Yehuda Agnon

A new approach to high-order Boussinesq models

An innite-order, Boussinesq-type dierential equation for wave shoaling over variable bathymetry is derived. Dening three scaling parameters { nonlinearity, the dispersion parameter, and the bottom slope { the system is truncated to a nite order. Using Pad e approximants the order in the dispersion parameter is eectively doubled. A derivation is made systematic by separately solving the Laplace equation in the undisturbed fluid domain and then addressing the nonlinear free-surface conditions. We show that the nonlinear interactions are faithfully captured. The shoaling and dispersion components are time independent. Boussinesq-type equations have been widely studied in recent years. The introduction of Pad e approximants greatly improves the dispersion and shoaling characteristics of these equations, making them an attractive tool for general coastal applications. A review of the subject is given by Madsen & Sch¨ (1998, referred to as MS in the following). The methods used most often for deriving high-order Boussinesq equations are based on two techniques: one is the choice of appropriate velocity variables which improves the dispersion characteristics of the resulting equation, and the other is enhancement of the equations by applying appropriate linear operators to the continuity equation and the momentum equation. In MS the two methods were combined leading to an improved dispersion relation. The variables used are normally a velocity variable (the velocity at some fraction of the depth or the mean velocity) and the free-surface elevation. This choice leads to a time-dependent system in which dispersion, nonlinearity and shoaling are all coupled. In water of intermediate depth, the highest order terms are required for representation of dispersion and shoaling. This is successfully achieved by using Pad e approximants. However, in previous studies there appears to be some trade o among the performance regarding nonlinearity, shoaling and dispersion in the dierent sets of equations. Our primary goal is to present an approach which performs well in all three respects. The present approach is based on decoupling the problem into two subproblems. One is the linear part of the problem, which involves solving the Laplace equation in the undisturbed fluid domain, and accounts for dispersion and shoaling. The idea of Boussinesq-type equations is to eliminate the vertical coordinate from the problem. Inx 2, this is achieved by the use of innite power series expansions. The kinematic bottom boundary condition then provides a relation between the horizontal and

A SIMPLIFIED ANALYTICAL MODEL FOR A FLOATING BREAKWATER IN WATER OF FINITE DEPTH

Abstract The performance of a box-type floating breakwater is studied. The implementation of simplifying assumptions concerning the flow beneath a pontoon-type floating breakwater, leads to an analytical solution of the two-dimensional linearized hydrodynamic problem. Comparison of the analytical results with a numerical solution of the full linear problem shows good agreement over a wide range of parameters.

Stochastic nonlinear shoaling of directional spectra

We derive a deterministic directional shoaling model and a stochastic directional shoaling model for a gravity surface wave field, valid for a beach with parallel depth contours accounting for refraction and nonlinear quadratic (three wave) interactions. A new phenomenon of non-resonant spectral evolution arises due to the medium inhomogeneity. The kernels of the kinetic equation depend on the bathymetry through an integral operator. Preliminary tests carried out on laboratory data for a unidirectional case indicate that the stochastic model also works rather well beyond the region where the waves may be regarded as nearly Gaussian. The limit of its applicability is decided by the dispersivity of the medium (relative to the nonlinearity). Good agreement with both laboratory data and the underlying deterministic model is found up to a value of about 1.5 for the spectral peak Ursell number. Beyond that only the deterministic model matches the measurements.

Evolution of a nonlinear wave field along a tank: experiments and numerical simulations based on the spatial Zakharov equation

Evolution of a nonlinear wave eld along a laboratory tank is studied experimentally and numerically. The numerical study is based on the Zakharov nonlinear equation, which is modied to describe slow spatial evolution of unidirectional waves as they move along the tank. Groups with various initial shapes, amplitudes and spectral contents are studied. It is demonstrated that the applied theoretical model, which does not impose any constraints on the spectral width, is capable of describing accurately, both qualitatively and quantitatively, the slow spatial variation of the group envelopes. The theoretical model also describes accurately the variation along the tank of the spectral shapes, including free wave components and the bound waves.

Long-period oscillations in a harbour induced by incident short waves

Progressive short waves with a narrow frequency band are known to be accompanied by long set-down waves travelling with the groups. In finite depth, scattering of short waves by a large structure or a varying coastline can radiate free long waves which propagate faster than the incident set-down. In a partially enclosed harbour attacked by short waves through the entrance, such free long waves can further resonate the natural modes of the harbour basin. In this paper an asymptotic theory is presented for a harbour whose horizontal dimensions are much greater than the entrance width, which is in turn much wider than the short wavelength.

Nonlinear evolution of a unidirectional shoaling wave field

Abstract Nonlinear energy transfer in the wave spectrum is very important in the shoaling region. Existing theories are limited to weakly dispersive situations (i.e. shallow water or narrow spectrum). A nonlinear evolution equation for shoaling gravity waves is derived, describing the process all the way from deep to shallow water. The slope of the bottom is taken to be smaller, or of the order of the wave steepness (ϵ). The waves are assumed unidirectional for simplicity. The shoaling domain extends up to, and excluding, the first line of breaking of the waves. Reflection by the shore is neglected. Dispersion is fully accounted for. The model equation includes terms due to quadratic interactions, which are effective over characteristic time and spatial scales of order ( T /ϵ) and (λ/ϵ), respectively, where λ and T are wavelength and period at the spectral peak. In the limit of shallow water, the quadratic interaction model tends to the Boussinesq model. By discretizing the wave spectrum, mixed initial and boundary value problems may be computed. The assumption of the existence of a steady state, transforms the problem into a boundary value one. For this case, solutions for a single triad of waves describing the subharmonic generation and for a full discretized spectrum were computed. The results are compared and found to be in good agreement with laboratory and field measurements. The model can be extended to directionally spread spectra and two dimensional bathymetry.

Accuracy and convergence of velocity formulations for water waves in the framework of Boussinesq theory

The objective of this paper is to discuss and analyse the accuracy of various velocity formulations for water waves in the framework of Boussinesq theory. To simplify the discussion, we consider the linearized wave problem confined between the still-water datum and a horizontal sea bottom. First, the problem is further simplified by ignoring boundary conditions at the surface. This reduces the problem to finding truncated series solutions to the Laplace equation with a kinematic condition at the sea bed. The convergence and accuracy of the resulting expressions is analysed in comparison with the target cosh- and sinh-functions from linear wave theory. First, we consider series expansions in terms of the horizontal velocity variable at an arbitrary

Nonparametric, nonlinear, short-term forecasting: theory and evidence for nonlinearities in the commodity markets

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Remote sensing of the roughness of a fractal sea surface

-level, which can be varied from the sea bottom to the still-water datum. Second, we consider the classical possibility of expanding in terms of the depth-averaged velocity. Third, we analyse the use of a horizontal pseudo-velocity determined by interpolation between velocities at two arbitrary

A non-periodic spectral method with application to nonlinear water waves

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