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Low-Frequency Models of Sound Transmission

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Acoustics

Abstract

Acoustic phenomena are often interpreted in terms of concepts based on the assumption that the acoustic wavelength is large compared with a characteristic length.

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Notes

  1. 1.

    The concept originated in major part with J. W. S. Rayleigh, The Theory of Sound, vol. 2, 1878, 2d ed., 1896, reprinted by Dover, New York, 1945, secs. 268, 340. Existence of higher-order modes was demonstrated experimentally by H. E. Hartig and C. E. Swanson, “ ‘Transverse’ acoustic waves in rigid tubes,” Phys. Rev. 54:618–626 (1938). Such modes are of interest in regard to noise generated by turbomachinery, fans, compressors, and jet engines. See, for example, J. M. Tyler and T. G. Sofrin, “Axial flow compressor noise studies,” Soc. Automot. Eng. Trans. 70:309–332 (1962).

  2. 2.

    J. W. S. Rayleigh, “Oscillations in cylindrical vessels,” Phil. Mag. (5)1:272–279 (1876); “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” ibid. 43:125–132 (1897). A related analysis for elastic waves in a solid cylinder was given by L. Pochhammer, “Concerning the velocities of small vibrations in an unlimited isotropic circular cylinder,” J. reine angew. Math., 81:324–336 (1876).

  3. 3.

    This follows from p. 173n. with ξ 1, ξ 2, ξ 3 = r, ϕ, x and with h r = 1, h ϕ = r, h x = 1.

  4. 4.

    Derived by L. Euler in 1764 in an analysis of vibrations of a stretched membrane. That J m(α n r) is a solution follows from an explicit substitution of its power-series expansion into the differential equation. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2d ed., Cambridge University Press, Cambridge, 1944, pp. 5, 6, 15–19.

  5. 5.

    J. McMahon, “On the roots of the Bessel and certain related functions,” Ann. Math. (Charlottesville, Va.) 9:23–30 (1894–1895).

  6. 6.

    First discussed by M. Taylor, “On the emission of sound by a source on the axis of a cylindrical tube,” Phil. Mag. (6)24:655–664 (1912).

  7. 7.

    Taylor, “On the emission of sound,” derives this when the source is on the axis of a circular tube. The generalization to a duct of arbitrary cross-sectional shape is given (although without details of derivation) by H. Lamb, “The propagation of waves of expansion in a tube,” Proc. Lond. Math. Soc. (2)37:547–555 (1934).

  8. 8.

    An extensive exposition of the concept is given by H. H. Woodson and J. R. Melcher, Electromechanical Dynamics, pt I, Discrete Systems, Wiley, New York, 1968, pp. 15–59.

  9. 9.

    G. W. Stewart, “Acoustic wave filters,” Phys. Rev. 20:528–551 (1922).

  10. 10.

    In electric-circuit theory, the term denotes any two-terminal-pair network. See, for example, H. H. Skilling, Electrical Engineering Circuits, Wiley, New York, 1957, pp. 537–572.

  11. 11.

    W. P. Mason, “A study of the regular combination of acoustic elements, with application to recurrent acoustic filters, tapered acoustic filters, and horns,” Bell Syst. Tech. J. 6:258–294 (1927).

  12. 12.

    This is the conventional acoustic analogy. An acoustic-mobility analogy in which pressure → current, volume velocity → voltage, is also occasionally used. The latter was introduced by F. A. Firestone, “A new analogy between mechanical and electrical systems,” J. Acoust. Soc. Am. 4:249–267 (1932–1933).

  13. 13.

    W. Lippert, “The measurement of sound reflection and transmission at right-angled bends in rectangular tubes,” Acustica 4:313–319 (1954); J. W. Miles, “The diffraction of sound due to right-angled joints in rectangular tubes,” J. Acoust. Soc. Am. 19:572–579 (1947). Lippert’s fig. 7 (based on his data) and Miles’ theory suggest that the continuity of pressure is a good approximation up to ka ≈ 1, where a is the width of the duct.

  14. 14.

