Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-14T22:30:41.578Z Has data issue: false hasContentIssue false
  • English
  • Français

Overturning of nonlinear acoustic waves. Part 2 Relaxing gas dynamics

Published online by Cambridge University Press:  26 April 2006

P. W. Hammerton  and
D. G. Crighton
P. W. Hammerton
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
D. G. Crighton
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider finite-amplitude acoustic disturbances propagating through media in which relaxation mechanisms, such as those associated with the vibration of polyatomic molecules, are significant. While the effect of these relaxation modes is to inhibit the wave steepening associated with nonlinearity, whether a particular mode is sufficient to prevent the occurrence of multi-valued solutions will depend on the form of the disturbance and on the characteristic parameters of the relaxation. Analysis of this condition is necessary in order to reveal which physical mechanisms actually determine the evolution of the wave profile. This then dictates the scaling of any embedded shock regions. Sufficient conditions for the occurrence of multi-valued solutions are obtained analytically for periodic waves, hence proving that in certain circumstances relaxation is in fact insufficient in fully describing the wave propagation. A much more precise criterion is then obtained numerically. This uses the techniques described in Part 1 for analysing the phenomenon of wave overturning using intrinsic coordinates. Illustrations are provided of the development of a harmonic signal for different classes of material parameters.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 601 - 615
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

Blythe, P. A. 1969 Nonlinear wave propagation in a relaxing gas. J. Fluid Mech. 37, 31–50.Google Scholar
Crighton, D. G. & Scott, J. F. 1979 Asymptotic solutions of model equations in nonlinear acoustics. Phil. Trans. R. Soc. Lond. A 292, 107–134.Google Scholar
Fisher, F. H. & Simmons, V. P. 1977 Sound absorption in sea water. J. Acoust. Soc. Am. 62, 558–564.Google Scholar
Hammerton, P. W. 1990 Nonlinear wave propagation with diffusion and relaxation. PhD thesis, University of Cambridge.
Hammerton, P. W. & Crighton, D. G. 1993 Overturning of nonlinear acoustic waves. Part 1. A general method. J. Fluid Mech. 252, 585–599.Google Scholar
Hunter, J. K. 1989 Numerical solutions of some nonlinear dispersive wave equations. Lect. Appl. Maths 26, 301–316.Google Scholar
Johannesen, N. H. & Scott, W. A. 1978 Shock formation distance in real air. J. Sound Vib. 61, 169–177.Google Scholar
Lighthill, M. J. 1956 Viscosity effects in sound waves of finite amplitude. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies), pp. 250–351. Cambridge University Press.
Naumkin, P. I. & Shishmarev, I. A. 1982 On the breaking of waves for the Whitham equation. Soviet Math. Dokl. 26, 150–152.Google Scholar
Naumkin, P. I. & Shishmarev, I. A. 1983 On the Cauchy problem for Whitham's equation. Soviet Math. Dokl. 28, 719–721.Google Scholar
Ockendon, H. & Spence, D. A. 1969 Nonlinear wave propagation in a relaxing gas. J. Fluid Mech. 39, 329–345.Google Scholar
Pierce, A. D. 1981 Acoustics: An Introduction to its Physical Principles and Applications. McGrawHill.
Polyakova, A. L., Soluyan, S. I. & Khokhlov, R. V. 1962 Propagation of finite disturbances in a relaxing medium. Soviet Phys. Acoust. 8, 78–82.Google Scholar
Southern, I. S. & Johannesen, N. H. 1980 Numerical analysis of the nonlinear propagation of plane periodic waves in a relaxing gas. J. Fluid Mech. 99, 343–364.Google Scholar
Whitham, G. B. 1967 Variational methods and applications to water waves. Proc. R. Soc. Lond. A 299, 6–25.Google Scholar