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A new form of Boussinesq equations for long waves in water of non-uniform depth

Szmidt J., Hedzielski B.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2012

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Vol. 60, nr 3

631-643

EN

The paper describes the non-linear transformation of long waves in shallow water of variable depth. Governing equations of the problem are derived under the assumption that the non-viscous fluid is incompressible and the fluid flow is a rotation free. A new form of Boussinesq-type equations is derived employing a power series expansion of the fluid velocity components with respect to the water depth. These non-linear partial differential equations correspond to the conservation of mass and momentum. In order to find the dispersion characteristic of the description, a linear approximation of these equations is derived. A second order approximation of the governing equations is applied to study a time dependent transformation of waves in a rectangular basin of water of variable depth. Such a case corresponds to a spatially periodic problem of sea waves approaching a near-shore zone. In order to overcome difficulties in integrating these equations, the finite difference method is applied to transform them into a set of non-linear ordinary differential equations with respect to the time variable. This final set of these equations is integrated numerically by employing the fourth order Runge - Kutta method.

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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.

3

A Lagrangian Finite Element Analysis of Gravity Waves in Water of Variable Depth

Staroszczyk R.

Archives of Hydro-Engineering and Environmental Mechanics

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2009

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Vol. 56, nr 1-2

43-61

EN

The paper is concerned with the problem of gravitational wave propagation in water of variable depth. The problem is formulated in the Lagrangian description, and the ensuing equations are solved numerically by a finite element method. In computations a convecting mesh that follows the material fluid particles is used. As illustrations, results of numerical simulations carried out for plane gravity waves propagating over bottoms of simple geometry are presented. For parameters typical of a laboratory flume, the transformation of a transient wave, generated by a single movement of a piston-like wave maker, is investigated. The results show the evolution of the free-surface elevation, displaying steepening of the wave over sloping beds and its gradual attenuation in regions of uniform depth.