Narrow Results

Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

A nonlocal spectral formulation of classic water waves is derived, and its connection to the classic water wave equations and the Dirichlet–Neumann operator is explored. The nonlocal spectral formulation is also extended to a two-fluid system with a free interface, from which long wave asymptotic reductions are obtained. Of particular interest is an asymptotically distinguished (2 + 1)-dimensional generalization of the intermediate long wave equation, which includes the Kadomtsev–Petviashvili equation and the Benjamin–Ono equation as limiting cases. Lump-type solutions to this (2 + 1)-dimensional ILW equation are obtained, and the speed versus amplitude relationship is shown to be linear in the shallow, intermediate, and deep water regimes.

A discussion and an improvement of the Neumann–Kelvin's model are suggested in this paper. This model is used in the simulation of progressive wave phenomenon. As mentioned by several authors ([7, 19, 23, 29]), this model is ill posed unless a capillary energy is introduced. The mathematical explanation is that a compactness inversions' property occurs if the capillary forces are omitted. Theoretical arguments and numerical simulations are used in the following which aims at giving an explanation of what happens from a mechanical point of view.

We consider a toy model for a damped water waves system in a domain ${\mathrm{\Omega }}_t\subset 𝕋\times ℝ$. The toy model is based on the paradifferential formulation of the water waves system derived in the work of Alazard–Burq–Zuily [On the water-waves equations with surface tension, Duke Math. J.158 (2011) 413–499]. The form of damping we consider is a modified sponge layer proposed for the $3d$ water waves system by Clamond et al. in [An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption, J. Comput. Phys.205 (2005) 686–705]. We show that, in the case of small Cauchy data, solutions to the toy model exhibit a quadratic lifespan. This is done via proving energy estimates with Klainerman vector fields.

A new approach to extend the applicability of different Boussinesq models [Nwogu, 1993; Schäffer and Madsen, 1995; Gobbi and Kirby, 1996] in deep water is introduced. This approach is based on optimizing the new defined performance index for linear wave dispersion property. The index measures the capability of a water wave model to simulate transformations of irregular wave over general topography. By optimizing this index at different relative water depths, the validity range for these models is extended. Optimizations process is based on calculating some selected free parameters in every model such that performance index is minimized. All optimized parameters are given in plotted figures verses the dimensionless peak frequency of the wave train. Comparisons among the optimized models are given. The approach applicability is investigated compared with Nwogu's model. Finite difference scheme over staggered grid is used to implement a high order numerical algorithm for the model.

This note marks the debut of a Mathcad worksheet, which has been developed to aid engineers in the calculation of properties of a surface wave propagating in water over a two-layer viscoelastic muddy bed. We first describe the problem formulation and then features of the worksheet. It is a very easy-to-use calculation tool. With the input of some basic parameters, such as the wave period and the fluid properties, one may get almost instantly the key results (wavenumber, wave damping rate, velocity and pressure fields, and so on) upon pressing a key. The worksheet has been extensively tested to ensure that it can produce reliable and accurate results.

A performance index is defined for linear wave dispersion property. The index measures the capability of a water wave model to simulate irregular wave transformation over general topography.

Optimization of this index leads to:

(1) the optimum vertical distribution functions for the nonlinear mild slope equations given by Isobe (1994) for different number of vertical distribution functions,

(2) the optimum values of wave numbers to be used for the fully dispersive nonlinear wave model given by Nadaoka et al. (1994) for different number of vertical distribution functions,

(3) the optimum values of the wave number to be used for the time dependent mild slope equation given by Kirby (1984).

All optimized parameters are given as functions of the dimensionless peak frequency of the energy spectrum. Comparisons among the optimized models are given.

The dynamics of solitary waves is studied in intricate domains such as open channels with sharp-bends and branching points. Of particular interest, the wave characteristics at sharp-bends is rationalized by using the Jacobian of the Schwarz–Christoffel transformation. It is observed that it acts in a similar fashion as a topography in other wave models previously studied. Previous numerical studies are revisited. A new very efficient algorithm is described, which computationally solves the problem in much more general channel configurations than presented in the literature. Also the conformal mapping naturally leads to a new strategy regarding the geometrical singularity at the sharp-bends. Finally preliminary results illustrate the use of the mapping's Jacobian at a channel's branching point. A future goal of this study regards deducing accurate reduced (one-dimensional) models for the reflection and transmission of solitary waves on graphs/networks.

