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Room Acoustics

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Acoustics

Abstract

The sound in a room consists of that coming directly from the source plus sound reflected or scattered (see Fig. 6.1) by the walls and by objects in the room.

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Notes

  1. 1.

    J. Duda, “Basic design considerations for anechoic chambers,” Noise Control Eng. 9:60–67 (1977); W. Koidan and G. R. Hruska, “Acoustical properties of the National Bureau of Standards anechoic chamber,” J. Acoust. Soc. Am. 64:508–516 (1978).

  2. 2.

    Standard design criteria are set forth in American National Standard Methods for the Determination of Sound Power Levels of Small Sources in Reverberation Rooms, ANSI S1.21-1972, American National Standards Institute, New York, 1972. See also the discussion by W. K. Blake and L. J. Maja, “Chamber for reverberant acoustic power measurements in air and in water,” J. Acoust. Soc. Am. 57:380–384 (1975).

  3. 3.

    W. C. Sabine, “Architectural acoustics,” Eng. Rec. 38:520–522 (1898); “Architectural acoustics,” ibid. 41:349–351, 376–379, 400–402, 426–427, 450–451, 477–478, 503–505 (1900); both the 1898 paper and the series of 1900 are also printed in Am. Archit. Build. News 62:71–73 (1898), ibid.68:3–5, 19–22, 35–37, 43–45, 59–61, 75–76, 83–84 (1900). All except that of 1898 are printed in W. C. Sabine, Collected Papers on Acoustics, Dover, New York, 1964. Historical sidelights are given by L. L. Beranek: “The Notebooks of Wallace C. Sabine,” J. Acoust. Soc. Am. 61:629–639 (1977).

  4. 4.

    G. Jaeger, “Toward a theory of reverberation,” Sitzungsber. Kais. Akad. Wiss. (Vienna), Math. Naturwiss. Kl., sec. IIa 120:613–634 (1911).

  5. 5.

    Various slightly different experimentally determined values for the numerical coefficient are mentioned in Sabine’s writings; 0.164 s/m is, for example, given in a 1906 paper (Collected Papers on Acoustics, p. 103). The value 0.161 is predicted by theory when the room temperature is 18.3∘C (65∘F); 0.164 corresponds to 9.4∘C (49∘F).

  6. 6.

    W. S. Franklin, “Derivation of equation of decaying sound in a room and definition of open window equivalent of absorbing power,” Phys. Rev. 16:372–374 (1903).

  7. 7.

    In the theory of radiative heat transfer, an intensity of radiation I is defined as the energy emitted by a surface per unit area of surface per unit time and per unit solid angle of propagation direction. The analog of the directional energy density defined in the text can be identified for volumes just outside such a surface and for directions pointing obliquely away from it as I∕c, where c is the speed at which the energy propagates. See, for example, F. Kreith, Principles of Heat Transfer, 3d ed., Intext, New York, 1973, p. 229.

  8. 8.

    That the absorption coefficient defined by Eq. (10) is not necessarily the same as what is required to yield the reverberation time via Eq. (5) is discussed at some length by T. F. W. Embleton, “Sound in large rooms,” in L. L. Beranek (ed.), Noise and Vibration Control, McGraw-Hill, New York, 1971, pp. 219–244.

  9. 9.

    W. B. Joyce, “Sabine’s reverberation time and ergodic auditoriums,” J. Acoust. Soc. Am. 58:643–655 (1975).

  10. 10.

    L. L. Beranek, “Audience and seat absorption in large halls,” J. Acoust. Soc. Am. 32:661–670 (1960); Music, Acoustics, and Architecture, Wiley, New York, 1962, pp. 541–554.

  11. 11.

    P. E. Sabine, Acoustics and Architecture, McGraw-Hill, New York, 1932, pp. 309–311. An earlier but dissimilar derivation leading to the same result was given by Jaeger, “Toward a theory of reverberation.”

  12. 12.

    A geometrical definition (not explicitly involving energy) leading also to 4V∕S has been given by C. W. Kosten, “The mean free path in room acoustics,” Acustica 10:245–250 (1960). Various proposed definitions are reviewed by F. V. Hunt, “Remarks on the mean free path problem,” J. Acoust. Soc. Am. 36:556–564 (1964).

  13. 13.

    C. F. Eyring, “Reverberation time in “‘dead’ rooms,” J. Acoust. Soc. Am., 1:217–241 (1930). The first conception of Eq. (5) is attributed to R. F. Norris by C. A. Andree, ibid. 3:549–550 (1932). Norris’ version of the derivation is given as appendix II in V. O. Knudsen’s Architectural Acoustics, Wiley, New York, 1932, pp. 603–605.

