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Nonlinear Effects in Sound Propagation

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Abstract

Acoustics is ordinarily concerned only with small-amplitude disturbances, so nonlinear effects are typically of minor significance.

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Notes

  1. 1.

    An alternate approach defines

    $$\displaystyle \begin{aligned} \lambda(\rho) = \int_{\rho_{o}}^{\rho}\frac{c(\rho)}{\rho} d\rho, \end{aligned}$$

    so that (subscripts denoting partial derivatives) ρt = (ρ∕c)λt, px = ρcλx, etc., and Eqs. (1) reduce to

    $$\displaystyle \begin{aligned} \lambda_t +v\lambda_{x}+ cv_{x} = 0, \qquad v_{t}+vv_{x} + c\lambda_{x}=0 , \end{aligned} $$
    (i)

    or

    $$\displaystyle \begin{aligned} (\lambda +v)_{t}+ (v +c)(\lambda+v)_{x} = 0, \qquad (\lambda-v)_{t}+(v-c)(\lambda-v)_{x} = 0 . \end{aligned} $$
    (ii)

    A particular solution (simple wave) results with v = λ, yielding

    $$\displaystyle \begin{aligned} v_{t}+(v+c)v_{x} =0 , \qquad p_{t}+(v +c)p_{x} = 0 , \end{aligned} $$
    (iii)

    which is the same as Eq. (3). [B. Riemann, “On the propagation of plane air waves of finite amplitude,” Abhandl. Ges. Wiss. Goettingen (1860), reprinted in The Collected Works of Bernhard Riemann, Dover, New York, 1953, pp. 156–175.]

  2. 2.

    S. Earnshaw, “On the mathematical theory of sound,” Phil. Trans. R. Soc. Land. 150:133–148 (1859). A similar result for a gas in which p is directly proportional to ρ had been obtained somewhat earlier by S. D. Poisson, “Memoir on the theory of sound,” J. Ec. Polytech. 7:319–392 (1808).

  3. 3.

    R. T. Beyer, “Parameter of nonlinearity in fluids,” J. Acoust. Soc. Am. 32:719–721 (1960); A. B. Coppens et al., “Parameter of nonlinearity in fluids, II,” ibid. 38:797–804 (1965); M. P. Hagelberg, G. Holton, and S. Kao, “Calculation of B∕A for water from measurements of ultrasonic velocity versus temperature and pressure to 10,000 kg/cm2,” ibid. 41:564–567 (1967).

  4. 4.

    For air, with ρ = 1.2kg∕m3, c = 340m∕s, and β = 1.2, the value of \(\bar {x}\) in meters is 6.3 × 106∕fPo when the frequency f is in hertz and Po is in pascals. For water, with ρ = 1000kg∕m3, c = 1500m∕s, β = 3.5, the corresponding value of \(\bar {x}\) is 15.3 × 1010∕fPo, so that, for example, a frequency of 200 kHz and a peak pressure amplitude of 104 Pa yield an \(\bar {x}\) of 77 m.

  5. 5.

    G. N. Watson, A Treatise on the Theory of Bessel Functions, 2d ed., Cambridge, 1944, pp. 16, 20. The identity in Eq. (12), which yields

    $$\displaystyle \begin{aligned} \sum_{n = 1}^{\infty} (n\sigma)^{-2}J_{n}^{2}(n\sigma)= \tfrac{1}{4} \end{aligned}$$

    is attributed by Watson, p. 572, to N. Nielsen (1901).

  6. 6.

    E. Fubini-Ghiron, “Anomalies in acoustic wave propagation of large amplitude,” Alta Freq. 4:530–581 (1935); D. T. Blackstock, “Propagation of plane sound waves of finite amplitude in nondissipative fluids,” J. Acoust. Soc. Am. 34:9–30 (1962).

  7. 7.

    J. Challis, “On the velocity of sound,” Phil. Mag. (3)32:494–499 (1848); G. G. Stokes, “On a difficulty in the theory of sound,” ibid. 33:349–356 (1848); G. B. Airy, “The Astronomer Royal on a difficulty in the problem of sound,” ibid., 34:401–405 (1849).

  8. 8.

    W. J. M. Rankine, “On the thermodynamic theory of waves of finite longitudinal disturbance,” Phil. Trans. R. Soc. Land. 160:277–288 (1870); H. Hugoniot, “On the propagation of movement through a body and especially through an ideal gas,” J. Ec. Polytech. 58:1–125 (1889); G. I. Taylor, “The conditions necessary for discontinuous motion in gases,” Proc. R. Soc. Land. A84:371–377 (1910). When the flow is not perpendicular to the shock front, the above still hold with v+ and v− interpreted at the normal components of v+ and v−. The tangential component of the velocity must be continuous across the shock surface. See, for example, L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, London, 1959, pp. 317–319.

