Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-19T01:19:04.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Resonance in nonlinear bubble oscillations

Published online by Cambridge University Press:  26 April 2006

Michael S. Longuet-Higgins
Michael S. Longuet-Higgins
Affiliation:
Center for Studies of Nonlinear Dynamics, La Jolla Institute, PO Box 1434, La Jolla, CA 92038, USA and Institute for Nonlinear Science, University of California, San Diego, R-002, La Jolla, CA 92093, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In two recent papers (Longuet-Higgins 1989a,b) the author showed that the shape oscillations of bubbles can emit sound like a monopole source, at second order in the distortion parameter ε. In the second paper (LH2) it was predicted that the emission would be amplified when the second harmonic frequency 2σn of the shape oscillation approaches the frequency ω of the breathing mode. This ‘resonance’ would however be drastically limited by damping due to acoustic radiation and thermal diffusion. The predictions were confirmed by further numerical calculations in Longuet-Higgins (1990a).

Ffowcs Williams & Guo (1991) have questioned the conclusions of LH2 on the grounds that near resonance there is a slow (secular) transfer of energy between the shape oscillation and the volumetric mode which tends to diminish the amplitude of the shape oscillation and hence falsify the perturbation analysis. They have also argued that the volumetric mode never grows sufficiently to produce sound of the stated order of magnitude. In the present paper we show that these assertions are unfounded. Ffowcs Williams & Guo considered only undamped oscillations. Here we show that when the appropriate damping is included in their analysis the secular transfer of energy becomes completely insignificant. The resulting pressure pulse (figure 5 below) is found to be essentially identical to that calculated in LH2, figure 3. Moreover, in the initial-value problem considered in LH2, the excitation of the volumetric mode takes place not by a secular energy transfer but by a resonance during the first few cycles of the shape oscillation. This accounts for the amplification near resonance found in Longuet-Higgins (1990a). Finally, it is pointed out that the initial energy of the shape oscillations is far greater than is required to produce the O(ε2) volume pulsations that were studied in LH2, and which were used for a comparison with field data. This acoustic radiation was not calculated by Ffowcs Williams & Guo.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 224 , March 1991 , pp. 531 - 549
Copyright
© 1991 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

Benney, D. J.: 1962 Non-linear gravity-wave interactions. J. Fluid Mech. 14, 577–584.Google Scholar
Brethebton, F. P.: 1964 Resonant interactions between waves; the case of discrete oscillations. J. Fluid Mech. 20, 457–479.Google Scholar
Williams, J. E. Ffowcs & Guo, Y. P. 1991 On resonant nonlinear bubble oscillations. J. Fluid Mech. 224, 507–529 (referred to herein as FWG).Google Scholar
Kerman, B. R.: 1988 Sea Surface Sound. Dordrecht: Kluwer, 639 pp.
Longuet-Hiooins, M. S.: 1989a Monopole emission of sound by asymmetric bubble oscillations. Part 1. Normal modes. J. Fluid Mech. 201, 525–541 (referred to herein as LH1).Google Scholar
Longuet-Higgins, M. S.: 1989b Monopole emission of sound by asymmetric bubble oscillations. Part 2. An initial-value problem. J. Fluid Mech. 201, 543–565 (referred to herein as LH2).Google Scholar
Longuet-higgins, M. S.: 1990a Bubble noise spectra. J. Acoust. Soc. Am. 87, 652–661 (referred to herein as LH3).Google Scholar
Longuet-Higgins, M. S.: 1990b An analytic model of sound production by raindrops. J. Fluid Mech. 214, 395–110.Google Scholar