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The spectral broadening of sound by turbulent shear layers. Part 1. The transmission of sound through turbulent shear layers

Published online by Cambridge University Press:  19 April 2006

L. M. B. C. Campos
L. M. B. C. Campos
Affiliation:
Engineering Department, University of Cambridge Present address: R. Rodrigo da Funseca 91-2°D, Lisbon 1, Portugal.
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Abstract

The transmission of sound through a turbulent shear layer is reviewed both with respect to the mathematical theory of the distortion of signals during propagation and also as the basis of the analysis in part 2 of the spectral broadening evinced by experimental results relevant to the study of aircraft noise. The reflexion and transmission coefficients, which involve both amplitude and phase changes, are obtained for scattering by an irregular and unsteady interface convected between two media; diffraction of high frequency sound by small-scale turbulence in a shear layer is accounted for by means of a phase shift and conservation laws for energy, wavenumber and amplitude. These results may be used to construct the sound field transmitted through a turbulent shear layer from a source in the interior of a jet; multiple internal reflexions are accounted for in the case of transmission through two parallel shear layers, i.e. a jet of finite width shielding a noise source. A statistical description is given of the process of attenuation of the transmitted wave as energy is diffracted by the turbulence, and of the partial compensation by interference between correlated components of the refracted wave; the reference case is a monochromatic point source, which, when placed behind a system of shear layers, has its energy redistributed directionally and spread over a range of frequencies. The expressions obtained for the energy flux include as particular cases the results of Howe for the plane (1975) and impedance (1976) layers, the effects of turbulence being shown to be consistent with the experiments of Schmidt & Tilmann (1970) and Ho & Kovasznay (1976a, b).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 89 , Issue 4 , 27 December 1978 , pp. 723 - 749
Copyright
© 1978 Cambridge University Press

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References

Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Born, M. & Wolf, E. 1970 Principles of Optics, 5th edn. Pergamon.
Blokhintsev, D. I. 1956 Acoustics of non-homogeneous moving medium. N.A.C.A. Tech. Memo. no. 1399.Google Scholar
Brekhovskikh, L. M. 1960 Waves in Layered Media. Academic Press.
Chandler, T. R. D., Colclough, A. R. & Quinn, T. J. 1976 A new determination of the gas constant by an acoustical method. Phil. Trans. Roy. Soc. A 283, 367–420.Google Scholar
Chernov, L. A. 1967 Wave Propagation in a Random Medium. Dover.
Clarke, R. H. 1973 Theory of acoustic propagation in a variable ocean. Saclant ASW Res. Centre, La Spezia, Italy, Memo. SM-28.Google Scholar
Crighton, D. G. & Leppington, F. G. 1973 Singular perturbation methods in acoustics: diffraction by a plate of finite thickness. Proc. Roy. Soc. A 335, 313–339.Google Scholar
D'Alembert, J. Le R.1747 Recherches sur les vibrations des cordes sonores. Mém. Acad. Sci. Paris Opuscules Mathématiques 1, 1–73.Google Scholar
Hamilton, W. R. 1827–32 Theory of systems of rays (with three supplements). Trans. Roy. Ir. Acad. 15, 69–174; 16, 1–61, 93–125; 17, 1–144. (See also Papers 1, 1–193.)Google Scholar
Ho, C. M. & Kovasznay, L. S. G. 1976a Propagation of a coherent acoustic wave through a turbulent shear flow. J. Acoust. Soc. Am. 60, 40–45.Google Scholar
Ho, C. M. & Kovasznay, L. S. G. 1976b Acoustical shadowgraph. Phys. Fluids 19, 1118–1123.Google Scholar
Howe, M. S. 1973a Multiple scattering of sound by turbulence and other inhomogeneities. J. Sound Vib. 27, 455–476.Google Scholar
Howe, M. S. 1973b On the kinetic theory of wave propagation in random media. Phil. Trans. Roy. Soc. A 274, 523–544.Google Scholar
Howe, M. S. 1975 Application of energy conservation to the solution of radiation problems involving uniformly convected source distributions. J. Sound Vib. 43, 77–86.Google Scholar
Howe, M. S. 1976 The attenuation of sound by a randomly irregular impedance layer. Proc. Roy. Soc. A 347, 513–535.Google Scholar
Jones, D. S. 1953 Diffraction by a thick semi-infinite plate. Proc. Roy. Soc. A 217, 153–175.Google Scholar
Jones, D. S. & Morgan, J. D. 1972 The instability of a vortex sheet on a subsonic stream under acoustic radiation. Proc. Camb. Phil. Soc. 72, 465–488.Google Scholar
Khinchin, S. I. 1948 Mathematical Foundations of Statistical Mechanics. Dover.
Kolmogorov, A. N. 1950 Foundations of the Theory of Probability. New York: Chelsea.
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.
Lighthill, M. J. 1953 On the energy scattered from the interaction of turbulence with sound and shock waves. Proc. Camb. Phil. Soc. 44, 531–551.Google Scholar
Lighthill, M. J. 1964 Group velocity. J. Inst. Math. Appl. 1, 1–28.Google Scholar
Lindeberg, J. W. 1922 Eine neue Erleitung der Exponentialgezetzes in der Wahrschein-lichtkeitsrechnung. Math. Z. 15, 211–225.Google Scholar
Liouville, J. 1837–8 Sur le dévelopement des fonctions ou des parties des fonctions en séries dont les divers termes sont à satisfaire à assujettis une même équation différentielle du second ordre contenant un paramètre variable. J. Math. Pures Appl. 1, 255–264; 2, 16–35; 3, 418–436.Google Scholar
Miles, J. W. 1958 On the disturbed motion of a plane vortex sheet. J. Fluid Mech. 4, 538–552.Google Scholar
Mises, R. Von 1960 The Mathematical Theory of Probability and Statistics (ed. H. Geiringer). Academic Press.
Rayleigh, Lord 1945 The Theory of Sound. Dover.
Schmidt, D. W. & Tilmann, P. M. 1970 Experimental study of sound phase fluctuations caused by turbulent wakes. J. Acoust. Soc. Am. 47, 1310–1324.Google Scholar
Sommerfeld, A. 1896 Matematische Theorie der Diffraction. Math. Ann. 47, 317–375.Google Scholar
Tatarski, V. I. 1967 Wave Propagation in a Turbulent Medium. Dover.
Whittaker, E. T. & Watson, G. N. 1927 A Course of Modern Analysis, 4th edn. Cambridge University Press.