Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-19T04:09:32.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Experiments on ripple instabilities. Part 1. Resonant triads

Published online by Cambridge University Press:  21 April 2006

Diane M. Henderson  and
Joseph L. Hammack
Diane M. Henderson
Affiliation:
Department of Engineering Sciences, University of Florida, Gainesville, FL 32611, USA
Joseph L. Hammack
Affiliation:
Department of Engineering Sciences, University of Florida, Gainesville, FL 32611, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Water waves for which both gravitation and surface tension are important (ripples) exhibit a variety of instabilities. Here, experimental results are presented for ripple wavetrains on deep water with frequencies greater than 19.6 Hz where a continuum of resonant triad interactions are dynamically admissible. The experimental wave-trains are indeed unstable, and the instability becomes more pronounced as non-linearity is increased. The unstable wavefield is characterized by significant spatial disorder while temporal measurements at fixed spatial locations remain quite ordered. In fact, for most experiments temporal measurements suggest that a selection process exists in which a single triad dominates evolution. The dominant triad typically does not involve a subharmonic frequency of the generated wave and persists over a wide range of amplitudes for the initial wave. Viscosity does not appear to be important in the selection process; however, it may be responsible for the lack of subsequent triad production by the excited waves of the initial triad. The presence of a selection process contradicts previous conjecture, based on the form of the interaction coefficients, that a broad-banded spectrum of waves should occur. The general absence of subharmonic growth also contradicts previously reported experiments. Results are also presented for wavetrains at the parametric boundary of 19.6 Hz and a degenerate case of resonant triads at 9.8 Hz (Wilton's ripples). In addition to resonant triads, the experiments show evidence of (generally) weaker narrow-band interactions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 184 , November 1987 , pp. 15 - 41
Copyright
© 1987 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

Bannerjee, P. P. & Korpel, A. 1982 Subharmonic generation by resonant three-wave interaction of deep-water capillary waves. Phys. Fluids 25, 1938–1943.Google Scholar
Benney, D. J. 1962 Non-linear gravity wave interactions. J. Fluid Mech. 14, 577–584.Google Scholar
Bloomfield, P. 1976 Fourier Analysis of Time Series: An Introduction. John Wiley.
Craik, A. D. D. 1986 Exact solutions of non-conservative equations for three-wave and second harmonic resonance. Proc. R. Soc. Lond. A 406, 1–12.Google Scholar
Faraday, M. 1831 On the forms and states assumed by fluid in contact with vibrating elastic surfaces. Phil. Trans. R. Soc. 31, 319.Google Scholar
Harrison, W. J. 1909 The influence of viscosity and capillarity on waves of finite amplitude. Proc. Lond. Math. Soc. 7, 107–121.Google Scholar
Hasselmann, K. 1967 A criterion for nonlinear wave stability. J. Fluid Mech. 30, 737–739.Google Scholar
Henderson, D. M. & Lee, R. C. 1986 Laboratory generation and propagation of ripples. Phys. Fluids 29, 619–624.Google Scholar
Hogan, S. J. 1984 Subharmonic generation of deep-water capillary waves. Phys. Fluids 27, 42–45.Google Scholar
Kaup, D. J. 1981 The solution of the general initial value problem for the full three dimensional three-wave resonant interaction. Physica 3D 1 and 2, 374–395.Google Scholar
Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.Google Scholar
Lamb, H. L. 1895 Hydrodynamics, 2nd edn. Cambridge University Press.
Lamb, H. L. 1932 Hydrodynamics. Dover Publication.
Lighter, S. & Chen, J. 1987 Subharmonic resonance of nonlinear cross-waves. J. Fluid Mech. 183, 451–465.Google Scholar
Lucassen-Reynders, E. H. & Lucassen, J. 1969 Properties of capillary waves. Adv. Colloid Interface Sci. 2, 347–395.Google Scholar
McGoldrick, L. F. 1965 Resonant interactions among capillary–gravity waves. J. Fluid Mech. 21, 305–331.Google Scholar
McGoldrick, L. F. 1970 An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40, 251–271.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.Google Scholar
Rayleigh, Lord 1890 On the tension of water surfaces, clean and contaminated, investigated by the method of ripples. Reprinted 1964 in Scientific Papers, vol. 3, pp. 397–425. Dover.
Rayleigh, Lord 1945 The Theory of Sound. Dover.
Scott, J. C. 1979 The preparation of clean water surfaces for fluid mechanics. In Surface Contamination: Genesis, Detection and Control, vol. 1 (ed. K. L. Mittal), pp. 477–497. Plenum.
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. R. Soc. Lond. A 309, 551–575.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Wilton, J. R. 1915 On ripples. Phil. Mag. 29(6), 688–700.Google Scholar