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Understanding Acoustics pp 533–601Cite as

One-Dimensional Propagation

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Abstract

Having already invested in understanding both the equation of state in Chap. 7 and in the hydrodynamic equations in Chap. 8, only straightforward algebraic manipulations will be required to derive the wave equation , justify its solutions, calculate the speed of sound in fluids, and derive the expressions for acoustic intensity and the acoustic kinetic and potential energy densities. The “machinery” developed to describe waves on strings will be sufficient to describe one-dimensional sound propagation in fluids, even though the waves on the string were transverse and the one-dimensional waves in fluids are longitudinal.

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Notes

  1. 1.

    Prof. Polack was on the faculty at the Danish Technical University (less than a 10 min drive from Lyngby to the Brüel & Kjær Headquarters in Nærum, Denmark) when he made this design. He is currently a professor at Université Pierre et Marie Curie and the Head of Doctoral School of Mechanics, Acoustics, Electronics and Robotics (SMAER, ED 391).

  2. 2.

    The resultant resonator was conical. It can be modeled easily in DeltaEC as a single CONE element (plus an electrodynamic driver VESPEAKER at one end and an OPENBRANCH radiation condition at the other end). This is a lot easier than 28 lumped elements once you accept that a cone will solve the problem.

  3. 3.

    Some say thermodynamics is the field where every partial derivative has its own name.

  4. 4.

    Analog Devices, Inc. sells a wonderful temperature sensing integrated circuit (AD 592) that sources one microampere of current for each degree of absolute temperature making electronic temperature compensation nearly trivial.

  5. 5.

    In acoustics, the equivalent of “optical trapping” is acoustical levitation superstability (see Sect. 15.4.7): M. Barmatz and S. L. Garrett, “Stabilization and oscillation of an acoustically levitated object,” U. S. Pat. No. 4,773,266 (Sept. 27, 1988).

  6. 6.

    In my estimation, one of the most significant engineering breakthroughs of the twentieth century was the invention, by Harold S. Black, of the negative feedback amplifier in 1928, now known as the “operational amplifier” (op-amp). In Black’s own words, “Our patent application was treated in the same manner as one for a perpetual-motion machine. In a climate where more gain was better, the concept that one would throw away gain to improve stability, bandwidth, etc., was inconceivable before that time.”

  7. 7.

    The “true” definition of an r.m.s. amplitude is always tied to the power associated with that amplitude. To account for nonsinusoidal waveforms, engineers introduce a dimensionless “crest factor,” CF, that is defined as the ratio of the peak value of a waveform to its rms value. For a sine waveform, CF (sine) = √2, for a triangular waveform, CF (triangle) = √3, and for a half-wave rectified sinewave, CF (half-wave rectified) = 2. Most instruments that measure the “true rms” value of a parameter (e.g., voltage) will exhibit an accuracy that decreases with increasing crest factor. For measurement of Gaussian noise, instruments that tolerate 3 < CF < 5 are usually adequate.

  8. 8.

    The weighting of A, B, C, and D (no weighting) levels are specified in several international standards, for example, the International Standards Organization ISO 3746:2010.

  9. 9.

    The transfer function for sound pressure at the eardrum to the sound pressure presented to the outer ear is above 14 dB from 2.5 to 3.2 kHz as shown in Table 2 of the ANSI-ASA S3.4 Standard.

  10. 10.

    When we include thermoviscous dissipation on the walls of the tube, the velocity will not be constant throughout the tube’s cross-section, since the no-slip boundary condition for a viscous fluid requires that the longitudinal velocity vanish at the tube’s walls. In many cases, the ratio of the viscous penetration depth, δ ν, to the tube’s radius, a, is small, δ ν ≪ a, so the flow velocity is nearly uniform throughout most of the tube’s cross-section.

  11. 11.

    The Reciprocity Theorem also applies in vector form. If we applied a vector force at some location, ①, on a flexible structure, and the vector displacement is measured at some other location, ②, we would observe the same vector displacement at ① if the same vector force were applied at ②.

  12. 12.

    The primary calibration of voltage is simpler (in principle, although it requires temperatures near absolute zero for the junction) since it is possible to relate the voltage across a superconducting Josephson junction to the resulting oscillation frequency , f, since their ratio is determined by Planck’s constant , h, and the charge on an electron, e: f = nhf/2e, where n is an integer. 2e/h = (483.5978525 ± 0.000003) MHz/μV.