  9. 9.

    W. D. Hayes, “The basic theory of gasdynamic discontinuities,” in H. W. Emmons (ed.), Fundamentals of Gas Dynamics, Princeton University Press, Princeton, N.J., 1958, pp. 416–481. The first of Eqs. (3) is the Hugoniot equation; the corresponding plot of p− versus 1∕ρ− for fixed p+ and 1∕ρ+ is a Hugoniot diagram.

  10. 10.

    R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948, pp. 142–144.

  11. 11.

    L. D. Landau, “On shock waves,” J. Phys. (USSR) 6:229–230 (1942), “On shock waves at large distances from the place of their origin,” ibid. 9:496–500 (1945); S. Chandrasekhar, “On the decay of plane shock waves,” Ballist. Res. Lab. Rep. 423, Aberdeen Proving Ground, Md., 1943; H. A. Bethe and K. Fuchs, “Asymptotic theory for small blast pressure,” in “Blast Wave,” Los Alamos Sci. Lab. Rep. LA 2000, August 1947, pp. 135–176.

  12. 12.

    L. D. Landau, “On shock waves,” “On shock waves at large distances,” G. B. Whitham, “The flow pattern of a supersonic projectile,” Commun. Pure Appl. Math. 5:301–348 (1952).

  13. 13.

    Z. A. Gol’berg, “On the propagation of plane waves of finite amplitude,” Sov. Phys. Acoust. 3:329–347 (1957).

  14. 14.

    D. T. Blackstock, “Connection between the Fay and Fubini solutions for plane sound waves of finite amplitude,” J. Acoust. Soc. Am. 39:1019–1026 (1966); Landau and Lifshitz, Fluid Mechanics, pp. 372–375.

  15. 15.

    I. Rudnick, “On the attenuation of a repeated sawtooth shock wave,” J. Acoust. Soc. Am. 25:1012–1013 (1953).

  16. 16.

    See, for example, the comments by W. Heisenberg, “Nonlinear problems in physics,” Phys. Today 20(5):27–33 (May 1967).

  17. 17.

    Whitham, “The flow pattern of a supersonic projectile”; Blackstock, “Connection between the Fay and Fubini solutions.”

  18. 18.

    The possibility that finite-amplitude effects may limit the acoustic efficiency of a sound source was suggested by L. V. King, “On the propagation of sound in the free atmosphere and the acoustic efficiency of fog-signal machinery: An account of experiments carried ut at Father Point, Quebec, September, 1913,” Phil. Trans. R. Soc. Lond. A218:211–293 (1919). The first correctly interpreted observation of saturation is due to C. H. Allen, Finite Amplitude Distortion in a Spherically Diverging Sound Wave in Air, Ph.D. thesis, Pennsylvania State University, 1950. For recent reviews, see J. A. Shooter, T. G. Muir, and D. T. Blackstock, “Acoustic saturation of spherical waves in water,” J. Acoust. Soc. Am. 55:54–62 (1974); D. A. Webster and D. T. Blackstock, “Finite-amplitude saturation of plane sound waves in air,” ibid. 62:518–523 (1977).

  19. 19.

    This technique for obtaining an approximate nonlinear equation for propagation in a dispersive medium is sometimes referred to as Whitham’s rule. Less heuristic derivations with various degrees of generality are given by P. A. Lagerstrom, J. D. Cole, and L. Trilling, “Problems in the theory of viscous compressible fluids,” Calif. Inst. Tech. Guggenheim Aeronaut. Lab. Rep. Of. Nav. Res., 1949; M. J. Lighthill, “Viscosity effects in sound waves of finite amplitude,” in G. K. Batchelor and R. M. Davies (eds.), Surveys in Mechanics, Cambridge University Press, London, 1956; Hayes, “The basic theory of gasdynamic discontinuities”; and H. Ockendon and D. A. Spence, “Non-linear wave propagation in a relaxing gas,” J. Fluid Mech. 39:329–345 (1969).

  20. 20.

    H. Bateman, “Some recent researches on the motion of fluids,” Mon. Weather Rev. 43:163–170 (1915); the equation later emerged in a mathematical model of turbulence proposed by J. M. Burgers (1939, 1940) and summarized in his “A mathematical model illustrating the theory of turbulence,” in R. von Mises and T. von Kármán (eds.), Adv. Appl. Mech., vol. 10, Academic, New York, 1948. Its recent extensive applications to nonlinear acoustics originated with the work of Lagerstrom, Cole, and Trilling, “Problems in the theory of viscous compressible fluids,” and with J. D. Cole, “On a quasi-linear parabolic equation occurring in aerodynamics,” Q. Appl. Math. 9:225–231 (1951).

