Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-22T07:35:19.747Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolution and decay of cylindrical and spherical nonlinear acoustic waves generated by a sinusoidal source

Published online by Cambridge University Press:  26 April 2006

P. L. Sachdev  and
K. R. C. Nair
P. L. Sachdev
Affiliation:
Department of Applied Mathematics, Indian Institute of Science, Bangalore-560012, India
K. R. C. Nair
Affiliation:
Department of Applied Mathematics, Indian Institute of Science, Bangalore-560012, India
Get access Rights & Permissions [Opens in a new window]

Abstract

The present work gives a comprehensive numerical study of the evolution and decay of cylindrical and spherical nonlinear acoustic waves generated by a sinusoidal source. Using pseudospectral and predictor–corrector implicit finite difference methods, we first reproduced the known analytic results of the plane harmonic problem to a high degree of accuracy. The non-planar harmonic problems, for which the amplitude decay is faster than that for the planar case, are then treated. The results are correlated with the known asymptotic results of Scott (1981) and Enflo (1985). The constant in the old-age formula for the cylindrical canonical problem is found to be 1.85 which is rather close to 2, ‘estimated’ analytically by Enflo. The old-age solutions exhibiting strict symmetry about the maximum are recovered; these provide an excellent analytic check on the numerical solutions. The evolution of the waves for different source geometries is depicted graphically.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 204 , July 1989 , pp. 389 - 404
Copyright
© 1989 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

Blackstock, D. T.: 1964 Thermovirus attenuation of plane, periodic, finite-amplitude sound waves. J. Acoust. Soc. Am. 36, 534–542.Google Scholar
Cole, J. D.: 1951 On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225–236.Google Scholar
Crighton, D. G. & Scott, J. F., 1979 Asymptotic solutions of model equations in nonlinear acoustics. Phil Trans. R. Soc. Lond. A 292, 101–134.Google Scholar
Douglas, J. & Jones, B. F., 1963 On predictor–corrector methods for nonlinear parabolic differential equations. J. Soc. Ind. & Appl. Maths 11, 195–204.Google Scholar
Enflo, B. O.: 1985 Saturation of a nonlinear cylindric sound wave generated by a sinusoidal source. J. Acoust. Soc. Am. 77, 54–60.Google Scholar
Fay, R. D.: 1931 Plane sound waves of finite amplitude. J. Acoust. Soc. Am. 3, 222–241.Google Scholar
Fubini, E.: 1935 Anamolie nella propagazione di onde acustache di grande ampiezza. Acta Freq. 4, 530–581.Google Scholar
Leibovich, S. & Seebass, A. R. (eds) 1974 Nonlinear Waves. Cornell University Press.
Lesser, M. B. & Crighton, D. G., 1975 Physical acoustics and the method of matched asymptotic expansions. In Physical Acoustics (ed. W. P. Mason & R. N. Thurston), vol. 11. Academic Press.
Lighthill, M. J.: 1956 Viscosity effects in sound waves of finite amplitude. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies). Cambridge University Press.
Nair, K. R. C.: 1988 Numerical and analytic studies of generalised Burgers equations. Ph.D. thesis, Indian Institute of Science, Bangalore, India.
Nimmo, J. J. C. & Crighton, D. G. 1986 Geometrical and diffusive effects in nonlinear acoustic propagation over long ranges. Phil. Trans. R. Soc. Lond. A 320, 1–35.Google Scholar
Parker, D. F.: 1980 The decay of saw-tooth solutions to the Burgers equation. Proc. R. Soc. Lond. A 369, 409–424.Google Scholar
Sachdev, P. L.: 1987 Nonlinear Diffusive Waves. Cambridge University Press.
Sachdev, P. L., Tikekar, V. G. & Nair, K. R. C. 1986 Evolution and decay of spherical and cylindrical N waves. J. Fluid Mech. 172, 347–371.Google Scholar
Scott, J. F.: 1981 Uniform asymptotics for spherical and cylindrical nonlinear acoustic waves generated by a sinusoidal source. Proc. R. Soc. Lond. A 375, 211–230.Google Scholar