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Propagation of nonlinear acoustic waves in a tunnel with an array of Helmholtz resonators

Published online by Cambridge University Press:  26 April 2006

N. Sugimoto
N. Sugimoto
Affiliation:
Department of Mechanical Engineering, Faculty of Engineering Science, University of Osaka, Toyonaka, Osaka 560, Japan
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Abstract

It is proposed that an array of Helmholtz resonators connected to a tunnel in its axial direction will suppress the propagation of sound generated by a travelling train and especially the emergence of shock waves in the far field. Under the approximation that the resonators may be regarded as continuously distributed, quasi-one-dimensional formulation is given for nonlinear acoustic waves by taking account of not only the resonators but also the wall friction due to the presence of a boundary layer and the diffusivity of sound. For a far-field propagation, the spatial evolution equation coupled with the equation for the response of the resonator is then derived. The linear dispersion relation suggests that the resonators, if appropriately designed, enhance the dissipation and give rise to the dispersion as well. By solving initial-value problems for the evolution equation, the array of resonators is proved to be very effective in suppressing shock waves in the far field. The resonators themselves fail to counteract shock waves once formed, but rather prevent their emergency by rendering acoustic waves dispersive. By this dispersion, it becomes possible, in a special case, for an acoustic soliton to be propagated in place of a shock wave.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 244 , November 1992 , pp. 55 - 78
Copyright
© 1992 Cambridge University Press

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References

Chester W. 1964 Resonant oscillations in closed tubes. J. Fluid Mech. 18, 44–64.Google Scholar
Drazin, P. G. & Johnson R. S. 1989 Solitons: An Introduction. Cambridge University Press.
Ingard U. 1953 On the theory and design of acoustic resonators. J. Acoust. Soc. Am. 25, 1037–1061.Google Scholar
Ingard, U. & Ising H. 1967 Acoustic nonlinearity of an orifice. J. Acoust. Soc. Am. 42, 6–17.Google Scholar
Ingard, U. & Labate S. 1950 Acoustic circulation effects and the nonlinear impedance of orifices. J. Acoust. Soc. Am. 22, 211–218.Google Scholar
Keller J. J. 1981 Propagation of simple non-linear waves in gas filled tubes with friction. Z. angew. Math. Phys. 1932, 170–181.Google Scholar
King L. V. 1936 On the electrical and acoustic conductivities of cylindrical tubes bounded by infinite flanges. Phil. Mag. 21 (7), 128–144.Google Scholar
Lighthill M. J. 1956 Viscosity effects in sound wave of finite amplitude. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies), pp. 250–351. Cambridge University Press.
Monkewitz P. A. 1985 The response of Helmholtz resonators to external excitation. Part 2. Arrays of slit resonators. J. Fluid Mech. 156, 151–166.Google Scholar
Monkewitz, P. A. & Nguyen-Vo N.-M. 1985 The response of Helmholtz resonators to external excitation. Part 1. Single resonators. J. Fluid Mech. 151, 477–497.Google Scholar
Pierce A. D. 1981 Acoustics: An Introduction to its Physical Principles and Applications. McGraw-Hill.
Sugimoto N. 1989 ‘Generalized’ Burgers equations and fractional calculus. In Nonlinear Wave Motion (ed. A. Jeffrey), pp. 162–179. Longman.
Sugimoto N. 1990 Evolution of nonlinear acoustic waves in a gas-filled pipe. In Frontiers of Nonlinear Acoustics, 12th Intl Symp. on Nonlinear Acoustics (ed. M. F. Hamilton & D. T. Blackstock), pp. 345–350. Elsevier.
Sugimoto N. 1991 Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. J. Fluid Mech. 225, 631–653.Google Scholar
Sugimoto N. 1992 Suppression of acoustic shock waves in a tunnel by an array of Helmholtz resonators. Proc. DGLR/AIAA Aeroacoustic conference, Aachen, Germany, pp. 571–575. Bonn: DGLR.
Whitham G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Wijngaarden L. van 1968 On the oscillations near and at resonance in open pipes. J. Engng Maths 2, 225–240.Google Scholar
Zinn B. T. 1970 A theoretical study of non-linear damping by Helmholtz resonators. J. Sound Vib. 13, 347–356.Google Scholar