scholarly journals Nonlinear long waves over a muddy beach

2013 ◽  
Vol 718 ◽  
pp. 371-397 ◽  
Author(s):  
Erell-Isis Garnier ◽  
Zhenhua Huang ◽  
Chiang C. Mei
Keyword(s):  
Boundary Layer ◽  
Thin Layer ◽  
Surface Waves ◽  
Nonlinear Waves ◽  
Water Waves ◽  
Long Waves ◽  
Fluid Mud ◽  
Weakly Nonlinear ◽  
Sea Bottom ◽  
Oscillatory Boundary Layer

AbstractWe analyse theoretically the interaction between water waves and a thin layer of fluid mud on a sloping seabed. Under the assumption of long waves in shallow water, weakly nonlinear and dispersive effects in water are considered. The fluid mud is modelled as a thin layer of viscoelastic continuum. Using the constitutive coefficients of mud samples from two field sites, we examine the interaction of nonlinear waves and the mud motion. The effects of attenuation on harmonic evolution of surface waves are compared for two types of mud with distinct rheological properties. In general mud dissipation is found to damp out surface waves before they reach the shore, as is known in past observations. Similar to the Eulerian current in an oscillatory boundary layer in a Newtonian fluid, a mean displacement in mud is predicted which may lead to local rise of the sea bottom.

Damping of weakly nonlinear shallow-water waves

1976 ◽  
Vol 76 (2) ◽  
pp. 251-257 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Boundary Layer ◽  
Solitary Wave ◽  
Water Waves ◽  
Maximum Error ◽  
Fourier Integral ◽  
Analytical Approximation ◽  
Fourth Power ◽  
Weakly Nonlinear ◽  
Power Spectral ◽  
Dimensional Gravity

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.

Amplification and decay of long nonlinear waves

1973 ◽  
Vol 58 (3) ◽  
pp. 481-493 ◽  
Author(s):  
S. Leibovich ◽  
J. D. Randall
Keyword(s):  
Shallow Water ◽  
Nonlinear Waves ◽  
Water Waves ◽  
Vries Equation ◽  
Variable Coefficients ◽  
Finite Amplitude ◽  
Rotating Fluids ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Waves

The interaction of weakly nonlinear waves with slowly varying boundaries is considered. Special emphasis is given to rotating fluids, but the analysis applies with minor modifications to waves in stratified fluids and shallow-water aves. An asymptotic solution of a variant of the Korteweg–de Vries equation with variable coefficients is developed that produces a ‘Green's law’ for the amplification of waves of finite amplitude. For shallow-water waves in water of variable depth, the result predicts wave growth proportional to the $-\frac{1}{3}$ power of the depth.

scholarly journals COMBINED REFRACTION-DIFFRACTION OF NONLINEAR WAVES IN SHALLOW WATER

1984 ◽  
Vol 1 (19) ◽  
pp. 68
Author(s):  
James T. Kirby ◽  
Philip L.F. Liu ◽  
Sung B. Yoon ◽  
Robert A. Dalrymple
Keyword(s):  
Shallow Water ◽  
Nonlinear Waves ◽  
Water Waves ◽  
Evolution Equations ◽  
Boussinesq Equations ◽  
Two Dimensions ◽  
Parabolic Approximation ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Dimensional Domain

The parabolic approximation is developed to study the combined refraction/diffraction of weakly nonlinear shallow water waves. Two methods of approach are taken. In the first method Boussinesq equations are used to derive evolution equations for spectral wave components in a slowly varying two-dimensional domain. The second method modifies the equation of Kadomtsev s Petviashvili to include varying depth in two dimensions. Comparisons are made between present numerical results, experimental data and previous numerical calculations.

scholarly journals APPROXIMATION DESCRIPTIONS OF THE FOCUSSING OF WATER WAVES

1986 ◽  
Vol 1 (20) ◽  
pp. 51
Author(s):  
D.H. Peregrine
Keyword(s):  
Theoretical Approach ◽  
Nonlinear Waves ◽  
Water Waves ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Accurate Solution ◽  
Wave Crest ◽  
Weakly Nonlinear ◽  
Alternative Approach ◽  
Weakly Nonlinear Waves

Underwater shoals and spurs focus water waves that propagate over them. The normal theoretical approach to finding a more accurate solution of the linear equations is to interpret the envelope of crossing rays as a cusp of caustics (see Figure 1) and to use Pearcey's function (Pearcey 1946). In practical cases the ray pattern is rarely sufficiently well defined to enable the cusp parameters to be deduced. An alternative approach is presented in which a length of wave crest heading towards the focussing region is approximated by an arc of a circle or parabola (Figure 2). Corresponding approximate solutions for linear and weakly nonlinear waves are described.

Non-Newtonian oscillatory layer flow, approximate solution

1979 ◽  
Vol 44 (10) ◽  
pp. 2908-2914 ◽  
Author(s):  
Ondřej Wein
Keyword(s):  
Boundary Layer ◽  
Approximate Solution ◽  
Galerkin Method ◽  
Horizontal Plane ◽  
Oscillatory Flow ◽  
Oscillatory Boundary Layer ◽  
Layer Flow ◽  
Pseudoplastic Liquid ◽  
The Galerkin Method

The problem of the oscillatory flow of pseudoplastic liquid in vicinity of the infinitely long horizontal plane is formulated in stresses. For Re i.e. for conditions of oscillatory boundary layer the problem is solved approximately by the Galerkin method.

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