Fractional view analytical analysis of generalized regularized long wave equation

An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma

Very little quantitative information exists on growth of wind waves in waters where relatively shallow depth could influence wave growth; results of laboratory investigation of growth of waves in shallow water is presented, showing experimental relationships between wind velocity, fetch, and wave parameters for two different water depths.

In this study, the potential Kadomtsev-Petviashvili (pKP) equation, which describes the oblique interaction of surface waves in shallow waters, is solved by the new extended direct algebraic method. The results of the study show that by applying the new direct algebraic method to the pKP equation, the behavior of the obliquely interacting surface waves in two dimensions can be analyzed. This article fairly clarifies the behaviors of surface waves in shallow waters. In the literature, several mathematical models have been developed in attempt to study these behaviors, with nonlinear mathematics being one of the most important steps; however, the investigations are still at a level that can be called ‘baby steps’. Therefore, every study to be carried out in this context is of great importance. Thus, this study will serve as a reference to guide scientists working in this field.

Numerous practical and mathematical techniques have been piloted to study ships’ behavior in deep water conditions with and without waves, and shallow water conditions without waves, while only limited investigations have been carried out to assess ships’ behavior in shallow waters with wave conditions as the flow around the stern region and appendages and the interaction effects are intricate. Therefore, this study attempts to understand the infrequently explored subset of a vessel’s behavior in regular waves in shallow water conditions (channel depth to ship draft ratio taken as 1.5). A container ship (S175) model scaled at 1:36 was the subject of a numerical study in which it was subjected to static and dynamic maneuver simulations in head sea conditions. The waves were induced using the dispersion relationship of waves in a given depth. The trends of forces and moments acting on the hull while undergoing maneuvering motions were obtained using a smooth particle hydrodynamics-based computational fluid dynamics solver. The resulting periodic trends of forces and moments were analyzed using the Fourier series method to extract the Fourier coefficients and, in turn, calculate the hydrodynamic derivatives. The trajectories in turning circle and zigzag maneuvers were also simulated using a MATLAB code. The results demonstrate an increase in trajectory parameters and improvement in counter maneuverability owing to the complex flow physics around the hull when encountering regular waves in shallow water conditions compared to waves in deep waters and a lack of waves in shallow waters.

<strong class="journal-contentHeaderColor">Abstract.</strong> Accurate description of the wind energy input into ocean waves is crucial to ocean wave modeling and a physics-based consideration on the effect of wave breaking is absolutely necessary to obtain such an accurate description. This study evaluates the performance of an improved formula recently proposed by Xu and Yu (2020), who took into account not only the effect of breaking but also the effect of air-flow separation on the leeside of steep wave crests in a reasonably consistent way. Numerical results are obtained through coupling an enhanced atmospheric wave boundary layer model with the ocean wave model WaveWatch III (v5.16). The coupled model has been extended to be valid in both deep and shallow waters. Duration-limited waves under controlled normal conditions and storm waves under practical hurricane conditions are studied in details to verify the improved model. Both the representative wave parameters and the parameters characterizing the wave spectrum are discussed. It is shown that the improved source-term package for the wind energy input and the wave energy dissipation leads to more accurate results under all conditions. It performs evidently better than other standard source-term options of ST2, ST4 and ST6 embedded in WaveWatch III. It is also demonstrated that the improvement is particularly important for waves at their early development stage and waves in shallow waters.

Assessing the performance of wave breaking parameterizations in shallow waters in spectral wave models

The purpose of this book is to present a self contained introduction to the mathematical and physical aspects of the theory of water waves. The book is aimed at undergraduate and graduate levels for engineers, physical scientists and mathematicians. Each chapter is concluded with practical problems expressed as exercises and accompanied by ample references for further studies. The book consists of ten chapters arranged into three parts: Part I: Basic Fluid Mechanics and Solutions Techniques which cover chapters 1-3. Part II: Water Waves covering chapters 4-7. Part III Advanced Water Waves which covers chapters 8-10. Parts I and II are elementary in nature; whereas Part III is more advanced. The first three chapters give the derivations of the fundamental mathematical equations. Chapter 2 outlines appropriate differential equations to describe the physical phenomena, and Chapter 3 reviews solution techniques of some simplified partial differential equations. Chapter 4 gives the developmental of wave equations, including the essential boundary conditons and describes small amplitude wave theory. Chapter 5 deals with finite amplitude wave theory and Chapter 6 outlines the study of tidal dynamics in shallow water. For random wave case, the deterministic methods described in previous chapters do not hold good. Therefore, chapter 7 is clearly devoted to wave statistics and wave energy spectrum. The application of wave theory is demonstrated in Chapter 8. Chapter 9 examines the nonlinear long waves in shallow water from a mathematical view point. The book concludes with Chapter 10 which illustrates the inverse scattering technique to solve solitary wave problem.

