Hostname: page-component-84b7d79bbc-4hvwz Total loading time: 0 Render date: 2024-07-31T20:06:40.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear acoustics in non-uniform infinite and finite layers

Published online by Cambridge University Press:  26 April 2006

W. Ellermeier
Get access Rights & Permissions [Opens in a new window]

Abstract

The propagation of weakly nonlinear acoustic waves in a non-uniform medium is treated. It is assumed that the waves are one-dimensional. Non-uniformities arising from variable cross-section and stratification are included. The effect of non-uniformities on unidirectional waves on an infinite interval and resonant waves on a finite interval is discussed for a near-uniform reference state (geometrical acoustics limit) and for stronger non-uniformities in the finite-interval case. Nonlinearities are taken into account up to quadratic and, wherever necessary, cubic order in the wave amplitude.

Unidirectional waves in the geometrical acoustics limit can formally be reduced to the behaviour in a uniform system described by a kinematic wave equation with constant coefficients. For illustration acceleration waves in a weakly non-uniform medium are treated. The resonance case in the geometrical acoustics limit is closely related to resonance in a uniform system so that the methods developed for that situation require only slight modification. For larger influence of non-uniformity the geometrical acoustics limit does not apply and the resonance problem may lead to a Duffing oscillator type of behaviour.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 257 , December 1993 , pp. 183 - 200
Copyright
© 1993 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blokhintsev, D. 1946 The propagation of sound in an inhomogeneous and moving medium I. J. Acoust. Am. 18, 322–328.Google Scholar
Brekhovskikh, L. 1960 Waves in Layered Media. Academic.
Chester, W. 1964 Resonant oscillations in closed tubes. J. Fluid Mech. 18, 44–64.Google Scholar
Chester, W. 1991 Acoustic resonance in spherically symmetric waves. Proc. R. Soc. Lond. A 434, 459–463.Google Scholar
Cox, E. A. & Mortell, M. P. 1983 The evolution of resonant oscillations in closed tubes. Z. Angew. Math. Phys. 34, 845–866.Google Scholar
Cox, E. A. & Mortell, M. P. 1986 The evolution of resonant wave-water oscillations. J. Fluid Mech. 162, pp. 99–116.Google Scholar
Crighton, D. G. 1992 Nonlinear acoustics. In: Modern Methods in Analytical Acoustics (ed. D. G. Crighton, A. P. Dowling, J. Ffows Williams et. al.) Springer.
Ellermeier, W. 1993 Resonant wave motion in a non-homogeneous system. Z. Angew. Math. Phys. (to appear.)Google Scholar
Hayes, W. D. & Runyan, H. L. 1970 Sonic-boom propagation through a stratified atmosphere. J. Acoust. Soc. Am. 51, 695–701.Google Scholar
Keller, J. B. & Lu Ting 1966 Periodic vibrations of systems governed by nonlinear partial differential equations. Commun. Pure Appl. Maths 19, 371–420.Google Scholar
Keller, J. J. 1977 Nonlinear acoustic resonances in shock tubes with varying cross sectional area. Z. Angew. Math. Phys. 28, 107–122.Google Scholar
Lettau, E. 1939 Messengen an Gasschwingungen großer Amplituden in Rohrleitungen. Dtsch. Kraftfahrforsch. 39, 961–971.Google Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear Oscillations. John Wiley & Sons.
Ockendon, H., Ockendon, J. R. & Johnson, A. D. 1986 Resonant sloshing in shallow water. J. Fluid Mech. 167, 465–479.Google Scholar
Ockendon, H., Ockendon, J. R., Peake, M. R. & Chester, W. 1993 Geometrical effects in resonant gas oscillations. J. Fluid Mech. 257, 201–217.Google Scholar
Pierce, A. D. 1981 Acoustics. McGraw Hill.
Saenger, R. A. & Hudson, G. E. 1960 Periodic shock waves in resonating gas columns J. Acoust. Soc. Am. 32, 961–971.Google Scholar
Seymour, B. R. Mortell, M. P. 1975 Nonlinear geometrical acoustics. In Mechanics Today, vol. 2 (ed. S. Nemat-Nasser), pp. 251–313. Pergamon.
Varley, E. & Cumberbatch, E. 1970 Large amplitude waves in stratified media: acoustic pulses. J. Fluid Mech. 43, 513–537.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley Interscience.