  14. 14.

    The variant on the derivation of taking the first interval as \(\tfrac {1}{2}l_{c}/c\), the rest as l c∕c, yields the same reverberation time.

  15. 15.

    T. W. F. Embleton, “Absorption coefficients of surfaces calculated from decaying sound fields,” J. Acoust. Soc. Am. 50:801–811 (1971).

  16. 16.

    H. C. Hottel, “Radiant heat transmission,” Mech. Eng. 52:699–704 (1930); D. C. Hamilton and W. R. Morgan, “Radiant-interchange configuration factor,” Nat. Adv. Comm. Aeronaut. Rep. NACA TN2836, Washington, 1952; Kreith, Principles of Heat Transfer, pp. 243–251.

  17. 17.

    E. Dietze and W. D. Goodale, Jr., “The computation of the composite noise resulting from random variable sources,” Bell Syst. Tech. J. 18:605–623 (1939); A. London, “Methods for determining sound transmission loss in the field,” J. Res. Natl. Bur. Stand. 26:419–453 (1941); Beranek, Acoustics, McGraw-Hill, New York, 1954, pp. 313–324; R. W. Young, “Sabine reverberation equation and sound power calculations,” J. Acoust. Soc. Am., 31:912–921 (1959).

  18. 18.

    See, for example, Beranek, Music, Acoustics, and Architecture; W. Furrer, Room and Building Acoustics and Noise Abatement, Butterworths, Washington, 1964; A. Lawrence, Architectural Acoustics, Elsevier, Amsterdam, 1970; Knudsen, Architectural Acoustics; A. F. B. Nickson and R. W. Muncey, “Criteria for Room Acoustics,” J. Sound Vib., 1:292–297 (1964); P. H. Parkin, W. E. Scholes, and A. C. Derbyshire, “The Reverberation Times of Ten British Concert Halls,” Acustica, 2:97–100 (1952); H. Bagenal and A. Wood, Planning for Good Acoustics, Methuen, London, 1931.

  19. 19.

    This originated with C. Zwikker, “Partitioning of loudspeaker intensities,” Ingenieur (The Hague) 44:39–45 (1929), and has subsequently been applied by a number of investigators, e.g., R. Thiele, “Directional distribution and chronological order of sound echoes in rooms,” Acustica 3:291–302 (1953); F. Santon, “Numerical prediction of echograms and the intelligibility of speech in rooms,” J. Acoust. Soc. Am. 59:1399–1405 (1976).

  20. 20.

    W. A. Munson, “The growth of auditory sensation,” J. Acoust. Soc. Am. 19:584–591 (1947); J. J. Zwislocki, “Temporal summation of loudness: An analysis,” ibid.46:431–441 (1969); M. J. Penner, “A power law transformation resulting in a class of short-term integrators That produce time-Intensity trades for noise bursts,” ibid. 63:195–201 (1978).

  21. 21.

    H. Haas, “On the influence of a simple echo on the comprehension of Speech,” Acustica 1:49–58 (1951). The value of 50 ms is what was chosen (with reference to speech) as the break point in the partitioning of acoustic energy density into a useful and a disturbing part in Thiele, “Directional Distribution. …”

  22. 22.

    S. Lifshitz, “Mean intensity of sound in an auditorium and optimum reverberation,” Phys. Rev., 27:618–621 (1926); W. A. MacNair, “Optimum reverberation time for auditoriums,” J. Acoust. Soc. Am. 1:242–248 (1930); J. P. Maxfield, “The time integral basic to optimum reverberation time,” ibid. 20:483–486 (1948).

  23. 23.

    See, for example, E. Meyer and H. Kuttruff, “Progress in architectural acoustics,” in E. G. Richardson and E. Meyer (eds.), Technical Aspects of Sound, vol. 3, Elsevier, Amsterdam, 1962, pp. 221–337.

  24. 24.

    M. R. Schroeder, “New method of Measuring reverberation time,” J. Acoust. Soc. Am. 37:409–412 (1965); W. T. Chu, “Comparison of reverberation measurements using Schroeder’s impulse method and decay-curve averaging method,” ibid. 63:1444–1450 (1978).

  25. 25.

    T. J. Schultz, “Sound power measurements in a reverberant room,” J. Sound Vib. 16:119–129 (1971).

  26. 26.