  21. 21.

    D. T. Blackstock, “Thermoviscous attenuation of plane, periodic, finite-amplitude sound waves,” J. Acoust. Soc. Am. 36:534–542 (1964).

  22. 22.

    F. H. Fisher and V. P. Simmons, “Sound absorption in sea water,” J. Acoust. Soc. Am. 62:558–564 (1977). (See Section 10-8 of the present text.)

  23. 23.

    J. S. Mendousse, “Nonlinear dissipative distortion of progressive sound waves at moderate amplitudes,” J. Acoust. Soc. Am. 25:51–54 (1953).

  24. 24.

    The study of shock structure dates back to Taylor, “The conditions necessary for discontinuous motion in gases,” and to R. Becker, “Shock waves and detonations,” Z. Phys. 8:321–362 (1922). The latter’s result for an ideal gas with finite viscosity and thermal conductivity reduces to that given here in the weak-shock limit.

  25. 25.

    A. L. Polyakova, S. I. Soluyan, and R. V. Khokhlov, “Propagation of finite disturbances in a relaxing medium,” Sov. Phys. Accoust. 8(1):78–82 (1962); Ockendon and Spence, “Nonlinear wave propagation”; O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics, Consultants Bureau, New York, 1977, pp. 88–96.

  26. 26.

    Cole, “On a quasi-linear parabolic equation”; E. Hopf, “The partial differential equation ut + uux = μuxx,” Commun. Pure Appl. Math. 3:201–230 (1950); the adaption to Eq. (11.6.7) of this technique for solving quasi-linear partial-differential equations is included in Mendousse, “Nonlinear dissipative distortion.”

  27. 27.

    G. N. Watson, A Treatise on the Theory of Bessel Functions, 2d ed., Cambridge University Press, London, 1944, pp. 77–80.

  28. 28.

    D. T. Blackstock, “Thermoviscous attenuation of plane, periodic, finite-amplitude sound waves,” J. Acoust. Soc. Am. 36:534–542 (1964). A heuristic justification of Eq. (13) proceeds from the recursion relation (Watson, p. 79)

    $$\displaystyle \begin{aligned} \frac{I_{\nu+1}( \varGamma/2)-I_{\nu-1}( \varGamma/2)}{(\nu+1)-(\nu-1)}= -\frac{2\nu}{ \varGamma}I_{\nu}\left(\frac{ \varGamma}{2}\right)\quad \mathrm{to}\quad \frac{d}{d\nu}I_{\nu}\left(\frac{ \varGamma}{2}\right) = -\frac{2\nu}{ \varGamma}I_{\nu}\left(\frac{ \varGamma}{2}\right) \end{aligned}$$

    which integrates to Eq. (13).

  29. 29.

    A proof is outlined by E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1927, p. 489. Applicable numerical results for the evaluation of the logarithmic derivative are given by L. M. Milne-Thomson, “Jacobian elliptic functions and Theta functions,” in M. Abramowitz and I. Stegun (eds.), Handbook of Mathematical Functions, Dover, New York, 1965, pp. 567–585.

  30. 30.

    W. D. Hayes, R. C. Haefeli, and H. E. Kulsrud, “Sonic boom propagation in a stratified atmosphere, with computer program,” NASA CR-1299, 1969; G. B. Whitham, “On the propagation of weak shock waves,” J. Fluid Mech. 1:290–318 (1956). The most notable circumstances for which the weak-shock-ray-theory wedding breaks down are those of caustics: W. D. Hayes, “Similarity rules for nonlinear acoustic propagation through a caustic” in 2d Conf. Sonic Boom Res., Washington, 1968, NASA SP-180, pp. 165–171; R. Seebass, “Nonlinear acoustic behavior at a caustic,” 3d Conf. Sonic Boom Res., NASA SP-255, 1971, pp. 87–120; F. Obermeier, “The behavior of asonic boom in the neighborhood of a caustic,” Max Planck Inst. Strömungsforschung, Rep. 28, Göttingen, 1976.

  31. 31.

    Landau, “On shock waves,” “On shock waves at large distances”; D. T. Blackstock, “On plane, spherical, and cylindrical sound waves of finite amplitude in lossless fluids,” J. Acoust. Soc. Am. 36:217–219 (1964).

  32. 32.

    H. L. Brode, “Numerical solutions of spherical blast waves,” J. Appl. Phys. 26:766–775 (1955).

  33. 33.

    W. M. Wright and N. W. Medendorp, “Acoustic radiation from a finite line source with N-wave excitation,” J. Acoust. Soc. Am. 43:966–971 (1968).

  34. 34.