  13. 13.

    This is equivalent to saying that the transducers are noncompliant in much the same way that the “constant displacement drive” for a sting does not “feel” the load of the string and preserves the “fixed” boundary condition. For reciprocity calibration of transducers that behave as an ultra-compliant driver (i.e., a constant force drive equivalent), see [36].

    Fig. 10.8
    figure 8

    Schematic representation of a gas-filled plane wave resonator that has a reversible transducer , R, at one end and a sound source, S, at the other end. A third transducer, that functions only as a microphone, M, can be located at either end of the resonator

    Full size image
  14. 14.

    The acoustic center of a reversible microphone or a sound source under free-field conditions is defined as the extrapolated center of the spherically diverging wave field.

  15. 15.

    Within the cone, the pressure amplitudes would be larger than those of a spherically spreading wave if the source’s volume velocity was the same in both cases. If the cone subtends a solid angle , Ω, then the pressures would be enhanced by a factor of 4π/Ω, assuming the additional load would not reduce the source’s volume velocity.

  16. 16.

    If the constant is negative, then there is a family of sinusoidal horn shapes (e.g., a globe terminated in a cusp) that describe the shape of the bell of the flute commonly associated with Indian snake charmers or the English horn, first used by Rossini, in 1829, in the opera, William Tell. [See B. N. Nagarkar and R. D. Finch, “Sinusoidal horns,” J. Acoust. Soc. Am. 50(1) 23–31 (1971).]

  17. 17.

    A nonuniform power transmission coefficient does not necessarily reduce the value of guided-wave enhancement of the coupling to an electrodynamic transducer, since human perception of low-frequency musical content is not particularly sensitive to such nonuniform response. The best example of such a psychoacoustic tolerance may be the success of the Bose Wave™ radio that employs a long serpentine duct that is driven by the rear of the forward-radiating speakers as shown in US Pat. No. 6,278,789 (Aug. 21, 2001).

  18. 18.

    Fritz Haber was also one of the world’s most infamous chemists, having developed the gas that was used to murder prisoners in the Nazi death camps.

References

  1. M.J. Lighthill, Waves in Fluids (Cambridge University Press, Cambridge, 1978); ISBN 0 521 29233 6 (paperback). See §2.6.

    Google Scholar 

  2. http://www.iso.org/iso/en/ISOOnline.frontpage

  3. V.A. Del Grosso, New equation for the speed of sound in natural waters (with comparisons to other equations). J. Acoust. Soc. Am. 56(4), 1084–1091 (1974)

    Article  ADS  Google Scholar 

  4. A.J. Zuckerwar, D.S. Mazel, Sound speed measurements in liquid oxygen-liquid nitrogen mixtures, NASA Technical Report 2464, 1985

    Google Scholar 

  5. M.R. Moldover, R.M. Gavioso, J.B. Mehl, L. Pitre, M. de Podesta, J.T. Zhang, Acoustic gas thermometry. Metrologia 51, R1–R19 (2014)

    Article  ADS  Google Scholar 

  6. M.R. Moldover et al., Measurement of the universal gas constant ℜ using a spherical acoustic resonator. J. Res. NBS 93, 85–144 (1988)

    Google Scholar 

  7. M. Greenspan, Comments on ‘Speed of sound in standard air’. J. Acoust. Soc. Am. 82(1), 370–372 (1987)

    Article  ADS  Google Scholar 

  8. E. Polturak, S.L. Garrett, S.G. Lipson, Precision acoustic gas analyzer for binary mixtures. Rev. Sci. Instrum. 57, 2837–2841 (1986)

    Article  ADS  Google Scholar 

  9. F.W. Giacobbe, Estimation of Prandtl numbers in binary mixtures of helium and other noble gases. J. Acoust. Soc. Am. 96(6), 3568–3680 (1994)

    Article  ADS  Google Scholar 

  10. Stanford Research Systems Model BGA244, Binary Gas Analyzer, www.thinkSRS.com

  11. S. Garrett, Sonic gas analyzer for hydrogen and methane, in Proceeding of Acoustics 08, Paris, 29 June–4 July, 2008, pp. 2767–2772