    J. Tichy, “Effects of source position, wall absorption, and rotating diffuser on the qualifications of reverberation rooms,” Noise Control Eng. 7:57–70 (1976); J. Tichy and P. K. Baade, “Effect of rotating diffusers and sampling techniques on sound-pressure averaging in reverberation rooms,” J. Acoust. Soc. Am. 56:137–143 (1974); C. E. Ebbing, “Experimental Evaluation of Moving Sound Diffusers for Reverberant Rooms,” J. Sound Vib., 16:99–118 (1971).

  27. 27.

    I. Pollack and J. M. Pickett, “Cocktail party effect,” J. Acoust. Soc. Am. 29:1262(A) (1957); W. R. MacLean, “On the acoustics of cocktail parties,” ibid. 31:79–80 (1959); L. A. Crum, “Cocktail party acoustics,” ibid. 57:S20 (1975).

  28. 28.

    E. Buckingham, “Theory and interpretation of experiments on the transmission of sound through partition walls,” Sci. Pap. Bur. Stand. (U.S.) 20:193–219 (1924–1926).

  29. 29.

    J. C. Maxwell, Theory of Heat, Longmans Green, London, 1871, p. 308. The demon is “a being whose faculties are so sharpened that he can follow every molecule in its course … who opens and closes [a] hole [connecting two portions of a vessel], so as to allow only the swifter molecules to pass from [side] A to [side] B, and only the slower ones to pass from [side] B to [side] A.”

  30. 30.

    For analyses of enclosures that are not large compared to source dimensions, see R. S. Jackson, “The performance of acoustic hoods at low frequencies,” Acustica 12:139–152 (1962), “Some aspects of the performance of acoustic hoods,” J. Sound Vib. 3:82–94 (1966); M. C. Junger, “Sound transmission through an elastic enclosure acoustically closely coupled to a noise source,” ASME Pap. 70-WA/DE-12, American Society of Mechanical Engineers, New York, 1970.

  31. 31.

    A. H. Davis, “Reverberation equations for two adjacent rooms connected by an incompletely soundproof partition,” Phil. Mag. (6)50:75–80 (1925).

  32. 32.

    Davis, “Reverberation equations …,”; H. Kuttruff, Room Acoustics, Applied Science, London, 1973, pp. 119–123.

  33. 33.

    See, for example, I. S. Sokolnikoff and R. M. Redheffer, Mathematics of Physics and Modern Engineering, 2d ed., McGraw-Hill, New York, 1966, pp. 148–151.

  34. 34.

    J. W. S. Rayleigh, The Theory Sound, vol. 2, 2d ed., reprinted by Dover, New York, 1945, sec. 267. Earlier work by J. M. C. Duhamel gave eigenfunctions and natural frequencies for finite segments of rectangular and circular tubes with rigid walls but ends that were pressure-release surfaces [“On the vibrations of a gas in cylindrical, conical, etc., tubes,” J. Math. Pures Appl. 14:49–110 (1849), especially pp. 84–86]. The basic concept per se of a vibration mode as a building block in the description of a vibrating system with more than 1 degree of freedom dates back to Daniel Bernoulli’s modal description of the vibrating string in 1753.

  35. 35.

    K. Schuster and E. Waetzmann, “On reverberation in closed spaces,” Ann. Phys. (5)1:671–695 (1929); M. J. O. Strutt, “On the acoustics of large rooms,” Phil. Mag. (7)8:236–250 (1929); P. M. Morse, “Some aspects of the theory of room acoustics,” J. Acoust. Soc. Am. 11:56–66 (1939).

  36. 36.

    J. W. S. Rayleigh, “On the fundamental modes of a vibrating system,” Phil. Mag. (5)46:434–439 (1873).

  37. 37.

    See, for example, R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1, Interscience, New York, 1953, p. 4.

  38. 38.

    The applicable theorem is that “the eigenfunctions of any self-adjoint differential system of the second order form a complete set.” That Eqs. (2) describe a self-adjoint system follows from the equivalence of Ψ ∇ 2 ϕ − ϕ ∇ 2 Ψ to the divergence of Ψ ∇ ϕ − ϕ ∇ Ψ and from the vanishing of the normal component of the latter at the walls when both Ψ and ϕ satisfy the boundary condition. For a general proof, see I. Stakgold, Boundary Value Problems of Mathematical Physics, vol. 1, Macmillan, New York, 1967, pp. 212–220.

  39. 39.

    P. M. Morse and K. U. Ingard, “Linear Acoustic Theory” in S. Flügge (ed.), Handbuch der Physik, vol. 11, pt. 2 (Akustik I), Springer, Berlin, 1961, p. 60.

  40. 40.

    E. T. Paris, “On the coefficient of sound-absorption measured by the reverberation method,” Phil. Mag. (7)5:489–497 (1928).

  41. 41.