    For results applicable to cylindrical ducts, see F. Karal, “The analogous acoustical impedance for discontinuities and constrictions of circular cross-section,” J. Acoust. Soc. Am. 25:327–334 (1953). Karal’s approximate low-frequency result in the present notation is that the acoustic inertance [equal to Z J∕(−iω)] associated with a junction between joined circular cylinders of radii b and a (with b < a) with a common axis is of the form

    $$\displaystyle \begin{aligned} M_{A}=\frac{8\rho}{3\pi^{2}b}H\left(\frac{b}{a}\right), \end{aligned}$$

    where H(b∕a) is 1 when b∕a → 0 and decreases monotonically to zero as b∕a → 1.0. The general theory for arbitrary ka is developed by J. W. Miles; “The reflection of sound due to a change in cross section of a circular tube,” ibid. 16:14–19 (1944). A derivation based on the Schwarz–Christoffel transformation applied to a rectangular duct, occupying the region 0 < y < a, 0 > z > d, with a rigid partition at x = 0 having a slit of width b in its middle extending from z = 0 to z = d, y = (a − b)∕2 to y = (a + b)∕2, yields an acoustic inertance

    $$\displaystyle \begin{aligned} M_{A}=\frac{2\rho}{\pi d}\ln\left[\csc\left(\frac{b}{a}\frac{\pi}{2}\right)\right], \end{aligned}$$

    which diverges logarithmically to ∞ as b → 0. J. W. Miles, “The Analysis of Plane Discontinuities in Cylindrical Tubes, II,” ibid., 17:272–284 (1946); P. M. Morse and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968, pp. 483–487.

  15. 15.

    H. Helmholtz, “Theory of air oscillations in tubes with open ends,” J Reine Angew. Math. 57:1–72 (1860); On the Sensations of Tone, 4th ed., 1877, trans. A. J. Ellis, Dover, New York, pp. 42–44, 55, 372–374; M. S. Howe, “On the Helmholtz resonator,” J. Sound Vib. 45:427–440 (1976); U. Ingard, “On the theory and design of acoustical resonators,” J. Acoust. Soc. Am. 25:1037–1062 (1953); A. S. Hersh and B. Walker, “Fluid mechanical model of the Helmholtz resonator,” NASA CR-2904 (1977). Applications to noise control are discussed by M. C. Junger, “Helmholtz resonators in load-bearing walls,” Noise Control Eng. 4:17–25 (1975).

  16. 16.

    As is explained in the next section, the pressure amplitude at moderate distances r from the opening is of the form \(\hat {A}( \boldsymbol {x})+\hat {B}/r\), where \(\hat {A}( \boldsymbol {x})\) is slowly varying with position x relative to the center of the opening, \(\hat {B}\) is independent of position; the identification for \(\hat {p}_{\mathrm {out}}\) is \(\hat {A}(0)\).

  17. 17.

    G. W. Stewart, “Acoustic transmission with a Helmholtz resonator or an orifice as a branch line,” Phys. Rev. 27:487–493 (1926).

  18. 18.

    For a number of similar examples, see L. L. Beranek, Acoustics, McGraw-Hill, New York, 1954, pp. 67–69, 437–442.

  19. 19.

    J. W. S. Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Phil. Mag. (5)43:259–272 (1897).

  20. 20.

    In the analysis (Sect. 4.8) of radiation from a vibrating circular plate, the range of ξ was taken to be from 0 to ∞ and the range of η to be between 0 and π. The distinction arises because we wish the coordinates to be continuous at all points not adjacent to solid boundaries. Here we wish ξ to be continuous at the orifice and accept the discontinuity of η at neighboring points on opposite sides of the plate.

  21. 21.

    H. Lamb, Hydrodynamics, 1879, 5th ed., 1932, sec. 108, pp. 144–145. Lamb’s expression in the present notation is \(\varPhi =-B \cot ^{-1}(\sinh \xi )\), which is \(-B[\pi /2-\tan ^{-1}(\sinh \xi )]\), so our result differs from his by a constant whose value is immaterial insofar as v = ∇ Φ is concerned. The solution dates back to E. Heine (1843).

  22. 22.

