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1
Content available Boussinesq-type Equations for Long Waves in Water of Variable Depth
Szmidt K.
Archives of Hydro-Engineering and Environmental Mechanics
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2011
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Vol. 58, nr 1-4
3-21
EN
The paper deals with the problem of the transformation of long gravitational waves propagating in water of variable depth. The main attention of the paper is focused on the derivation of equations describing this phenomenon. These equations are derived under the assumption that the non-viscous fluid is incompressible and rotation free, and that the fluid velocity components may be expressed in the form of the power series expansions with respect to the water depth. This procedure makes it possible to transform the original two-dimensional problem into a one-dimensional one, in which all unknown variables depend on time and a horizontal coordinate. The partial differential equations derived correspond to the conservation of mass and momentum. The solution of these equations is constructed by the finite difference method and an approximate discrete integration in the time domain. In order to estimate the accuracy of this formulation, theoretical results obtained for a specific problem were compared with experimental measurements carried out in a laboratory flume. The comparison shows that the proposed theoretical formulation is an accurate description of long waves propagating in water of variable depth.
2
Content available On the Description of Long Water Waves in Material Variables
Szmidt K., Hedzielski B.
Archives of Hydro-Engineering and Environmental Mechanics
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2009
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Vol. 56, nr 1-2
63-83
EN
Shallow water equations formulated in material variables are presented in this paper. In the model considered, a three-dimensional physical problem is substituted by a two-dimensional one describing a transformation of long waves in water of variable depth. The latter is obtained by means of the assumption that a vertical column of water particles remains vertical during the entire motion of the fluid. Under the assumption of small, continuous variation of the water depth, the equations for gravity waves are derived through Hamilton's principle formulated in terms of the material coordinates. This formulation ensures the conservation of mechanical energy. The approximation depends on the wave parameters as well as on the bed bathymetry. The latter may influence a solution of the model decisively; thus, one should be careful in applying the description to complicated geometries of fluid domains encountered in engineering practice.
3
Content available remote Dwuwymiarowe nieliniowe fale długie w strefie zmiennej głębokości wody
Chybicki W.
Inżynieria Morska i Geotechnika
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2007
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nr 4
215-219
PL
Dwuwymiarowy model propagacji nieliniowych, dyspersyjnych fal długich nad dnem o zmiennym profilu w opisie materialnym. Metoda korektor - predyktor Adamsa-Bashfortha-Moultona. Porównanie obliczeń numerycznych z doświadczeniami w kanale falowym. Przykłady obliczeń falowania dwuwymiarowego.
EN
2D model of propagation of non-linear dispersional long waves over the uneven bottom profile in material description. Corrector - predictor Adamas-Bashforth-Moulton's method. Numerical calculations versus test results from the wave flume. Examples of numerical calculations for 2D cases.
4
Content available Simple water waves in Lagrangian description
Chybicki W.
Archives of Hydro-Engineering and Environmental Mechanics
|
2006
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Vol. 53, nr 4
381-387
EN
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.
5
Content available remote Fale długie nad dnem o zmiennym profilu
Chybicki W.
Inżynieria Morska i Geotechnika
|
2004
|
nr 6
285-288
PL
Nowy model fal długich nad zmiennym dnem w opisie Langrange'a. Równania ruchu w zapisie wariacyjnym z funkcją Langrange' a. Porównanie wyników obliczeń numerycznych z doświadczeniami w kanale falowym.
EN
New model for long waves over the uneven bottom described in Lagrangian description. Equations of motion in variational notation with Lagrange's function. Comparison of the results of numerical calculations with experiments made in wave flume.
6
Content available Long water waves as a structure fluid interaction problem
Wilde P., Chybicki W.
Archives of Hydro-Engineering and Environmental Mechanics
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2004
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Vol. 51, nr 2
95-118
EN
The paper describes a new formulation of the theory of long shallow water waves, which is based on the fundamental assumption that vertical material lines of fluid remain vertical during the entire motion. To make the problem consistent from the point of view of physics the case of waves in a flume due to the motion of a piston type generator is considered. At the piston the material line of water particles remains vertical during the entire motion and thus the generation follows the assumption in the description of the motion of water in the flume. Wave equations are derived with the help of a variational formulation of the problem in a material description. The Lagrangian is the difference between the kinetic and potential energies of the fluid and the mechanical system that describes a very simplified wave generator. The basic assumption simplifies the geometry of the displacement field. The definitions of generalized forces follow from variational calculus. The procedure ensures that the energy is preserved. A simple discrete formulation of the problem is based on the finite element method and the corresponding approximate expressions for energies.
7
Content available Certain solutions of periodic long waves with non-linear dispersion
Chybicki W.
Archives of Hydro-Engineering and Environmental Mechanics
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2004
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Vol. 51, nr 4
307-315
EN
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.
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