other distortion mechanisms can verify. I do hope that is being done and that an unequivocal protocol is used that is not too terribly difficult or expensive to duplicate. The calculation of these coeficients doesn't seem to depend on the geometry of the Tx and Rx, including separation, and should apply equally in the tube or free standing and at sundry separations. This is a prediction that should be tested by the same experiment. What I would like to see, of course, is a derivation of (5) and the algorithm that produces the coeficients based on first acoustic principles and accepted mathematical procedure. It is well and good to postulate a basis function and a purely mathematical algorithm for making it fit the driving function but it is a leap from there to why those same coeficients via (4) yield the acoustic prediction they do. I can see awaiting experimental confirmation before attempting that if it is in the offing. Thanks, Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein > I have verified with my own Matlab calculation that, with the > coeficients you gave here and v=10, a pure sin of magnitude 10 and any > frequency can be _extremely_ well approimated by (4). Aargh. Of course I meant "by (5)." Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein Hi Bob: Lots of snips, I won't bother to indicate them all > I have verified with my own Matlab calculation that, with > the coeficients you gave here and v=10, a pure sin of [quoted text clipped - 11 lines] > (4). The coeficients depend only on the magnitude of the > sinusoid and not its frequency. Correct? For the single frequency case that is basically correct. By the way, the Matlab version I sent you loses accuracy for high frequency, but that is just due to the fact that I set it up to work for a frequency of 1Hz. I have a revised version that works with high accuuracy at any frequency. The only quibble I have is your use of the word "approximation" My equations (4) and (5) are intended to represent the initial terms of an infinite series, and in the limit of an infinite number ot terms they are exact, not approximate. This is no different than the situation with a Fourier series. > With those coeficients (4) does not converge to (5) for x=0 > as required. Can you reconcile that? Yes, that is the whole point!!!! It's supposed to converge to the piston velocity at the piston face. Equation (4) not only "converges" to (5) where it is supposed to, it is exactly equal to equation (5) at the piston face. That is precisely how the bondary value solution is set up. > The b0 component adds a DC offset to (4) which vanishes when > calculating (5) with your coeficients. You have said > elsewhere that purely alternating oscilation of a piston in > a tube produces a DC air current. That seems to me to be a > matter generator but we shall see. No, it doen't vanish. the b0 term is in both equations. And contrary to being a matter generator, I originally predicted that this term must exist precisely to conserve matter. If you follow my link on the my web site there is a full explanation of this. > What I would like to see, of course, is a derivation of (5) > and the algorithm that produces the coeficients based on [quoted text clipped - 5 lines] > they do. I can see awaiting experimental confirmation > before attempting that if it is in the offing. If the wave equation is not "first acoustic principles" what is??? It is an absolutely standard practice in physics to postulate the form of a solution, as I do in equation (5), and then verify that it satisfies the boundary conditions. You seem to make a big deal out of the fact that I solve the equation numerically instead of analytically. So what? I could solve the equation analytically if I wanted to spend the time, but what is gained? Its obvious that it simply involves lots of mindless expansions and equating like terms, and I am happy to let Matlab do the work for me. I am spending a lot of time on these posts, and I would like to believe that it is of benefit to someone besides Bob. To quote Pink Floyd, "Is anybody out there?" > I am spending a lot of time on these posts, and I would like to believe that > it is of benefit to someone besides Bob. To quote Pink Floyd, "Is anybody > out there?" Yes Art. I've been following this thread (in its various manifestations in various NG's) but I have not been able to study the physics as closely as I would like... I'm really busy at the moment, and I shouldn't really be spending time posting on NG's... Art and Bob, despite the ups and downs, you do seem to be making good progress in this discussion now - you're not just "bricks in the wall" :-) Chris W SignatureThe voice of ignorance speaks loud and long, but the words of the wise are quiet and few. -- > My equations > (4) and (5) are intended to represent the initial terms of an infinite > series, and in the limit of an infinite number ot terms they are exact, not > approximate. This is no different than the situation with a Fourier series. Understood. I'll just accept that the terms