Edge waves over a shelf: full linear theory

1984 ◽  
Vol 142 ◽  
pp. 79-95 ◽  
Author(s):  
D. V. Evans ◽  
P. Mciver
Keyword(s):  
Shallow Water ◽  
Wide Class ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Explicit Solution ◽  
Mode Shapes ◽  
Edge Waves ◽  
Wave Solutions ◽  
Linearized Theory ◽  
Sloping Beach

Edge-wave solutions to the linearized shallow-water equations for water waves are well known for a variety of bottom topographies. The only explicit solution using the full linearized theory describes edge waves over a uniformly sloping beach, although the existence of such waves has been established for a wide class of bottom geometries. In this paper the full linearized theory is used to derive the properties of edge waves over a shelf. In particular, curves are presented showing the variation of frequency with wavenumber along the shelf, together with some mode shapes for a particular shelf geometry.

scholarly journals Conservation laws for shallow water waves on a sloping beach

1986 ◽  
Vol 9 (2) ◽  
pp. 387-396
Author(s):  
Yilmaz Akyildiz
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Water Waves ◽  
Simple Wave ◽  
Shallow Water Waves ◽  
Linear Partial Differential Equations ◽  
Wave Solutions ◽  
Hodograph Plane ◽  
Sloping Beach ◽  
Linear Nature

Shallow water waves are governed by a pair of non-linear partial differential equations. We transfer the associated homogeneous and non-homogeneous systems, (corresponding to constant and sloping depth, respectively), to the hodograph plane where we find all the non-simple wave solutions and construct infinitely many polynomial conservation laws. We also establish correspondence between conservation laws and hodograph solutions as well as Bäcklund transformations by using the linear nature of the problems on the hodogrpah plane.

scholarly journals Approximate Traveling Wave Solutions of Coupled Whitham-Broer-Kaup Shallow Water Equations by Homotopy Analysis Method

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
M. M. Rashidi ◽  
D. D. Ganji ◽  
S. Dinarvand
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Homotopy Analysis Method ◽  
Shallow Water Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Analysis Method ◽  
Wave Solutions ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is applied to obtain the approximate traveling wave solutions of the coupled Whitham-Broer-Kaup (WBK) equations in shallow water. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the systems of nonlinear partial differential equations.

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.

Water waves of finite amplitude on a sloping beach

1958 ◽  
Vol 4 (1) ◽  
pp. 97-109 ◽  
Author(s):  
G. F. Carrier ◽  
H. P. Greenspan
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Finite Amplitude ◽  
Explicit Solutions ◽  
Shallow Water Theory ◽  
Non Linear ◽  
Wave Forms ◽  
Time Histories ◽  
Sloping Beach

In this paper, we investigate the behaviour of a wave as it climbs a sloping beach. Explicit solutions of the equations of the non-linear inviscid shallow-water theory are obtained for several physically interesting wave-forms. In particular it is shown that waves can climb a sloping beach without breaking. Formulae for the motions of the instantaneous shoreline as well as the time histories of specific wave-forms are presented.

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.

The generation of surface waves over a sloping beach by an oscillating line source

1976 ◽  
Vol 79 (3) ◽  
pp. 573-585 ◽  
Author(s):  
Clare A. N. Morris
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Line Source ◽  
Short Wave ◽  
Long Waves ◽  
Edge Waves ◽  
Wave Regime ◽  
Wave Solutions ◽  
Sloping Beach

AbstractA line source whose strength varies sinusoidally with time and also with the co-ordinate measured along its length is situated parallel to the shoreline of a beach of angle ¼π0. Both long-and short-wave solutions are found. It is shown that for certain positions of the source, long waves are not radiated to infinity, while in the short-wave regime, the solutions take the form of edge-waves, with resonances occurring at certain wavenumbers. Computations of the free-surface contours are presented for a range of wavenumbers.

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