    A suggested guide to the voluminous early literature on sonic booms is L. J. Runyan and E. J. Kane, “Sonic boom literature survey,” vol. 2, “Capsule summaries,” Fed. Av. Admin. Rep. FAA-RD-73-129-II, AD771-274, 1973, available from Nat. Tech. Inf. Serv., Springfield, Va.

  35. 35.

    The topic discussed here is essentially that of linearized supersonic flow about a body of revolution, the theory of which originated with T. von Kármán and N. B. Moore, “Resistance of slender bodies moving with supersonic velocities with special reference to projectiles,” Trans. Am. Soc. Mech. Eng., sec. APM 54:303–310 (1932). Antecedents date back to J. Ackeret (1925, 1928) and earlier. While the sonic boom has intrinsic nonlinear features, i.e., shock waves, it was demonstrated by G. B. Whitham that a viable theory of the sonic boom could be developed taking the linearized flow solution as a starting point: “The behavior of supersonic flow past a body of revolution, far from the axis,” Proc. R. Soc. Land. A201:89–109 (1950); “The flow pattern of a supersonic projectile,” Commun. Pure Appl. Math. 5:301–348 (1952).

  36. 36.

    The analogy of the homogeneous version of (3) to that for sound propagation in two dimensions is known as von Kármán’s acoustic analogy: T. von Kármán, “Supersonic aerodynamics: principles and applications,” J. Aeronaut. Sci. 14:373–409 (1947); J. W. Miles, “Acoustical methods in supersonic aerodynamics,” J. Acoust. Soc. Am. 20:314–323 (1948).

  37. 37.

    The model represented by Eq. (6) is inapplicable if A′(ξ) should be discontinuous since it would lead to a singular prediction for p. A method of treating such contingencies is given by M. J. Lighthill, “Supersonic flow past slender bodies of revolution, the slope of whose meridian section is discontinuous,” Q. J. Mech. Appl. Math. 1:90–102 (1948).

  38. 38.

    The version (10a) is inapplicable for a projectile of infinite length. The modification to allow for discontinuities in \(A_{B}^{\prime }(\xi )\) proposed by Whitham (1962) proceeds from Lighthill’s (1948) result and yields the Riemann–Stieltjes integral

    $$\displaystyle \begin{aligned} F_{W}(\xi,\alpha) = \frac{1}{2\pi}\int\left(\frac{2}{\alpha R}\right)^{1/2}h \left(\frac{\xi-\mu}{\alpha R}\right)dA_{B}^{\prime}(\mu), \end{aligned}$$

    where R(μ) is the body radius and α is (M2 − 1)1∕2. The function h(X) decreases monotonically from 1 at X = −1, passes through 0.73, 0.56, 0.48 at X = 0, 1, 2, and asymptotically approaches 1∕(2X)1∕2. The integration extends up to \(\mu _{\max }(\xi )\), where \(\xi =\mu _{\max } - \alpha R(\mu _{\max })\) determines \(\mu _{\max }\).

  39. 39.

    D. L. Lansing, “Calculated effects of body shape on the bow-shock overpressures in the far field of bodies in supersonic flow,” NASA Tech. Rep. R-76, Langley Research Center, Hampton, Va., 1960. Lansing introduces a body shape constant Cb related to the K above such that

    $$\displaystyle \begin{aligned} C_{b} = \frac{\gamma\sqrt{\pi}}{2^{5/4}\beta^{1/2}} K, \end{aligned}$$

    so Eq. (18) becomes

    $$\displaystyle \begin{aligned} p_{\mathrm{fs}} = \frac{p_{o}(M^2 - 1)^{1/8}}{(r/L)^{3/4}}\frac{2R_{\max}}{L} C_{b}. \end{aligned}$$

    Typical values of Cb range from 0.54 to 0.81, so K ranges from 0.57 to 0.85.

  40. 40.

    When lift contributions are taken into account, this is no longer exactly the case, as explained by R. Seebass and F. E. McLean, “Far-field sonic boom waveforms,” Am. Inst. Aeronaut. Astronaut. J. 6:1153–1155 (1968).

  41. 41.

    W. D. Hayes, “Linearized supersonic flow,” Ph.D. thesis, California Institute of Technology, 1947; H. Lomax, “The wave drag of arbitrary configurations in linearized flow as determined by areas and forces in oblique planes,” NACA RM A55A18, National Advisory Committee for Aeronautics, Washington, 1955; F. Walkden, “The shock pattern of a wing-body combination, far from the flight path,” Aeronaut. Q. 9:169–194 (1958); J. Morris, “An investigation of lifting effects on the intensity of sonic booms,” J. R. Aeronaut. Soc. 64:610–616 (1960).

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    Allan D. Pierce

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Pierce, A.D. (2019). Nonlinear Effects in Sound Propagation. In: Acoustics. Springer, Cham. https://doi.org/10.1007/978-3-030-11214-1_11

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