on the rhs of (5) are an appropriate basis for an infinite expansion of a sinusoid. I wouldn't have believed that on inspection (because I can't see yet how either of the summation terms can have a DC component to cancel b0) but you've clearly proved it by example and I'll believe you can do it again for any v. >>With those coeficients (4) does not converge to (5) for x=0 >>as required. Can you reconcile that? [quoted text clipped - 3 lines] > it is supposed to, it is exactly equal to equation (5) at the piston face. > That is precisely how the bondary value solution is set up. I can't buy that at an infinitessimal dx beyond the piston (4) applies but exactly at the piston suddenly a pure sinusoid applies. It says that the air at the piston face moves differently than the piston does. That kind of a discontinuity is just not reasonable nor is it allowed by the wave equation. > No, it doen't vanish. the b0 term is in both equations. And contrary to > being a matter generator, I originally predicted that this term must exist > precisely to conserve matter. If you follow my link on the my web site > there is a full explanation of this. Agreed that the coeficient doesn't vanish, but in (5) any DC component it might seem to imply does vanish whereas in (4) it doesn't because all the components of (4) are orthogonal. That the basis set chosen for (5) is not orthogonal is what allows b0 to be cancelled whether I like it or not. If there is a DC flow of air down the tube, as (4) clearly states, and there is no drift of the piston, as (5) clearly states then matter is being created. I don't need to examine any explanation to know the truth of this. One need not examine any process in the tube, only boundry condtions to see that it violates conservation of mass for a non-drifting, non-leaking piston to produce a net flow of mass down a tube and that's what (4) and (5) together state. > If the wave equation is not "first acoustic principles" what is??? That's not where you started. You have postulated (4), (5) and an algorithm, taken together, as axioms (without proof, that is) and predicted results as a consequence of the axioms. The wave equation doesn't enter into those axioms at all except at the point of piston/air interface and the discontinuity there says that the wave equation is violated at that point. > It is an > absolutely standard practice in physics to postulate the form of a solution, > as I do in equation (5), and then verify that it satisfies the boundary > conditions. Which it doesn't at the piston/air interface. I agree that there is a long history in science of postulating equations that fit a phenomenon rather than deriving them when new physics is required to do that. I don't think there is any new physics hiding in this problem. Theoretical issues aside, the question here is whether a properly designed experiment validates all the predictions that arise from your conjecture. If they do then there will of course be motivation to derive them rather than postulate them. The predictions I see are: 1) The spectral content of (4) will be measured by an omnidirectional microphone when using the coeficients for (5) that give the piston motion defined on its lhs for any v for which they are calculated. 2) For any v the spectral content of the wave will scale with frequency as predicted by (4). 3) The same results will obtain whether the measurement is done with a piston in a tube, a free standing piston, a piston in a baffle or any other configuration. 4) Except for an appropriate delay the results are independant of the separation between piston and microphone in any configuration. 3) and 4) are a consequence of configuration entering nowhere into your conjecture. Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein I am out of patience. The math speaks for itself, and I am no longer going to participate in this discussion. > I am out of patience. The math speaks for itself, and I am no longer going > to participate in this discussion. Which also speaks for itself. Bob Signature"Things should be described as simply as possible, but no simpler." A. Einstein > > I am out of patience. The math speaks for itself, and I am no longer going > > to participate in this discussion. > > Which also speaks for itself. > Bob Who the hell do you think you are? You come to this newsgroup asking knowledgeable professional scientists and engineers for their opinions, but because you don't like what you are told, because you are too technically inept to understand what you are told and because you are too arrogant to accept that you are worng, you go on a rampage. First you unleash against me one of the most slanderous and foul-mouthed diatribes yet. Next you tell Ang, a nationally-recognized acoustics consultant to f.ck off. And, if that weren't enough, you thank Art Ludwig, a well-respected physicist/mathematician, for all of the work that he has done at your request, by publicly spitting in his face. Clearly, you have no dignity, self respect or integrity. You are a despicable human