Damping of weakly nonlinear shallow-water waves

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.