Surface Waves In Shallow Water Research Articles

Obliquely incident surface waves in shallow water of slowly varying depth

The propagation properties of obliquely incident, weakly nonlinear surface waves in shallow water of slowly varying depth are studied analytically. The depth changes slowly in a direction that makes a constant angle with the propagation direction of the incident wave, initially travelling in a region of uniform depth. In the adjacent inhomogeneous region, the depth variations occur on a scale shorter than that on which the wave evolves. It is shown that the problem then reduces to an evolution equation with constant coefficients. Since weak three-dimensional effects are also taken into account, this equation is related to the KP equation. The effect of oblique incidence on mass transfer is studied in detail. In addition, it is shown that the conditions for fission of an incident solitary wave differ from the case of normal incidence.

One-Dimensional Model of Shallow Water Surface Waves Generated by Landslides

A numerical model is presented as the basis for the study of surface waves generated by bank and bottom landslides in rivers. The flow is assumed to be properly modeled by the shallow water equations. The solid movement is introduced in the model as an external action, and assumed rigid and impervious. Two situations are identified in the flow subsequent to a solid movement: longitudinal and transversal sliding. A discussion on the modeling difficulties associated with the latter is included. The flow equations are solved by means of an upwind scheme specially adapted to advancing fronts over dry irregular beds.

Stability of solitary waves and weak rotation limit for the Ostrovsky equation

Ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation. In this model dispersion is taken into account while dissipation is neglected. Existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter (which can be either positive or negative). It is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations. The issue of passing to the limit as the rotation parameter tends to zero for solutions of the Cauchy problem is investigated on a bounded time interval.

Lie Group of Transformations for a KdV–Boussinesq Equation

We consider a combined Korteweg–deVries and Boussinesq equation governing long surface waves in shallow water. Considering traveling wave solutions, the basic equations will be reduced to a second order ordinary differential equation. Using the Lie group of transformations we reduce it to a first order ordinary differential equation and employ a direct method to derive its periodic solutions in terms of Jacobian elliptic functions and their corresponding solitary wave and explode decay mode solutions.

Modeling spectra of breaking surface waves in shallow water

Predictions from Boussinesq‐equation‐based models for the evolution of breaking surface gravity waves in shallow water are compared with field and laboratory observations. In the majority of the 10 cases investigated, the observed spectral evolution across the surf zone is modeled more accurately by a dissipation that increases at high frequency than by a frequency‐independent dissipation. However, in each case the predicted spectra are qualitatively accurate for a wide range of frequency‐dependent dissipations, apparently because preferential reduction of high‐frequency energy (by dissipation that increases with increasing frequency) is largely compensated by increased nonlinear energy transfers to high frequencies. In contrast to the insensitivity of predicted spectral levels, model predictions of skewness and asymmetry (statistical measures of the wave shapes) are sensitive to the frequency dependence of the dissipation. The observed spatial evolution of skewness and asymmetry is predicted qualitatively well by the model with frequency‐dependent dissipation, but is predicted poorly with frequency‐independent dissipation. Although the extension of the Boussinesq equations to breaking waves is ad hoc, a dissipation depending on the frequency squared (as previously suggested) reproduces well the observed evolution of wave frequency spectra, skewness, and asymmetry.

Alternative Form of Boussinesq Equations for Nearshore Wave Propagation

Boussinesq‐type equations can be used to model the nonlinear transformation of surface waves in shallow water due to the effects of shoaling, refraction, diffraction, and reflection. Different linear dispersion relations can be obtained by expressing the equations in different velocity variables. In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth‐averaged velocity. This significantly improves the linear dispersion properties of the Boussinesq equations, making them applicable to a wider range of water depths. A finite difference method is used to solve the equations. Numerical and experimental results are compared for the propagation of regular and irregular waves on a constant slope beach. The results demonstrate that the new form of the equations can reasonably simulate several nonlinear effects that occur in the shoaling of surface waves from deep to shallow w...

Bragg scattering of sound by surface waves in shallow water

We present a theory of Bragg scattering of sound by surface gravity waves which are nearly periodic with a slowly varying envelope. Through several examples we illustrate how the resonant scattering of sound from a CW source in shallow water can reveal the characteristics of weak surface waves. Only plane sound waves of a guided mode are studied in this paper. Specific examples include the scattering by surface waves which are modulated in amplitude and/or phase. The incident sound waves are sinusoidal but detuned. Applications to short surface waves riding on long surface waves are examined.

The second harmonic resonance for the shallow water surface-wave in a rectangular trough

By using the perturbation method of multiple scales, this paper deals with the phenomenon of the second harmonic resonance for shallow water surface-wave in a rectangular trough. The results show that the envelope of the wave only depends on slow-variables of time. Eqs. of wave envelope are strictly solved and the results are discussed.

Interior and boundary difference equations for hyperbolic differential equations

AbstractInterior and boundary difference equations are derived for several hyperbolic partial differential equations by means of an integral method. The method is applied to a simple transport equation, to waves in a compressible, isentropic fluid, and to surface waves in shallow water. Boundary conditions treated are (a) a perfectly reflecting boundary, (b) an open boundary with outgoing waves and a specified incoming wave, and (c) a partially reflecting boundary. For open boundaries, the major assumption for the algorithms to be valid is that outgoing waves can be defined, an assumption equivalent to the most general statement of Sommerfeld's radiation condition. The difference equations obtained are conservative, second‐order accurate, two time‐level, explicit, and stable (for one‐dimensional, time‐dependent problems) for cΔt/Δx ⩽ 1 where c is the wave speed, Δt is the temporal grid size, and Δx is the spatial grid size. Numerical calculations demonstrate the excellent accuracy of the procedure.

Canonical formulation of shallow water waves

The author introduces a new potential and use it with the velocity potential to construct a new variational principle for long surface waves in shallow water. Applying Dirac's theory of constraints, the author casts it into canonical form and obtain an explicit expression for the exact Hamiltonian.

Erratum for “Shallow Water Surface Wave Elevation Distributions”

Erratum for “Shallow Water Surface Wave Elevation Distributions”

Shallow Water Surface Wave Elevation Distributions

Many widely used engineering formulas dealing with wind-generated waves have been derived with the assumption that the distribution of instantaneous sea surface elevations is described by the Gaussian distribution law. When real wave conditions are not well described by the Gaussian law, the propriety of theses formulas and designs based upon these formulas is questionable. The validity of the Gaussian assumption for shallow water surface wave elevation distributions is examined in this not.