Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-20T20:13:43.716Z Has data issue: false hasContentIssue false
  • English
  • Français

Subharmonic resonance of short internal standing waves by progressive surface waves

Published online by Cambridge University Press:  26 April 2006

D. F. Hill  and
M. A. Foda
D. F. Hill
Affiliation:
Environmental Water Resources, Department of Civil Engineering, University of California at Berkeley, Berkeley, CA 94720, USA
M. A. Foda
Affiliation:
Environmental Water Resources, Department of Civil Engineering, University of California at Berkeley, Berkeley, CA 94720, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Experimental evidence and a theoretical formulation describing the interaction between a progressive surface wave and a nearly standing subharmonic internal wave in a two-layer system are presented. Laboratory investigations into the dynamics of an interface between water and a fluidized sediment bed reveal that progressive surface waves can excite short standing waves at this interface. The corresponding theoretical analysis is second order and specifically considers the case where the internal wave, composed of two oppositely travelling harmonics, is much shorter than the surface wave. Furthermore, the analysis is limited to the case where the internal waves are small, so that only the initial growth is described. Approximate solution to the nonlinear boundary value problem is facilitated through a perturbation expansion in surface wave steepness. When certain resonance conditions are imposed, quadratic interactions between any two of the harmonics are in phase with the third, yielding a resonant triad. At the second order, evolution equations are derived for the internal wave amplitudes. Solution of these equations in the inviscid limit reveals that, at this order, the growth rates for the internal waves are purely imaginary. The introduction of viscosity into the analysis has the effect of modifying the evolution equations so that the growth rates are complex. As a result, the amplitudes of the internal waves are found to grow exponentially in time. Physically, the viscosity has the effect of adjusting the phase of the pressure so that there is net work done on the internal waves. The growth rates are, in addition, shown to be functions of the density ratio of the two fluids, the fluid layer depths, and the surface wave conditions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 321 , 25 August 1996 , pp. 217 - 233
Copyright
© 1996 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

Ball, K. 1964 Energy transfer between external and internal gravity waves. J. Fluid Mech. 19, 465–478.Google Scholar
Brekhovskikh, L. M., Goncharov, V. V., Kurtipov, V. M. & Nangol'nykh, K. A. 1972 Resonant excitation of internal waves by nonlinear interaction of surface waves. Izv. Akad. Nauk SSSR, Fiz-Atmos. Okeana 8, 192–203. (English Transl. Atmos. Ocean. Phys. 8, 192–197).Google Scholar
Foda, M. A. & Tzang, S. Y. 1994 Resonant fluidization of silty soil by water waves. J. Geophys. Res. 99 (C10), 20463–20475.Google Scholar
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. R. Soc. Lond. A 309, 551–575.Google Scholar
Thorpe, S. A. 1968 On standing internal gravity waves of finite amplitude. J. Fluid Mech. 24, 737–751.Google Scholar
Ting, C. K. F. & Raichlen, F. 1986 Wave interaction with a rectangular trench. J. Waterway. Port. Coastal, and Ocean Engng ASCE 112 (3), 454–460.Google Scholar