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Natural Science Forum / Physics / Acoustics / September 2004More on Mass Transport
In a recent post I pointed out that the standard plane wave solution of the linear wave equation for sound creates a time-average mass transport in the direction of propagation, which (I postulated) must be corrected by adding a constant velocity in the opposite direction. Ken Plotkin pointed out that this is a second order effect, and maybe the mass flow would disappear if the more accurate solution to the non-linear acoustic equations were used. This prompted me to investigate (thanks Ken, non-linear acoustics is extremely interesting!). The book by Enflo and Hedberg, "Theory of Nonlinear Acoustics in Fluids," gives an exact solution to Burgers' equation for the non-dissipative case, the Bessel-Fubini solution. (Pierce calls it the Fubini-Ghiron solution). I went through all of the numerical hoops and calculated the time-average product of density and velocity, using the full non-linear solution, at a series of x values along the direction of propagation. The result is that the mass flow is the same for both the non-linear and linear solutions. In retrospect this is not surprising, since the first term of the non-linear solution, the only non-zero term at x=0, is essentially the linear solution. The non-linear solution primarily differs in that larger and larger harmonics are generated as the wave propagates away from x=0. Since the mass flow at x=0 must be the same as the flow at any other value of x, it follows that the result must be the same. However it was still nice to numerically confirm this. I stated that I have never seen this mass transport mentioned in any text. That is still true for the particular case under consideration, but I did find a discussion of a related phenomenon in one of Beyer's books, "Nonlinear Acoustics" published in 1974. He has a whole chapter devoted to "streaming," which is a mass transport phenomenon noted by Faraday in 1831. (Beyer even includes experimental data). However Beyer's equation 7.18 on page 245 shows that this phenomenon is only present when the wave is attenuated. The mass transport I am referring to occurs in a wave that is not attenuated at all, so it is a different phenomenon than the one analyzed by Beyer. > The book by Enflo and Hedberg, "Theory of Nonlinear Acoustics in > Fluids," gives an exact solution to Burgers' equation for the [quoted text clipped - 11 lines] > of x, it follows that the result must be the same. However it was > still nice to numerically confirm this. Art, do you have a handle on the distance at which the level of air- produced distortion is similar to the level of Doppler produced distortion? If the equivalence distance is very close to the piston, it may be very difficult, if not impossible, to separate air-produced distortion from Doppler distortion in an actual measurement situation (assuming that distortion in actual piston motion is negligible). Also, if this is the case, the issue of the existence of dynamic Doppler distortion produced by a vibrating piston is mute for practical listening distances with the possible execption of headphone and insert-earphone listening. With regard to the mass transport issue, I know from practical experience that both mass transport and severe waveform distortion occur at high ultrasonic sound pressure levels in water. A 3MHz focused ultrasonic transducer facing upward underwater makes an interesting drinking fountain. It also turns a sinusoid into a sawtooth as the propagating wave moves forward. Also, the same transducer when excited simultaneously by 3.0000MHz + 3.0001MHz allows one to create a dynamic 1KHz force that is confined to a 1mm^2 area. I doubt that the nonlinear mechanisms are the the same in water and air, but one never knows what clues might materialize from looking into the mass transport issue in the case of waterborne ultrasound. [snip] >With regard to the mass transport issue, I know from practical experience >that both mass transport and severe waveform distortion occur at high [quoted text clipped - 7 lines] >from looking into the mass transport issue in the case of waterborne >ultrasound. Nonlinear propagation in air and water should have strong similarities. The fundamental mechanism is that the propagation speed changes with amplitude. (delta sound speed)/(sound speed) in air is (gamma+1/2gamma) times delta-P/P. For a given amplitude and wavelength, it's straightforward to compute when things will steepen into shocks. Sawtooth waves form in air. The two most common examples are sonic booms and buzzsaw noise from turbines. I'm puzzled about where the flowing mass comes from in the 1-D case in a tube. Might the air be heating from energy added by the piston? Ken Plotkin > [snip] >>With regard to the mass transport issue, I know from practical [quoted text clipped - 21 lines] > > Ken Plotkin What about the cause of mass flow in water which could not be more evident. When illuminated by a laser vibrometer, the effects of streaming around a small target at the focal point is clearly evident. Do you have any experience in this area? >What about the cause of mass flow in water which could not be more evident. >When illuminated by a laser vibrometer, the effects of streaming around a >small target at the focal point is clearly evident. Do you have any >experience in this area? No experience at all. Is that in three dimensions? I'm sure sound can induce flows. Without that, thermoacoustic devices would not work. But in those, the induced flow circulates. I'm just puzzled about how Art's 1-D solution has a mean flow. Ken Plotkin > I'm just puzzled about how Art's 1-D solution has a mean flow. I believe that this is acoustic 'radiation pressure'. Angelo Campanella >> I'm just puzzled about how Art's 1-D solution has a mean flow. > > I believe that this is acoustic 'radiation pressure'. > > Angelo Campanella In the waterborne case, that is exactly what it is. Also, measuring the dc component of pulsed radiation pressure with a sensitive force balance is a standard way of determining the acoustic output power of ultrasonic transducers. http://nvl.nist.gov/pub/nistpubs/jres/101/5/j5fick.pdf >>What about the cause of mass flow in water which could not be more >>evident. When illuminated by a laser vibrometer, the effects of [quoted text clipped - 6 lines] > Without that, thermoacoustic devices would not work. But in those, > the induced flow circulates. Yes, in three dimensions. > I'm just puzzled about how Art's 