Surface Waves In Shallow Water Research Articles

Long time well-posedness of Whitham–Boussinesq systems

Consideration is given to three different full dispersion Boussinesq systems arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water. We prove that, under a non-cavitation condition on the initial data, these three systems are well-posed on a time scale of order , where ɛ is a small parameter measuring the weak nonlinearity of the waves. For one of the systems, this result is new even for short time. The two other systems involve surface tension, and for one of them, the non-cavitation condition has to be sharpened when the surface tension is small. The proof relies on suitable symmetrizers and the classical theory of hyperbolic systems. However, we have to track the small parameters carefully in the commutator estimates to get the long time well-posedness. Finally, combining our results with the recent ones of Emerald provide a full justification of these systems as water wave models in a larger range of regimes than the classical (a, b, c, d)-Boussinesq systems.

Analysis of the Magnetic Signature of Surface Waves Measured in a Laboratory Experiment

Abstract A magnetic signature is created by secondary magnetic field fluctuations caused by the phenomenon of seawater moving in Earth’s magnetic field. A laboratory experiment was conducted at the Surge Structure Atmosphere Interaction (SUSTAIN) facility to measure the magnetic signature of surface waves using a differential method: a pair of magnetometers, separated horizontally by one-half wavelength, were placed at several locations on the outer tank walls. This technique significantly reduced the extraneous magnetic distortions that were detected simultaneously by both sensors and additionally doubled the magnetic signal of surface waves. Accelerometer measurements and local gradients were used to identify magnetic noise produced from tank vibrations. Wave parameters of 4-m-long waves with a 0.56-Hz frequency and a 0.1-m amplitude were used in this experiment. Freshwater and saltwater experiments were completed to determine the magnetic difference generated by the difference in conductivity. Tests with an empty tank were conducted to identify the noise of the facility. When the magnetic signal was put through spectral analysis, it showed the primary peak at the wave frequency (0.56 Hz) and less pronounced higher-frequency harmonics, which are caused by the nonlinearity of shallow water surface waves. The magnetic noise induced by the wavemaker and related vibrations peaked around 0.3 Hz, which was removed using filtering techniques. These results indicate that the magnetic signature produced by surface waves was an order of magnitude larger than in traditional model predictions. The discrepancy may be due to the magnetic permeability difference between water and air that is not considered in the traditional model.

Higher-dimensional extended shallow water equations and resonant soliton radiation

Higher order equations governing long surface waves in shallow water beyond the standard Korteweg-de Vries and Kadomtsev-Petviashvili equations are derived from the full water wave equations. The higher order dispersive and nonlinear terms in these equations lead to resonance between nonlinear wave structures and dispersive radiation. This resonance between solitary waves and dispersive shock waves (undular bores) is studied numerically and analytically using exponential asymptotic theory. A key result is that there is a near node in the resonant wave amplitude for the higher order equations with the water wave coefficients.

Visualisation of inviscid transonic flows using a water table

The analogue of compressible inviscid flows and shallow water surface waves are studied via water table experiments. In particular, the non-uniqueness problems of transonic flows in nozzles and around airfoils are examined numerically and experimentally. Qualitative comparisons of the two results are shown for several cases.

Stable soliton solutions to the shallow water waves and ion-acoustic waves in a plasma

The Kawahara equation (KE) and the modified Kawahara equation (MKE) are important modeling equations to interpret shallow water waves with surface tension, magneto-acoustic wave in plasma and gravity waves. In this study, the stable broad-ranging soliton solutions including the well-known bell-shape soliton, anti-bell shape soliton, periodic soliton, compacton, kink, etc. are established through the sine-Gordon expansion (SGE) method. The effect of the physical parameters, namely the dispersion and perturbed dispersive coefficients in the surface elevation are shown through the figure. The surface elevations remain ascertainable for diverse values of the physical and the associated free parameters. The 3D and contour plot of the ascertained solutions clarifies the surface wave properties. The achieved results for each modeling equation provide a significant contribution in analyzing the ion-acoustic waves in plasma, gravity waves, the surface waves in shallow water. This study asserts that the SGE method is reliable, competent, easy and powerful for extracting closed form soliton solutions.

Simulations of shallow water surface waves and comparison with water table experiments

The hyperbolic systems of equations describing the shallow water surface waves are solved numerically using finite difference schemes and an explicit time integration procedure. The equations are similar to isentropic Euler equations (with γ = 2) and can be simplified using a potential flow model. The results of isentropic Euler, potential model, transonic small disturbance, and full Euler equations are compared for typical flows around pointed and blunt bodies for several Mach numbers. On the other hand, water table experiments are described and the flow over obstacles is studied. Using the theory of hydraulic analogy, the relations between compressible flows and shallow water surface waves are discussed. Flow patterns, including formation of shock waves and expansion fans are presented. It is demonstrated that the water table can be an inexpensive educational tool for demonstration of transonic and supersonic flow phenomena.