    This was pointed out to the author by Preston W. Smith, Jr.

  42. 42.

    R. H. Lyon, “Statistical analysis of power injection and response in structures and rooms,” J. Acoust. Soc. Am., 45:545–565(1969).

  43. 43.

    G. C. Maling, Jr., “Calculation of the acoustic power radiated by a monopole in a reverberation chamber,” J. Acoust. Soc. Am. 42:859–865 (1967). The analogous result for a point dipole is given by S. N. Yousri and F. J. Fahy, “An analysis of the acoustic power radiated by a point dipole source into a rectangular reverberation chamber,” J. Sound Vib. 25:39–50 (1972).

  44. 44.

    D.-Y. Maa, “Distribution of eigentones in a rectangular chamber at low frequency range,” J. Acoust. Soc. Am. 10:235–238 (1939).

  45. 45.

    H. Weyl, “The asymptotic distribution law for the eigenvalues of linear partial differential equations (with application to the theory of black body radiation)”, Math. Ann. 71:441–479 (1912). A general proof is given by Courant and Hilbert, Methods of Mathematical Physics, vol. 1, pp. 429–445.

  46. 46.

    M. Schroeder, “The statistical parameters of frequency curves of large rooms,” Acustica, 4:594–600 (1954); M. R. Schroeder and K. H. Kuttruff, “On frequency response curves in rooms: comparison of experimental, theoretical, and Monte Carlo results for the average frequency spacing between maxima,” J. Acoust. Soc. Am. 34:76–80 (1962). The first reference placed the transitional peak spacing at \(\tfrac {1}{10}(\varDelta f)_{\mathrm {res}}\), but this was changed to \(\tfrac {1}{3}(\varDelta f)_{\mathrm {res}}\) in the 1962 paper.

  47. 47.

    R. V. Waterhouse, “Output of a sound source in a reverberation chamber and other reflecting environments,” J. Acoust. Soc. Am. 30:4–13 (1958).

  48. 48.

    See, for example, Maling, “Calculation of the acoustic power.”

  49. 49.

    M. Schroeder, “The statistical parameters of frequency curves of large rooms,” Acustica 4:594–600 (1954).

  50. 50.

    J. L. Doob, Stochastic Processes, Wiley, New York, 1953, pp. 71–72, 141.

  51. 51.

    From (5) one has, for a bivariate Gaussian distribution with q 1 = x, q 2 = y, r = 〈xy〉∕〈y 2〉,

    $$\displaystyle \begin{aligned} \sum_{i,j} [M^{-1}]_{ij}q_i q_j = \frac{\langle y^2 \rangle x^2 - 2\langle xy\rangle xy + \langle x^2\rangle y^2} {\langle x^2\rangle \langle y^2\rangle - \langle xy\rangle^2} = \frac{(x - ry)^2}{\langle (x-ry)^2\rangle} + \frac{y^2}{\langle y^2\rangle} , \end{aligned}$$

    so w(x, y) factors into a product of probability density functions for the statistically independent quantities x − ry and y. Also, Eq. (5) yields 〈y 4〉 = 3〈y 2〉2. Consequently, algebraic manipulation of the expression 〈[(x − ry) + ry]2 y 2〉 leads to Eq. (11).

  52. 52.

    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, London, 1973, pp. 235–236, 243.

  53. 53.

    H. Cramer, Mathematical Methods of Statistics, Princeton University Press, Princeton, N.J., 1946, p. 376. [Our w(z) is Cramer’s j 1(z) with ν = 1, such that his S 1, and S 2 are both zero.]

  54. 54.

    This is in accord with measurements reported by P. Doak, “Fluctuations of the sound pressure level in rooms when the receiver position Is varied,” Acustica, 9:1–9 (1959).

  55. 55.

    M. R. Schroeder, “Effect of frequency and space averaging on the transmission responses of multimode media,” J. Acoust. Soc. Am. 46:277–283 (1969).

  56. 56.

    R. K. Cook, R. V. Waterhouse, R. D. Berendt, S. Edelman, and M. C. Thompson, Jr., “Measurement of correlation coefficients in reverberant sound fields,” J. Acoust. Soc. Am. 27:1072–1077 (1955).

  57. 57.

    D. Lubman, “Spatial averaging in a diffuse sound field,” J. Acoust. Soc. Am. 46:532–534 (1969).

  58. 58.

    Cook et al., “Measurement of correlation coefficient ….”

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    Allan D. Pierce

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Pierce, A.D. (2019). Room Acoustics. In: Acoustics. Springer, Cham. https://doi.org/10.1007/978-3-030-11214-1_6

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