being, and you epitomize what it means to be a contemptible piece of human waste. atlab do the work for me. > I am spending a lot of time on these posts, and I would like to believe that > it is of benefit to someone besides Bob. To quote Pink Floyd, "Is anybody > out there?" I think it is safe to say that there are two groups "out there." One group consists mostly of the audio people who are incapable of understanding either the concepts, the math, or both. The other group consists mostly of technically-competent scientists and engineers who understand what you have done and accept your results, but don't have any questions at this time. The natural result is mostly silence from both groups, but for different reasons. > It is a mental experiment. > My "experiment" does not involve FAD. Once the wave is launched, the peaks On closer scrutiny... noting for instance that a harmonic is predicted that is some 80 dB down from the fundamental, It is easy to see intuitively that the slight curvature of the pressure vs volume curve (an hyperbola) is enough to cause this, owing to the precision of your math model.... good thinking. > and valleys maintain exactly the same spatial relationship as they continue > down the tube. I'm talking about an effect that occurs when the air is > totally linear, and the effect is due simply to the position of the piston > when the wave is generated In > http://www.silcom.com/~aludwig/Physics/Exact_piston/dopdist.htm I'm looking for a clear representation of the doppler effect. The side bands should be in a clear pattern otherwise known to be associated with frequency modulation. Do I see that in your Equation (8)? Perhaps in (7)? > Personally I hate to get bogged down in semantics. If the signal coming out > of the speaker is different than the electrical signal going in, I call it > distortion. Good point. Though others might also call it modulation. In the case of audio sounds emitted by a loudspeaker, it is fair to use distortion since it detracts from perfect reproduction. > undistorted sine wave is not a valid solution, despite the fact that the air > is perfectly linear, and the piston is moving in a perfect sinusoidal That is due to the pressure-volume curvature, I think. Your citing the apparent superior clarity of the large surface electrostatic loudspeaker is very important. If true, then we will see a decided shift to such units for all sound reproduction above 1 kHz (where clarity and intelligibility are really evident and important. > P.S. I sincerely hope that this is my last post, and even if provoked (I > don't mean you, Angelo), I will probably not reply. The only exception I > would make is that I would participate in further discussion of my > boundary-value solution. Just relax for a while; give the world a chance to catch up to you! Ang. C. Signature--------- www.CampanellaAcoustics.com --------- > Just relax for a while; give the world a chance to catch up to you! Hi Angelo: Good advice. I have posted a plot of the harmonics generated in the case of a single frequency here http://www.silcom.com/~aludwig/Physics/Exact_piston/ddist1.gif and in the case of two frequencies here http://www.silcom.com/~aludwig/Physics/Exact_piston/ddist2.gif Finally here is a plot of the piston position vs. time for one cycle at the low frequency, for the two frequency case http://www.silcom.com/~aludwig/Physics/Exact_piston/Dopdistvp.gif There are actually two curves in the last figure: black is the piston velocity, and red is the sound wave velocity at the piston face. The required boundary condition is that the two curves agree. Obviously they do. I really wish this were not called "Doppler distortion" because it raises all kinds of irrelevant arguments. What it has in common with the Doppler effect is that it can be modeled as a phase modulation due to the cone position, or FM due to the cone velocity. (Either approach gives the same result). The conventional Doppler effect can also be derived as a phase modulation: phase shift is a linear function of time for a constant relative velocity. However the conventional Doppler effect derivation is only valid for a constant relative velocity; it is not necessarily valid when there is accelerated motion. In any case, I think the effect for a loudspeaker should be analyzed for what it is, without worrying too much about how it compares with train whistles. Regarding experimental confirmation, this distortion is directly proportional to the high frequency; ordinary quadratic distortion is independent of the high frequency. Ordinary quadratic distortion is directly proportional to the ampitude of the high frequency; this distortion is independent of the amplitude of the high frequency. In all cases I am refering to the magnitude of the sidebands around the high frequency, relative to the fundemantal magnitude at the high frequency. This very clear distinction should make experimental proof very solid. I hope to see some experimental data in the near future. |
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