    For the more general case of an elliptical orifice of area A and eccentricity e [defined such that (1 − e 2)1∕2 is ratio of minor axis to major axis] the result is

    $$\displaystyle \begin{aligned} \frac{M_{A,\mathrm{or}}}{\rho}=\frac{1}{2}\left(\frac{\pi}{A}\right)^{1/2}\frac{2}{\pi}K(e^2)(1-e^2)^{1/4}\\ \approx\frac{1}{2}\left(\frac{\pi}{A}\right)^{1/2}\left(1-\frac{e^4}{64}-\frac{e^6}{64}-\ldots\right), \end{aligned} $$

    where K(e 2) is the complete elliptical integral of the first kind defined by Eq. (5.3.8). This is derived by Rayleigh, Theory of Sound, vol. 2, sec. 306. Rayleigh’s discussion is in terms of a conductivity, which is the same as ρ divided by the acoustic inertance. His conclusion based on the above result is that it is a good approximation to take the conductivity as 2(A∕π)1∕2 [or to take M A,or as (ρ∕2)(π∕A)1∕2]. For a general review, see C. L. Morfey, “Acoustic properties of openings at low frequencies,” J. Sound Vib. 9:357–366 (1969).

  23. 23.

    The theorem is due to W. Thomson (Lord Kelvin), “On the vis-viva [kinetic energy] of a liquid in motion,” Camb. Dublin Math. J., 1849; reprinted in Mathematical and Physical Papers, vol. 1, Cambridge University Press, Cambridge, 1882, pp. 107–112.

  24. 24.

    The proof begins with the requirement Φ∇2 Φ = 0. With a vector identity and with v = ∇ Φ, this leads to

    $$\displaystyle \begin{aligned} \tfrac{1}{2}\rho{\boldsymbol\nabla}\boldsymbol{\cdot}(\varPhi \boldsymbol{v})=\tfrac{1}{2}\rho v^{2} . \end{aligned}$$

    Integration over the volume and subsequent application of Gauss’s theorem yields

    $$\displaystyle \begin{aligned} \tfrac{1}{2}\rho\varPhi_{2}U_{12}-\tfrac{1}{2}\rho\varPhi_{1}U_{12}=\mathrm{KE}, \end{aligned}$$

    so the definition, M A∕ρ = (Φ 2 − Φ 1)∕U 12, requires that \(2\mathrm {KE}/U_{12}^{2}\) also be M A.

  25. 25.

    J. W. S. Rayleigh, “On the theory of resonance,” Phil. Trans. R. Soc. Lond. 161:77–118 (1870); Theory of Sound, vol. 2, sec. 305. Rayleigh’s statement of the theorem, paraphrased in the terminology of the present text, was that if the ambient density is diminished in any region, the acoustic inertance should also be decreased. The inertance would be the M A,I + M A,II in Eq. (4) if ρ were formally considered to go to zero in a thin layer about the surface S mid. Consequently, the actual inertance should be greater than or equal to M A,I + M A,II. In terms of the electrical analog, Rayleigh’s assertion seems obvious, but the physical realization of such a limiting case in a fluid-dynamic context presents conceptual difficulties, so the theorem is here demonstrated without consideration of cases where the ambient density is nonuniform.

  26. 26.

    L. V. King, “On the electrical and acoustic conductivities of cylindrical tubes bounded by infinite flanges,” Phil. Mag. (7)21:128–144 (1936).

  27. 27.

    H. Levine and J. Schwinger, “On the radiation of sound from an unflanged circular pipe,” Phys. Rev. 73:383–406 (1948). The case when the tube walls are of finite thickness is analyzed by Y. Ando, “On the sound radiation from semi-infinite pipe of certain wall thickness,” Acustica 22:219–225 (1970).

  28. 28.

    W. P. Mason, “The approximate networks of acoustic filters,” Bell Syst. Tech. J. 9:332–340 (1930).

  29. 29.

    Helmholtz, “Theory of air oscillations”; Rayleigh, The Theory of Sound, vol. 2, sec. 314. The necessity for an end correction emerged with the experimental discovery by Felix Savart (1823) that the first velocity node is less than \(\frac {1}{4}\) wavelength from the open end. The boundary condition of p = 0 at the open end (without end correction) was adopted by Daniel Bernoulli, Euler, and Lagrange in the eighteenth century.

  30. 30.