Simulations of shallow water surface waves and comparison with water table experiments

Simulations of shallow water surface waves and comparison with water table experiments

On the weak solutions and persistence properties for the variable depth KDV general equations

Considered in this paper is a nonlinear evolution equation for surface waves in shallow water over uneven bottom. Due to the dependence on the bottom function of the coefficients and its involved complicated structure of the equation, the generalized Gronwall’s inequalities are proposed to be used to derive the required priori estimates, with which, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs(R) with 1<s≤3∕2 is obtained. Moreover, the persistence properties in weighted Lϕp≔Lp(R,ϕpdx) spaces on strong solutions are also investigated.

An improved Serre model: Efficient simulation and comparative evaluation

• A new Serre model with improved dispersion characteristics is considered. • A careful development of the new model allows an efficient numerical implementation. • A splitting scheme based on high order finite volume and finite difference is used. • Numerical experiments illustrate the superiority of the proposed model. The so-called Serre or Green and Naghdi equations are a well-known set of fully nonlinear and weakly dispersive equations that describe the propagation of long surface waves in shallow water. In order to extend its range of application to intermediate water depths, some modifications have been proposed in the literature. In this work, we analyze a new Serre model with improved linear dispersion characteristics. This new Serre system, herein denoted by Serre α, β , presents additional terms of dispersive origin, thus extending its applicability to more general depth to wavelength ratios. A careful development of the Serre α, β model allows a straightforward and efficient numerical implementation. This model is suitable for numerical integration by a splitting strategy which requires the solution of a hyperbolic problem and a dispersive problem. The hyperbolic part is discretized using a high-order finite volume method. For the dispersive part standard finite differences are used. A set of numerical experiments are conducted to validate the Serre α, β model and to test the robustness of our numerical scheme. Theoretical solutions and benchmark experimental data are used. Moreover, comparisons against the classical Serre equations and against another well established Serre model with improved dispersion characteristics are also made.

Local well-posedness and persistence properties for the variable depth KDV general equations in Besov space $B^{ \frac 32 }_{2,1}$

In this paper, we consider a nonlinear evolution equation for surface waves in shallow water over uneven bottom. First, the local well-posedness is obtained in Besov space $B^{ \frac 32 }_{2,1}$. Then, persistence properties on strong solutions are also investigated.

On the multi-symplectic structure of the Serre–Green–Naghdi equations

In this short note, we present a multi-symplectic structure of the Serre–Green–Naghdi (SGN) equations modelling nonlinear long surface waves in shallow water. This multi-symplectic structure allow the use of efficient finite difference or pseudo-spectral numerical schemes preserving exactly the multi-symplectic form at the discrete level.

Mechanical Balance Laws for Boussinesq Models of Surface Water Waves

Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion η and the horizontal velocity w at a given level in the fluid.

Radiation stress and mean drift in continental shelf waves

The time- and depth-averaged mean drift induced by barotropic continental shelf waves (CSW’s) is studied theoretically for idealized shelf topography by calculating the mean volume fluxes to second order in wave amplitude. The waves suffer weak spatial damping due to bottom friction, which leads to radiation stress forcing of the mean fluxes. In terms of the total wave energy density E̅̅ over the shelf region, the radiation stress tensor component S̅11 for CSW’s is found to be different from that of shallow water surface waves in a non-rotating ocean. For CSW’s, the ratio S̅11/E̅ depends strongly on the wave number. The mean Lagrangian flow forced by the radiation stress can be subdivided into a Stokes drift and a mean Eulerian drift current. The magnitude of latter depends on ratio between the radiation stress and the bottom stress acting on the mean flow. When the effect of bottom friction acts equally strong on the waves and the mean current, calculations for short CSW’s show that the Stokes drift and the friction-dependent wave-induced mean Eulerian current varies approximately in anti-phase over the shelf, and that the latter is numerically the largest. For long CSW’s they are approximately in phase. In both cases the mean Lagrangian current, which is responsible for the net particle drift, has its largest numerical value at the coast on the shallow part of the shelf. Enhancing the effect of bottom friction on the Eulerian mean flow, results in a general current speed reduction, as well as a change in spatial structure for long waves. Applying realistic physical parameters for the continental shelf west of Norway, calculations yield along-shelf mean drift velocities for short CSW’s that may be important for the transport of biological material, neutral tracers, and underwater plumes of dissolved oil from deepwater drilling accidents.

