Damping of weakly nonlinear shallow-water waves

Author:

Miles John W.

Abstract

Boundary-layer damping of one-dimensional gravity waves of slowly varying amplitude a(t), characteristic wavenumber k, and characteristic frequency ω in water of depth d and kinematic viscosity v is calculated for a [Lt ] d, d [Lt ] 1/k and (2ν/ω)½ [Lt ] d. General results are given for the temporal evolution of the power spectral density determined by either a Fourier-integral (spatially aperiodic) or Fourier-series (spatially periodic) representation of the wave. Solitary and cnoidal waves are considered as examples. Keulegan's (1948) inverse-fourth-power decay for the solitary wave is recovered, and the numerical parameter therein is evaluated by reduction to a Riemann zeta function. A universal decay curve is obtained for the Stokes-scaled amplitude S = a/k2d3 of the cnoidal wave as a function of the boundary-layer-scaled time (νω)½t/d; the result is both more flexible and more compact than that obtained by Isaacson (1976). The decay is within 5% of that for a solitary wave (inverse fourth power) for S > 2 or that for an infinitesimal wave (exponential) for S < 2. An analytical approximation with a maximum error of less than 1% is obtained by joining an asymptotic approximation for S > 1 to the exponential approximation for S < 1.

Publisher

Cambridge University Press (CUP)

Subject

Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics

Reference8 articles.

1. Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics .Pergamon.

2. Keulegan, G. H. 1948 Gradual damping of solitary waves.J. Res. Nat. Bur. Stand 40,487–498.

3. Cayley, A. 1895 An Elementary Treatise on Elliptic Functions .London:Deighton, Bell & Co.

4. Ott, E. & Sudan, R. N. 1970 Damping of solitary waves.Phys. Fluids,13,1432–1434.

5. Lamb, H. 1932 Hydrodynamics .Dover.

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