    Google Scholar 

  12. S.L. Garrett, G.W. Swift, R.E. Packard, Helium gas purity monitor for recovery systems. Physica 107B, 601–602 (1981)

    Google Scholar 

  13. M.V. Golden, R.M. Keolian, S.L. Garrett, Sonic gas analyzers, in Proceedings 16th International Congress on Acoustics and 135th Meeting of Acoustical Society of America, vol. III, Seattle, WA, 20-26 June 1998, pp. 1705–1706

    Google Scholar 

  14. G.M. Sessler, J.E. West, Electret transducers: a review. J. Acoust. Soc. Am. 53(6), 1589–1600 (1973)

    Article  ADS  Google Scholar 

  15. National Semiconductor Model LM2907

    Google Scholar 

  16. J.N. Tjøtta, S. Tjøtta, Nonlinear equations of acoustics, in Frontiers of Nonlinear Acoustics: Proceedings of 12th International Symposium on Nonlinear Acoustics, ed. M.F. Hamilton, D.T. Blackstock (Elsevier, London, 1990)

    Google Scholar 

  17. L.D. Landau, E.M. Lifshitz, Fluid Mechanics, 2nd edn. (Butterworth-Heinemann, New York, 1987), ISBN 0 7506 2767 0, §81

    Google Scholar 

  18. F.M.F. Chapman, Decibels, SI units, and standards, J. Acoust. Soc. Am. 108(2), 480 (2000)

    Google Scholar 

  19. H. Fletcher, W.A. Munson, Loudness, its definition, measurement, and calculation. J. Acoust. Soc. Am. 5(2), 82–108 (1933)

    Article  ADS  Google Scholar 

  20. D.W. Robinson, R.S. Dadson, A re-determination of the equal-loudness relations for pure tones. Br. J. Appl. Phys. 7, 166–188 (1956)

    Article  ADS  Google Scholar 

  21. ISO 226: Acoustics—normal equal-loudness contours (International Standards Organization, Geneva, 2003)

    Google Scholar 

  22. B.R. Glasberg, B.C.J. Moore, Prediction of absolute thresholds and equal loudness contours using a modified loudness model. J. Acoust. Soc. Am. 120(2), 585–588 (2006)

    Article  ADS  Google Scholar 

  23. American National Standard for Sound Level Meters, ANSI-ASA S1.4-1983, most recently reaffirmed by the American National Standards Institute (ANSI), 21 Mar 2006

    Google Scholar 

  24. S. Fidell, A review of US aircraft noise regulatory policy. Acoust. Today 11(4), 26–34 (2015)

    Google Scholar 

  25. P.D. Schomer, Criteria for assessment of noise annoyance. Noise Control Eng. J. 53(4), 132–144 (2005)

    Article  Google Scholar 

  26. American National Standard Method for the Calculation of the Speech Intelligibility Index, ANSI-ASA S3.5-1997

    Google Scholar 

  27. G.W. Swift, Thermoacoustic engines. J. Acoust. Soc. Am. 84(4), 1145–1180 (1988)

    Article  ADS  Google Scholar 

  28. W.R. MacLean, Absolute measurement of sound without a primary standard. J. Acoust. Soc. Am. 12(1), 140–146 (1940)

    Article  ADS  Google Scholar 

  29. J.W. Strutt (Lord Rayleigh), Proc. London Math. Soc. 4, 357 (1983); Scientific Papers, vol. I, (Dover, 1964), pp. 140. Note: The Scientific Papers are available from the Acoustical Society of America in a fully-searchable version on CD-ROM

    Google Scholar 

  30. R.K. Cook, Absolute pressure calibrations of microphones, J. Res. Nat’l. Bur. Standards 25, 489 (1940) and J. Acoust. Soc. Am. 12(1), 415–420 (1941)

    Google Scholar 

  31. G.S.K. Wong, T.F.W. Embleton (eds.), AIP Handbook of Condenser Microphones (Am. Inst. Phys., 1995); ISBN 1-56396-284-5. Chapters 8–12

    Google Scholar 

  32. L.L. Foldy, H. Primakoff, A general theory of passive linear electroacoustic transducers and the electroacoustic reciprocity theorem. I, J. Acoust. Soc. Am. 17(2), 109–120 (1945) and H. Primakoff, L.L. Foldy, A general theory of passive linear electroacoustic transducers and the electroacoustic reciprocity theorem. II, J. Acoust. Soc. Am. 19(1), 50–58 (1947)