N-fold Darboux transformation and solitonic interactions of a variable-coefficient generalized Boussinesq system in shallow water

Under consideration in this paper is a variable-coefficient generalized Boussinesq system for the long weakly-nonlinear and weakly-dispersive surface waves in shallow water. With the aid of symbolic computation, N-fold Darboux transformation ( N-DT) is constructed for that system. Analytic solutions of the system are obtained via the N-DT. Elastic interactions of three bell-shaped and periodic bell-shaped solitons are obtained. Fusion interactions and periodic fusion–fission interactions of the solitary waves are graphically analyzed, which are inelastic.

Interactions of solitons in a variable-coefficient generalized Boussinesq system in shallow water

Under consideration in this paper is a variable-coefficient generalized Boussinesq (gBq) system, which can model the propagation of long weakly nonlinear and weakly dispersive surface waves in shallow water. With the aid of the Darboux transformation and symbolic computation, soliton solutions of the gBq system are obtained that do not have singularities under a selection of the spectral parameters. Interactions of the solitons with the elastic properties are discussed. The outcome of this paper might be of some help for the investigation of nonlinear and dispersive problems in fluid dynamics.

Propagation of water wave over a periodically perforated bottom and the band structure

The band structures of shallow water surface wave propagating over a periodically perforated bottom are calculated by the plane wave expansion method，and complete band gaps appear. Through considering three drilling patterns （square holes arrayed in square lattice，circular holes arrayed in square and hexagonal lattices），we find that the number，width，frequency and location of band gaps are related to the area，shape，orientation of drilled holes and the lattice. These rules are similar to those in photonic/ phononic crystals.

A simple LBE wave runup model

The suitability of a Lattice Bhatnagar-Gross-Krook (LBGK) modelling approach is tested to examine the behaviour of free surface waves in shallow water. The free surface model is fully non-linear. Mathematically, this is taken into account through the collision integral. The present LBGK model discretises the non-linear shallow water equation in rotational flow and approximates the collision between particles using a single time relaxation. It is further assumed that the waves do not overturn. The test case includes a long-wave run-up study on a sloping beach. Two methods have been evaluated to treat the wet-dry interface: • a thin film in dry areas of the beach slope • a linear extrapolation scheme. The standard LBGK solutions are second-order accurate in space and first-order accurate in time. The uniform grid solutions compared well with other findings reported in the literature when using a shoreline algorithm.

Wavefront modelling of pulse propagation

Wavefront modelling has been developed for the rapid calculation of receiver waveforms in underwater sound pulse propagation. The method uses a solution of the wave equation which expresses the field as a sum over phase integrals. Each phase integral corresponds to a particular sequence of reflections or turning points and is evaluated analytically by using stationary phase approximations. Each point of stationary phase corresponds to a ray path. Pairs or triplets of stationary phase points correspond to caustics or foci and the phase and amplitude of the field is found directly. The method can handle rapid range dependence such as occurs with surface waves in shallow water and double scattering from the same surface. The method is fast and accurate and examples will be discussed. [Work supported by ONR.]

Obliquely incident surface waves in shallow water of slowly varying depth

Obliquely incident surface waves in shallow water of slowly varying depth

Modelling of wave damping at Guyana mud coast

The Guyana coastal system is characterized by very thick deposits of Amazon mud and high mud concentrations in its coastal waters. The mud deposits can be quite soft and may liquefy under incoming waves. Subsequently, the liquefied mud damps the incoming waves effectively. This paper presents a simple model to predict wave attenuation over soft (fluid) mud beds. This model is based on the two-layer approach by Gade [Gade, H.G., 1958, Effects of a non-rigid, impermeable bottom on plane surface waves in shallow water, Journal of Marine Research, 16 (2) 61–82.] which is implemented in the standard version of the state-of-the-art wave-prediction model SWAN. Input to the mud wave damping module consists of the extension, thickness, density and viscosity of the liquefied (fluid) mud layer. The model is validated against small-scale wave attenuation measurements carried out in a laboratory wave flume. The model predictions agree favourably with the experimental data. Next, the model is applied to predict wave height and wave attenuation in the Guyana coastal system. Extension and thickness of the liquefiable layer could be assessed from dual-frequency echo soundings. In the absence of field data, the density of the liquefied mud layer is obtained from literature, whereas the value of the mud's viscosity had to be established by trial and error — the selected value, though, is in the range of literature values. The model predicts significant wave attenuation. The computed changes in wave energy spectrum agree qualitatively with measurements in Surinam, whereas the decrease in significant wave height agrees more or less with historic observations along the Guyana coast.