1-D solution has a mean flow. > > Ken Plotkin Perhaps its simply a dc component associated with waveform asymmetry. = > > I'm just puzzled about how Art's 1-D solution has a mean flow. > > > > Ken Plotkin > > Perhaps its simply a dc component associated with waveform asymmetry. That is my opinion. The plane wave solution is a solution to a differential equation, no more and no less. It is subject to the boundary conditions of any given problem. For the case of a piston in an infinitely long tube, one of the boundary conditions is that there cannot be a time-average mass flow. Adding a DC term, an integration constant if you will, introduces a slight asymmetry in the velocity, as you state, and then both the differential equation and boundary conditions are satisfied. The standard acoustic power flow is 1/2 of pressure squared divided by the impedance of air. But at a more fundamental level the power flow is the net time-average flow of kinetic energy of the molecules, including the rotational energy in the case of a diatomic molecule. If you calculate this for the plane wave solution, the flow of molecular kinetic energy exceeds the acoustic power flow by the very significant factor of 7/2. If you include the DC velocity term the flow of molecular kinetic energy is exactly equal to the acoustic power flow, and equal to the work done by the piston. So conservation of mass and conservation of energy both require exactly the same DC term. > Art, do you have a handle on the distance at which the level of air- > produced distortion is similar to the level of Doppler produced distortion? [quoted text clipped - 5 lines] > a vibrating piston is mute for practical listening distances with the > possible execption of headphone and insert-earphone listening. Gary: At the moment I only have a comparison for a piston vibrating at a single frequency. It is not obvious to me what the non-linear result will be for two frequencies, but I will try to develop the answer for that case (or perhaps someone else has the answer and will post it). In any case, for one frequency, and a piston peak velocity of .040 meters/sec the "Doppler" 1st harmonic is -85 dB below the fundamental - which may be too small for you to measure anyway. This is independent of frequency. The non-linear first harmonic is roughly proportional to frequency and distance from the piston face. At 100 Hz and 0.5 meters from the piston the level is -84 dB. (Incidentally the equation in my 1993 edition of Breanek is incorrect for the non-linear 1st harmonic. The equation in Pierce is correct, as well as the equations in Enflo & Hedberg, and the book by Beyer). The non-linear ist harmonic is in phase with the fundamental, the "Doppler" harmonic is 90-degrees out of phase, so that is a finger print as well. [several snips] > > Art, do you have a handle on the distance at which the level of air- > > produced distortion is similar to the level of Doppler produced [quoted text clipped - 3 lines] > At the moment I only have a comparison for a piston vibrating at a single > frequency. Fortunately Enflo and Hedberg give the generalization of the Bessel-Fubini solution for the case of 2 sinusoidal frequencies, which I now have programmed. As an example, for frequencies of f1=100 Hz and f2=8000 Hz, with peak velocities of .04 and .0002 meters/second respectively, the intermods at f2-f1 and f2+f1 due to air non-linearity are -68 dB relative to the fundamental at f2, at a distance of 1 meter from the piston. The Doppler intermods are -46.6 dB. Both levels are, to the first order, independent of the velocity at the high frequency (of course the experimental SNR is sensitive to the absolute level rather than the relative level). If the velocity at the low frequency is halved, to the first order the non-linear intermods drop 12 dB and the Doppler intermods drop 6 dB. The Doppler intermods are independent of the distance from the piston, whereas the non-linear are directly proportional to the distance, again to the first order. So the non-linear distortion does not appear to preclude a measurement of the Doppler distortion in a tube. It should be noted that the distortion due to air non-linearity is much higher in a tube (i.e. plane wave behavior) than it is for a piston in a baffle, because in free space the 1/r amplitude drop rapidly reduces the generation of harmonics. The Doppler distortion will be about the same. > [several snips] >> [quoted text clipped - 29 lines] > rapidly reduces the generation of harmonics. The Doppler distortion > will be about the same. Art, this is very good news for several reasons. The required piston velocities not at all excessive and in my setup can be produced with IM distortion in measured piston motion that is 75-80dB below the level of the high frequency carrier. Additionally, by appropriate choice of measuring distance from the piston, it looks like it will be possible to measure predominantly Doppler distortion at a short but reasonable distance from the piston and predominantly nonlinear air distortion at a long but reasonable distance from the piston. Lastly because of the differences in phase behavior of the two distortion mechanisms, there will be no question that Doppler distortion is being measured at the short distance and that nonlinear air distortion is being measured at the other. Too bad I am not yet retired like you, or I would have the measurement results for you by tomorrow. > The book by Enflo and Hedberg, "Theory of Nonlinear Acoustics in Fluids," > gives an exact solution to Burgers' equation for the non-dissipative case, [quoted text clipped - 10 lines] > that the result must be the same. However it was still nice to numerically > confirm this. Bravo for unearthing this. You are right about Fubini-Ghiron. It's been so long ago that I forgot about them. > (Beyer even includes experimental data). However Beyer's equation 7.18 on > page 245 shows that this phenomenon is only present when the wave is > attenuated. The mass transport I am referring to occurs in a wave that is > not attenuated at all, so it is a different phenomenon than the one analyzed > by Beyer. I strongly suspect that Beyer's math is correct, and that the amount of energy required to set the air in motion (as you have seen so far) is so tiny that the decibel SPL reduction is not measurable by common sound level meters. The basic energy needed is only that to accelerate the moving air mass from rest to the velocity you see... pico-joules, maybe? Angelo Campanella |
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