In order to model the evolution of a solitary wave near an obstacle or over an uneven bottom, the long-wave equations including curvature effects are introduced to describe the deformation and fission of a barotropic solitary wave passing over a shelf or an obstacle. The numerical results obtained from these equations are shown to be in good agreement with an analytical model derived by Germain (1984) in the framework of a generalized shallow-water theory, and with experimental results collected in a large channel equipped with a wave generator. Given the initial conditions, i.e. amplitude of the incident solitary wave, water depth in the deep region, and height of the shelf or the barrier, it is possible to predict the amplitude and number of the transmitted solitary waves as well as the amplitude of the reflected wave, and to describe the shape of the free surface at any time.

A Petrov-Galerkin finite element method (FEM) for the regularized long wave (RLW) equation is proposed. Finite elements are used in both the space and the time domains. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weight functions. An implicit, conditionally stable, one-step predictor-corrector time integration scheme results. The accuracy and stability are investigated by means of local expansion by Taylor series and the resulting equivalent differential equation. An analysis based on a linear Fourier series solution and the Von Neumann's stability criterion is also performed. Based on the order of the analytical approximations and of the domain discretization it is concluded that the scheme is of third order in the nonlinear version and of fourth order in the linear version. Three numerical experiments of wave propagation are presented and their results compared with similar ones in the literature: solitary wave propagation, undular bore propagation, and cnoidal wave propagation. It is concluded that the present scheme possesses superior conservation and accuracy properties. IntroductionThe regularized long wave (RLW) equation was first proposed by Peregrine [23] for modelling the propagation of unidirectional weakly nonlinear and weakly dispersive water waves. Later on, Benjamin et al. [4] proposed the use of the RLW equation as a preferred alternative to the more classical Korteweg-de Vries (KdV) equation [20], to model a larger class of physical phenomena. These authors showed that the RLW equation is better posed than the KdV equation. Abdulloev et al.[1] numerically exposed the inelastic behaviour of solitary waves modelled by the RLW equation, which results from the fact that this model only possesses three conserved quantities, as demonstrated by Olver [22].Numerical solutions of the RLW equation based on the finite difference method were proposed by several authors (see, e.g. [13,14,19,23]). Spectral methods for the same equation were presented by Ben-Yu and Manoranjan [3] and Sloan [25]. Several finite element schemes based on spline Galerkin techniques have been applied to the RLW equation, (see, e.g. [2,5,8,9,15,16,17,18,24]). Luo and Liu [21] proposed a mixed finite element formulation. These finite element formulations were usually approximations with C 1 continuity. A finite volume method was introduced for a generalized KdV-RLW equation by Bradford and Sanders [7]. Finally, Durán and López-Marcos [11,12] showed the importance of conservative numerical methods for the long run simulation and the solitary wave interactions of the RLW model.Our purpose here is to formulate a Petrov-Galerkin finite element method based on a space-time finite element with C 0 continuity. The weight functions are derived from those proposed by Yu and Heinrich [26] for the convective-diffusion equation. It can be extended to bidimensional, multivariate systems [27] and thus amenable to be extended to the Boussinesq equations [6].The r...

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In recent years, several authors have conducted experimental studies on the scour around piles fixed in the wave flume. In addition, the settlement of structures has been studied for the case of pipelines, due either to steady currents [International Journal of Offshore and Polar Engineering 4 (1) (1994) 30] or to wave forcing [Coastal Engineering 42 (2001) 313]. However, the settlement of a pile due to scour in waves has so far not been investigated.This paper reports an experimental study on the settlement of vertical cylindrical piles, either surface-piercing or entirely submerged, exposed to waves.The primary objective of the study was to investigate: (i) the scour-settlement process with piles of differing heights and densities with their bases initially lying on the seabed and (ii) the influence of the piles height and weight on the hydrodynamics vertical settlement.A series of experiments was carried out using isolated vertical cylinders of a given diameter (30 mm), different materials (PVC, aluminium, brass, copper and bronze) and different heights (30, 6 and 3 cm). The characteristics of the waves remained constant in all tests (water height = 15 cm, wave period = 2.1 s, local wave height = 4.0 cm, Keulegan -Carpenter number = 12).The cylinders were set up over a sand bed, so that they could move vertically. Both the initial settlement of each cylinder (when it occurs), due to its weight, and the evolution of additional settlement caused by flow-induced scour were measured. The tests were conducted over a period of 4000 wave cycles. D

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