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3 Archives of Hydro-Engineering and Environmental Mechanics

3 Chybicki W.

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Simple water waves in Lagrangian description

Chybicki W.

Archives of Hydro-Engineering and Environmental Mechanics

2006

Vol. 53, nr 4

381-387

Details of the model of long water waves in the Lagrangian description are presented. The equation of motion is derived from variational formulation of the problem. Only two important cases are considered: when the water depth changes uniformly in space or the depth is constant. For quasilinear hyperbolic system obtained in this description the Riemann invariants and equation of simple waves are found. For constant depth, the Riemann invariants are exactly the same as in the Euler description, however, the velocity of wave propagation is different. In case of uniform slope the velocity, as well as the Riemann invariants are different. In the Lagrangian description the free surface is described in parametric form.

Run-Up of Linear Long Waves on Uniform Slope in Lagrangian Description

Chybicki W.

Archives of Hydro-Engineering and Environmental Mechanics

2006

Vol. 53, nr 2

97-104

The case of linear, two-dimensional long waves on a uniform slope is considered. It is assumed that the fluid is nonviscous and incompressible. In the present paper the description of the long wave proposed by Wilde (Wilde, Chybicki 2004) is based on the fundamental assumption that the vertical material lines of fluid remain vertical during the entire motion. The equations of motion are derived with the help of a variational formulation of the problem. The Lagrangian is the difference between the kinetic and potential energy of the fluid. In the paper a correction followed from dispersion to the results obtained by Shuto is presented.

Certain solutions of periodic long waves with non-linear dispersion

Chybicki W.

Archives of Hydro-Engineering and Environmental Mechanics

2004

Vol. 51, nr 4

307-315

The paper describes some special solutions of the long water waves theory proposed by Wilde. The wave equation is derived with the help of a variational formulation of the problem with the Langrangian being the difference between the kinetic and potential energies. In order to look for travelling wave solutions the simple transformation is made. The solutions have been found in the same way as in the KdV equation. Solutions for different wave amplitudes are presented in the paper. The special cases of solutions are solitary waves. It is proved that bounded solutions of an equation can represent periodic or solitary waves and both length and velocity of waves increase when the height of waves increases.