The classical water-wave problem is described, and two parameters (ε-amplitude; δ-long wave or shallow water) are introduced. We describe various nonlinear problems involving weak nonlinearity (ε → 0) associated with equations of integrable type ("soliton" equations), but with vorticity. The familiar problem of propagation governed by the Korteweg–de Vries (KdV) equation is introduced, but allowing for an arbitrary distribution of vorticity. The effect of the constant vorticity on the solitary wave is described. The corresponding problem for the Nonlinear Schrödinger (NLS) equation is briefly mentioned but not explored here. The problem of two-way propagation (admitting head-on collisions), as described by the Boussinesq equation, is examined next. This leads to a new equation: the Boussinesq-type equation valid for constant vorticity. However, this cannot be transformed into an integrable Boussinesq equation (as is possible for the corresponding KdV and NLS equations). The solitary-wave solution for this new equation is presented. A description of the Camassa–Holm equation for water waves, with constant vorticity, with its solitary-wave solution, is described. Finally, we outline the problem of propagation of small-amplitude, large-radius ring waves over a flow with vorticity (representing a background flow in one direction). Some properties of this flow, for constant vorticity, are described.

An overview is presented of recent progress on the relation between the pressure at the bottom of the flat water bed and the elevation of the free water boundary within the context of the one-dimensional, irrotational water wave problem. We present five different approaches to this problem. All are compared to (1) numerical data for Stokes waves, one-dimensional traveling wave solutions of the full irrotational water wave problem, and (2) experimental data for high-amplitude waves in a long, narrow (i.e. one-dimensional) wave tank.

Large surface water waves can be triggered in moraine-dammed lakes during earthquakes and may lead to the overtopping failure of moraine dams. In the earthquake-prone Himalayas, there are thousands of moraine-dammed lakes; their outburst may lead to catastrophic disasters (e.g. floods and debris flow), posing severe threats to humans and infrastructures downstream. This paper experimentally studied earthquake-induced water waves (EWWs) in moraine-dammed lakes and examined the effects of several factors (e.g. water depth, earthquake parameters, and uneven lake basin). The experimental results suggest that the EWWs positively correlate to the earthquake wave, and the maximum height of the EWWs increases by 10%–15% when the effect of the uneven lake basin is considered. Based on the experiment data, we derived a calculation equation to estimate the maximum amplitude of EWWs considering the basin effect, and proposed a fast risk assessment method for moraine lakes due to overtopping EWWs. Finally, based on the method, we assessed the failure risk of the moraine lakes located in the Gyirong river basin where the China–Nepal corridor crosses. The study broadens understandings of the risk source of moraine-dammed lakes.

When a charge explodes underwater near a free surface, a bubble would be generated and the surface pushed up very high. Experiments have shown that the motion of the spike lags a lot behind the bubble motion. Many studies only focus on the nonlinear interaction between the bubble and free surface while the water waves afterward is mainly studied based on the linear theory. The nonlinear motion of the water wave after the bubble pulsation is seldom studied. In this study, we concerns the interaction between underwater explosion generated bubble and a free surface and its bursting at a free surface in a closed domain. Suppose that the fluid outside the bubble is incompressible, non-viscous and irrotational and the velocity potential satisfies the Laplace equation. Boundary integral method is used to solve the Laplace equation for the velocity potential. The bubble content is described by an adiabatic law. The whole process of the bubble motion and subsequently the water wave propagation will be simulated in this paper. Particular attention will be focused on the phenomenon of water wave propagation in a closed domain.

New types of two and three-dimensional free surface flows are presented and discussed. The solutions are obtained by boundary integral equation methods. The discretized equations are solved by Newton's iterations coupled with continuation methods. Appropriate initial guesses for the iterations are found by first solving a 'modified problem' and then taking an appropriate limit.

Due to refraction, the oceanic ridge acts as a waveguide forcing long-period waves to propagate along the topography. Based on the linear shallow water equations, analytical solutions of trapped waves over a hyperbolic-cosine squared ocean ridge are obtained, which are described by combining the associated Legendre functions of the first and second kinds. The spatial distribution pattern for each mode is discussed, and the wave amplitude gets the maximum at the ridge top and decays gradually towards both sides. The higher the mode number, the slower the rate of amplitude decreases, so that more energy is distributed over more of the ridge. The trapped wave number is not only related to the frequency, but also to the varying water depth parameters.

Using techniques from the theory of oscillatory integrals, we prove rigorous estimates which show that the linearization of the vortex sheet equations of motion about a quiescent state disperse under certain circumstances. Such dispersion is only possible only through the joint effects of surface tension (which damps high frequency modes) and gravitation (which damps low frequency modes).