scholarly journals Mass Transport Velocity in Shallow-Water Waves Reflected in a Rotating Ocean with a Coastal Boundary

2007 ◽  
Vol 37 (10) ◽  
pp. 2429-2445 ◽  
Author(s):  
Frode Hoydalsvik
Keyword(s):  
Mass Transport ◽  
Shallow Water ◽  
Water Waves ◽  
Traditional Approach ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Transport Velocity ◽  
Uniform Solution ◽  
Mass Transport Velocity ◽  
Sloping Bottom

Abstract The mass transport velocity in shallow-water waves reflected at right angles from an infinite and straight coast is studied theoretically in a Lagrangian reference frame. The waves are weakly nonlinear and monochromatic, and propagate in a homogenous, viscous, and rotating ocean. Unlike the traditional approach where the domain is divided into thin boundary layers and a core region, the uniform solution is obtained here without constraints on the thickness of the bottom wave boundary layer. It is shown that the mass transport velocity is not only sensitive to topography, but depends heavily on the interplay between the vertical length scales. Similarities and differences between the cases of a constant depth, a linearly sloping bottom, and a wavy and linearly sloping bottom are discussed. The mass transport velocity can be divided into two main categories—that induced by waves with a frequency close to the inertial frequency, and that induced by waves with a much larger frequency. For waves significantly affected by rotation to first order, the cross-shore mass transport velocity is very small relative to the alongshore mass transport velocity, and the direction of the mass transport velocity is reversed relative to that in waves of much higher frequencies.

Mass transport in water waves over an elastic bed

1995 ◽  
Vol 450 (1939) ◽  
pp. 371-390 ◽  
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Mass Transport ◽  
Water Waves ◽  
Viscous Boundary Layer ◽  
Curvilinear Coordinate ◽  
Transport Velocity ◽  
Mass Transport Velocity ◽  
Orthogonal Curvilinear Coordinate System ◽  
Viscous Boundary

The mass transport velocity in water waves propagating over an elastic bed is investigated. Water is assumed to be incompressible and slightly viscous. The elastic bed is also incompressible and satisfies the Hooke’s law. For a small amplitude progressive wave perturbation solutions via a boundary-layer approach are obtained. Because the wave amplitude is usually larger than the viscous boundary layer thickness and because the free surface and the interface between water and the elastic bed are moving, an orthogonal curvilinear coordinate system (Longuet-Higgins 1953) is used in the analysis of free surface and interfacial boundary layers so that boundary conditions can be applied on the actual moving surfaces. Analytical solutions for the mass transport velocity inside the boundary layer adjacent to the elastic seabed and in the core region of the water column are obtained. The mass transport velocity above a soft elastic bed could be twice of that over a rigid bed in the shallow water.

Three-Dimensional Weakly Nonlinear Shallow Water Waves Regime and its Traveling Wave Solutions

2018 ◽  
Vol 15 (03) ◽  
pp. 1850017 ◽  
Author(s):  
Aly R. Seadawy
Keyword(s):  
Free Surface ◽  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Waves ◽  
Three Dimensional ◽  
Traveling Wave Solutions ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Wave Solutions

The problem formulations of models for three-dimensional weakly nonlinear shallow water waves regime in a stratified shear flow with a free surface are studied. Traveling wave solutions are generated by deriving the nonlinear higher order of nonlinear evaluation equations for the free surface displacement. We obtain the velocity potential and pressure fluid in the form of traveling wave solutions of the obtained nonlinear evaluation equation. The obtained solutions and the movement role of the waves of the exact solutions are new travelling wave solutions in different and explicit form such as solutions (bright and dark), solitary wave, periodic solitary wave elliptic function solutions of higher-order nonlinear evaluation equation.

scholarly journals The shallow water equations: conservation laws and symplectic geometry

1987 ◽  
Vol 10 (3) ◽  
pp. 557-562 ◽  
Author(s):  
Yilmaz Akyildiz
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Symplectic Form ◽  
Symplectic Geometry ◽  
Nonlinear Differential Equations ◽  
Shallow Water Waves ◽  
Hodograph Method ◽  
Sloping Bottom

We consider the system of nonlinear differential equations governing shallow water waves over a uniform or sloping bottom. By using the hodograph method we construct solutions, conservation laws, and Böcklund transformations for these equations. We show that these constructions are canonical relative to a symplectic form introduced by Manin.

Characteristic of the algebraic traveling wave solutions for two extended (2 + 1)-dimensional Kadomtsev–Petviashvili equations

2019 ◽  
Vol 35 (07) ◽  
pp. 2050028 ◽  
Author(s):  
Jian-Gen Liu ◽  
Xiao-Jun Yang ◽  
Yi-Ying Feng
Keyword(s):  
Shallow Water ◽  
Traveling Wave ◽  
Water Waves ◽  
Traveling Wave Solutions ◽  
Rational Solutions ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Restoring Forces ◽  
First Time

With the aid of the planar dynamical systems and invariant algebraic cure, all algebraic traveling wave solutions for two extended (2 + 1)-dimensional Kadomtsev–Petviashvili equations, which can be used to model shallow water waves with weakly nonlinear restoring forces and to describe waves in ferromagnetic media, were obtained. Meanwhile, some new rational solutions are also yielded through an invariant algebraic cure with two different traveling wave transformations for the first time. These results are an effective complement to existing knowledge. It can help us understand the mechanism of shallow water waves more deeply.

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.

Mass transport in water waves

1953 ◽  
Vol 245 (903) ◽  
pp. 535-581 ◽  
Keyword(s):  
General Theory ◽  
Mass Transport ◽  
Boundary Layers ◽  
Water Waves ◽  
Equations Of Motion ◽  
Maximum Velocity ◽  
Field Equations ◽  
Transport Velocity ◽  
Mass Transport Velocity ◽  
General Method

It was shown by Stokes that in a water wave the particles of fluid possess, apart from their orbital motion, a steady second-order drift velocity (usually called the mass-transport velocity). Recent experiments, however, have indicated that the mass-transport velocity can be very different from that predicted by Stokes on the assumption of a perfect, non-viscous fluid. In this paper a general theory of mass transport is developed, which takes account of the viscosity, and leads to results in agreement with observation. Part I deals especially with the interior of the fluid. It is shown that the nature of the motion in the interior depends upon the ratio of the wave amplitude a to the thickness d of the boundary layer: when a 2 / d 2 is small the diffusion of vorticity takes place by viscous ‘conduction’; when a 2 / d 2 is large, by convection with the mass-transport velocity. Appropriate field equations for the stream function of the mass transport are derived. The boundary layers, however, require separate consideration. In part II special attention is given to the boundary layers, and a general theory is developed for two types of oscillating boundary: when the velocities are prescribed at the boundary, and when the stresses are prescribed. Whenever the motion is simple-harmonic the equations of motion can be integrated exactly. A general method is described for determining the mass transport throughout the fluid in the presence of an oscillating body, or with an oscillating stress at the boundary. In part III, the general method of solution described in parts I and II is applied to the cases of a progressive and a standing wave in water of uniform depth. The solutions are markedly different from the perfect-fluid solutions with irrotational motion. The chief characteristic of the progressivewave solution is a strong forward velocity near the bottom. The predicted maximum velocity near the bottom agrees well with that observed by Bagnold.

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.

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