A variational formulation of water waves is developed, based on the Hamiltonian theory of surface waves. An exact and unified description of the two-dimensional problem in the vertical plane is obtained in the form of a Hamiltonian functional, expressed in terms of surface quantities as canonical variables. The stability of the corresponding canonical equations can be ensured by using positive definite approximate energy functionals. While preserving full linear dispersion, the method distinguishes between short-wave nonlinearity, allowing the description of Stokes waves in deep water, and long-wave nonlinearity, applying to long waves in shallow water. Both types of nonlinearity are found necessary to describe accurately large-amplitude solitary waves.

The scientific developments carried out in the last years with spaceborne synthetic aperture radar (SAR), as well as the comparisons with in-situ sensors, have demonstrated that SAR is a reliable remote sensing tool to study wave fields on the open ocean. In their so-called SAR image mode, SAR systems on board satellites are able to scan ocean areas about 100 × 100 km2 with a spatial resolution about 20 × 20 m2. These SAR images are able to provide information about the spatial variability of wave fields, as well as other phenomena that occur on the ocean surface, such as the local wind field. This work investigates the capabilities of spaceborne SAR to extract sea state information for those areas close to coastal locations, where the incoming wave fields present high spatial inhomogeneities. For this purpose, ERS-2 SAR and ENVISAT Advanced SAR (ASAR) images acquired over the North Sea and the Bay of Biscay are used to study ocean waves in shallow waters under different climate and oceanographic conditions. In addition, the potential of ENVISAT ASAR capability to scan the ocean surface with dual polarizations (horizontal, HH, and vertical, VV) is analyzed in this work. The relevance of these investigations for different practical applications concerning the analysis of ocean waves is discussed.

It is investigated that there exists the possibility of the false target signals induced by reverberation in an active sonar system due to the internal waves in shallow water. The rays down-refracted from the internal waves may generate strong bottom-reverberation signals, which can result in false target signals. Sound waves emitted from a source propagate 3-dimensionally. Therefore, the study of internal waves on the reverberation should be studied for azimuthal direction as well as 2-dimensional (r-z) plane. Internal-wave modelling was conducted, based on solitons which were predicted with the various conditions such as, the range of source-soliton, horizontal widths of soliton. Variable depth sonar (VDS) was assumed as a source, of which the depth was located in the minimum sound speed layer in a simulation environment. Finally, the simulation on the reverberation level with time was made based on ray-based reverberation model, and the results implied that several false-target signals could be displayed on the PPI(Plan Position Indicator) scope simultaneously with range from source to soliton, and the horizontal width of soliton.

The canonical Hamiltonian structure of the equations of fluid dynamics obtained in the Boussinesq approximation are considered. New variational formulations of these equations are proposed and it is found that, as in the case of the KdV equation and the equations governing long waves in shallow water, they are degenerate Lagrangian systems. Therefore, in order to cast these equations into canonical form it is again necessary to use Dirac’s theory of constraints. It is found that there are primary and secondary constraints which are second class and it is possible to construct the Hamiltonian in terms of canonical variables. Among the examples of Boussinesq equations that are discussed are the equations of Whitham–Broer–Kaup which Kupershmidt has recently expressed in symmetric form and shown to admit tri-Hamiltonian structure.

Ship waves in shallow water are considerably different from those in deep water.They are greatly influenced by the depth Froude number. In the present paper, the numerical method proposed by Tanimoto et al.(1999) has been confirmed to be applicable to predict ship waves in a shallow water by comparing with experimental data by Kurata et al.(1983).Variations of the maximum wave height by the mid ship section coefficient, the distance from the sailing line and the depth Froude number are discussed together with experimental data.The maximum wave height increases as the depth Froude number increases up to the critical speed condition. In the range of super critical speed, the maximum wave height becomes almost constant or slightly decreasing tendency.

Based on a multidimensional Riemann theta function, the Hirota bilinear method is extended to explicitly construct multiperiodic (quasiperiodic) wave solutions for the (2+1) -dimensional Bogoyavlenskii breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves in shallow water. The two-periodic (biperiodic) waves are a direct generalization of one-periodic waves, their surface pattern is two dimensional, that is, they have two independent spatial periods in two independent horizontal directions. The two-periodic waves may be considered to represent periodic waves in shallow water without the assumption of one dimensionality. A limiting procedure is presented to analyze asymptotic behavior of the one- and two-periodic waves in details. The exact relations between the periodic wave solutions and the well-known soliton solutions are established. It is rigorously shown that the periodic wave solutions tend to the soliton solutions under a "small amplitude" limit.

Analytical behavior of soliton solutions to the couple type fractional-order nonlinear evolution equations utilizing a novel technique

Under investigation in this paper is the Whitham–Broer–Kaup (WBK) system, which describes the dispersive long wave in shallow water. Through a variable transformation, the WBK system is casted into a general Broer–Kaup system whose Lax pair can be derived by the Ablowitz–Kaup–Newell–Segur technology. With symbolic computation, based on the aforementioned Lax pair, the N-fold Darboux transformation is constructed with a gauge transformation and the multi-soliton solutions are obtained. Finally, the elastic interactions of the two-soliton solutions (including the head-on and overtaking collisions) for the WBK system are graphically studied. Those multi-soliton collisions can be used to illustrate the bidirectional propagation of the waves in shallow water.