    P. O. A. L. Davies, “The design of silencers for internal combustion engines,” J. Sound Vib. 1:185–201 (1964); T. F. W. Embleton, “Mufflers,” in L. L. Beranek (ed.), Noise and Vibration Control, McGraw-Hill, New York, 1971, pp. 362–405; E. K. Bender and A. J. Bremmer, “Internal-combustion engine intake and exhaust system noise,” J. Acoust. Soc. Am. 58:22–30 (1975).

  31. 31.

    D. D. Davis, G. M. Stokes, D. Moore, and G. L. Stevens, “Theoretical and experimental investigation of mufflers with comments on engine-exhaust muffler design,” Nat. Advis. Comm. Aeronaut. Rep. 1192, Washington, 1954; G. W. Stewart, “Acoustic wave filters,” Phys. Rev. 20:528–551 (1922).

  32. 32.

    A detailed discussion along similar lines but with nonlinear orifice impedance and ambient flow taken into account is given by J. W. Sullivan, “A method of modeling perforated tube muffler components,” J. Acoust. Soc. Am. 66:772–788 (1979).

  33. 33.

    For a historical overview, see J. K. Hilliard, “Historical review of horns used for audience-type sound reproduction,” J. Acoust. Soc. Am. 59:1–8 (1976).

  34. 34.

    This follows from Eqs. (7.7.1) and (7.7.8) with α set to 0, with \(\hat {p}_{H}/\hat {U}_{H}\) set to Z end, and with \(\hat {p}_{G}/\hat {U}_{G}\) set to Z.

  35. 35.

    C. T. Molloy, “Response peaks in finite horns,” J. Acoust. Soc. Am. 22:551–557 (1950); H. Levine and J. Schwinger, “On the radiation of sound from an unflanged circular pipe,” Phys. Rev. 73:383–406 (1948).

  36. 36.

    Beranek, Acoustics, p. 268.

  37. 37.

    C. R. Hanna and J. Slepian, “The function and design of horns for loud speakers,” Trans. Am. Inst. Elec. Eng. 43:393–411 (1924): “Variations in acoustic power of the order of ten to one between 200 and 4000 cycles are not noticed by the ear, however, and the departure from a uniform response can be kept within this range by a proper design of the horn.” The 10:1 is at variance with the original conception of the decibel as the minimum increment of sound level detectable by the human ear but may be appropriate for broadband sound. Beranek (Acoustics, p. 280) chooses a design in one of his examples for which the variation is 2:1 and refers to such as “fairly well damped” resonances.

  38. 38.

    A. G. Webster, “Acoustical impedance, and the theory of horns and of the phonograph,” Proc. Natl. Acad. Sci. (U.S.) 5:275–282 (1919).

  39. 39.

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

  40. 40.

    One can also set it to + m 2, in which case r(x) is \(r_{\mathrm {th}}(\cos mx+T\sin {mx})\). This is discussed by B. N. Nagarkar and R. D. Finch, “Sinusoidal horns,” J. Acoust. Soc. Am. 50:23–31 (1971), who point out that the bell of an English horn is a sinusoidal horn.

  41. 41.

    Similar examples are exhibited by H. F. Olson, “Horn loud speakers,” RCA Rev. 1(4):68–83, April, 1937. Examples for the catenoidal horn are given by G. J. Thiessen, “Resonance characteristics of a finite catenoidal horn,” J. Acoust. Soc. Am. 22:558–562 (1950).

  42. 42.

    R. W. Carlisle, “Method of improving acoustic transmission in folded horns,” J. Acoust. Soc. Am. 31:1135–1137 (1959).

  43. 43.

    A. L. Thuras, R. T. Jenkins, and H. T. O’Neil, “Extraneous frequencies generated in air carrying intense sound waves,” J. Acoust. Soc. Am. 6:173–180 (1935); S. Goldstein and N. W. McLachlan, “Sound waves of finite amplitude in an exponential horn,” ibid. 275–278 (1935).

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    Allan D. Pierce

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Pierce, A.D. (2019). Low-Frequency Models of Sound Transmission. In: Acoustics. Springer, Cham. https://doi.org/10.1007/978-3-030-11214-1_7

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