    Google Scholar 

  33. F.V. Hunt, Electroacoustics: The Analysis of Transduction, and its Historical Background (Acoust. Soc. Am, 1982). Originally published by Wiley, 1954. See Ch. 5

    Google Scholar 

  34. I. Rudnick, Unconventional reciprocity calibration of transducers. J. Acoust. Soc. Am. 63(3), 1923–1925 (1978)

    Article  ADS  Google Scholar 

  35. S.L. Garrett, S. Adams, S. Putterman, I. Rudnick, Resonant nonlinear mode conversion in He II. Phys. Rev. Lett. 41, 413–416 (1978)

    Article  ADS  Google Scholar 

  36. G.W. Swift, S.L. Garrett, Resonance reciprocity calibration of an ultracompliant transducer. J. Acoust. Soc. Am. 81(5), 1619–1623 (1987)

    Article  ADS  Google Scholar 

  37. G.S.K. Wong, Primary pressure calibration by reciprocity, in AIP Handbook of Condenser Microphones, eds. by G.S.K. Wong, T.F.W. Embleton (Am. Inst. Phys., 1995); ISBN 1-56396-284-5

    Google Scholar 

  38. V. Nedzelintsky, Primary method for calibrating free-field response, in AIP Handbook of Condenser Microphones, eds. G.S.K. Wong, T.F.W. Embleton (Am. Inst. Phys., 1995); ISBN 1-56396-284-5

    Google Scholar 

  39. I. Rudnick, M.N. Stein, Reciprocity free field calibration of microphones to 100 Kc in air. J. Acoust. Soc. Am. 20(6), 818–825 (1948)

    Article  ADS  Google Scholar 

  40. R.S. Wakeland, Use of electrodynamic drivers in thermoacoustic refrigerators. J. Acoust. Soc. Am. 107(2), 827–832 (2000)

    Article  ADS  Google Scholar 

  41. C.H. Allen, I. Rudnick, A powerful high frequency siren. J. Acoust. Soc. Am. 19(5), 857–865 (1947)

    Article  ADS  Google Scholar 

  42. W.L. Nyborg, Acoustic streaming, in Physical Acoustics, vol. IIB, ed. by W.P. Mason (Academic Press, New York, 1964)

    Google Scholar 

  43. J.K. Hilliard, Historical review of horns used for audience-type sound reproduction. J. Acoust. Soc. Am. 59(1), 1–8 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  44. P.M. Morse, Vibration and Sound, 2nd edn. (McGraw-Hill, New York, 1948). Reprinted by the Acoust. Soc. Am., 1995; ISBN 0-88318-876-7. See Fig. 58

    Google Scholar 

  45. V. Salmon, Generalized plane wave horn theory, J. Acoust. Soc. Am. 17(3), 199–211 (1946) and A new family of horns, J. Acoust. Soc. Am. 17(3), 212–218 (1946)

    Google Scholar 

  46. P. Bèquin, C.L. Morfey, Weak nonlinear propagation of sound in a finite exponential horn. J. Acoust. Soc. Am. 109(6), 2649–2659 (2001)

    Article  ADS  Google Scholar 

  47. L.L. Beranek, Acoustics (Acoust. Soc. Am., 1996); ISBN 0-88318-494-X. Chapter 9

    Google Scholar 

  48. M. Kleiner, Electroacoustics (Taylor & Francis, 2013); ISBN 978-1-4398-3618-70. Chapter 19

    Google Scholar 

  49. H. Witschi, Profiles in toxicology, Fritz Haber: 1868-1934. Toxicol. Sci. 55, 1–2 (2000)

    Article  Google Scholar 

  50. P.W. Mullen, Modern Gas Analysis (Interscience, New York, 1955)

    Google Scholar 

  51. L.L. Beranek, Acoustic Measurements (Wiley, New York, 1949), pp. 161–171

    Google Scholar 

  52. A.E. Aliev et al., Underwater sound generation using carbon nanotube projectors. Nano Letters (Am. Chem. Soc.) 10, 2374–2380 (2010)

    Article  ADS  Google Scholar 

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    Steven L. Garrett

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Garrett, S.L. (2017). One-Dimensional Propagation. In: Understanding Acoustics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-49978-9_10

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