High-Order Models of Nonlinear and Dispersive Wave in Water of Varying Depth with Arbitrary Sloping Bottom

High-order models with a dissipative term for nonlinear and dispersive wave in water of varying depth with an arbitrary sloping bottom are presented in this article. First, the formal derivations to any high order of mu(= h/lambda, depth to deep-water wave length ratio) and epsilon(= a/h, wave amplitude to depth ratio) for velocity potential, particle velocity vector, pressure and the Boussinesq-type equations for surface elevation eta and horizontal velocity vector (U) over right arrow at any given level in water are given. Then, the exact explicit expressions to the fourth order of mu are derived. Finally, the linear solutions of eta, (U) over right arrow, C (phase-celerity) and C-g (group velocity) for a constant water depth are obtained. Compared with the Airy theory, excellent results can be found even for a water depth as large as the wave legnth. The present high-order models are applicable to nonlinear regular and irregular waves in water of any varying depth (from shallow to deep) and bottom slope (from mild to steep).

ON THE SINGULARITIES OF SOLUTIONS TO 4-D SEMILINEAR DISPERSIVE WAVE EQUATIONS

In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz＇s inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal

ABUNDANT LOCALIZED COHERENT STRUCTURES FOR THE (2+1)-DIMENSIONAL LONG DISPERSIVE WAVE SYSTEM

In the present paper, a simple and direct method was proposed to solve the (2+ 1)-dimensional long dispersive wave equations. A variable-dependent transformation was intorducedto convert the equations into the simpler forms, which are coupled and linear partial differentialequations, then obtain its general solution. Some special types of the localized excitations, suchas oscillating dromion, multi-solitoff, multi-dromion, multi-lump and multi-ring soliton solutionsare derived by selecting the arbitrary functions appropriately.

The inverse problem based on a full dispersive wave equation

The inverse problem for harmonic waves and wave packets was studied based on a full dispersive wave equation. First, a full dispersive wave equation which describes wave propagation in nondissipative microstructured linear solids is established based on the Mindlin theory, and the dispersion characteristics are discussed. Second, based on the full dispersive wave equation, an inverse problem for determining the four unknown coefficients of wave equa- tion is posed in terms of the frequencies and corresponding wave numbers of four different harmonic waves, and the inverse problem is demonstrated with rigorous mathematical theory. Research proves that the coefficients of wave equation related to material properties can be uniquely determined in cases of normal and anomalous dispersions by measuring the frequen- cies and corresponding wave numbers of four different harmonic waves which propagate in a nondissipative microstructured linear solids.

Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model

Before going further with fractional derivative which is constructed by Rabotnov exponential kernel,there exist many questions that are not addressed.In this paper,we try to recapitulate all the fundamental calculus,which we can obtain with this new fractional operator.The problems in this paper are to determine the solutions of the fractional differential equations where the second members are constant functions,polynomial functions,exponential functions,trigonometric functions,or Mittag-Leffler functions.For all the fractional differential equations,the obtained solutions are represented graphically.The Laplace transform of the fractional derivative with Rabotnov exponential kernel is the primary tool in the investigations.Finally,we give the fundamental solution to the nonlinear time-fractional modified Degasperis-Procesi equation by considering the fractional operator with Rabotnov exponential kernel.

An efficient computational technique for time-fractional modified Degasperis-Procesi equation arising in propagation of nonlinear dispersive waves

In this paper,an efficient hybrid numerical scheme which is based on a joint venture of the q-homotopy analysis method and Sumudu transform is applied to investigate the time-fractional modified Degasperis-Procesi(DP)equation.The present study considers the Caputo fractional derivative.The fractional order modified DP model is very important and plays a great role in study of ocean engineering and science.The proposed scheme provides a beautiful opportunity for proper selection of the auxiliary parameter h and the asymptotic parameterρ(≥1)to handle mainly the differential equations of nonlinear nature.The offered scheme produces the solution in the shape of a convergent series in a large admissible domain which is helpful to regulate the region of convergence of a series solution.The proposed work computes the approximate analytical solution of the fractional modified DP equation systematically and also presents graphically the variation of the obtained solution for diverse values of the fractional parameterβ.

Exact travelling wave solutions for （1＋ 1）-dimensional dispersive long wave equation

A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.

Bifurcation analysis and exact traveling wave solutions for (2+1)-dimensional generalized modified dispersive water wave equation

We investigate (2+1)-dimensional generalized modified dispersive water wave (GMDWW) equation by utilizing the bifurcation theory of dynamical systems. We give the phase portraits and bifurcation analysis of the plane system corresponding to the GMDWW equation. By using the special orbits in the phase portraits, we analyze the existence of the traveling wave solutions. When some parameter takes special values, we obtain abundant exact kink wave solutions, singular wave solutions, periodic wave solutions, periodic singular wave solutions, and solitary wave solutions for the GMDWW equation.

Bifurcation of travelling wave solutions for (2+1)-dimension nonlinear dispersive long wave equation

In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.

Non-completely elastic interactions in a(2+1)-dimensional dispersive long wave equation

With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the （24-1）-dimensional dispersive long wave equation. When selecting appropriate multi-valued functions in the variable separation solution, we investigate the interactions among special multi-dromions, dromion-like multi-peakons, and dromion-like multi-semifoldons, which all demonstrate non-completely elastic properties.

Large-scale inverted-V channels of upflowing oxygen ions pumped by Alfvén waves

Large-scale inverted-V channels of upflowing oxygen ions are frequently identified in data collected by Cluster,at all local times,near the open-closed field line boundary over Earth’s high-latitude ionosphere-occur with downward propagating MHD Alfvén waves which have cascaded into kinetic regimes of plasma.The transverse acceleration of the oxygen ions in the center of these structures is interpreted as the integrated energization by these waves along the channels.Also observed within the channels are upward parallel electric fields,a key characteristic of kinetic Alfvén waves,which may contribute not only to lifting the ions but also to precipitating aurora electrons that might initiate ion upflow in the ionosphere below.Statistics on five-year observations of Cluster show that the channels typically form during geomagnetic perturbations,particularly when solar-wind dynamic pressure is high or highly fluctuated.Near the open-closed field line boundary,the stronger the wave power,the higher the upward oxygen flux and the higher the beam energy,indicating that these waves provide a simple but efficient way to drive oxygen upflows.

Stress Wave Dispersion in Large-Diameter SHPB and Its Manifold Manifestations

The wave dispersion due to the lateral inertia in the split Hopkinson pressure bar(SHPB) with large-(diameter) bar is numerically analyzed by means of the LS-DYNA3D code. The results show that, ① the stress distribution across the bar section is non-uniform along the radius direction and such non-uniformity depends on the material Poisson ratio and propagation distance; ② with increasing the bar diameter, the high frequency oscillations are notably enhanced and the rise time of wave front becomes longer, meanwhile the amplitude of the stress wave attenuates; ③ with decreasing the rise time of wave front, the wave dispersion markedly enhanced, particularly in the large diameter bar. All of those effects should not be neglected in order to obtain accurate results by the SHPB test..

Laboratory Observations on Internal Solitary Wave Evolution over A Submarine Ridge

Laboratory experiments were conducted in a wave flume on internal solitary wave （ISW） of depression and elevation types propagating over a submarine ridge in semicircular/triangular shape. Tests were arranged in series for combinations of submarine ridges of different heights and ISW of different amplitudes. The resuhant wave motions were found differing from thee of surface gravity waves. In deeper water, where an ISW of depression-type prevailed, the process of wave breaking displayed downward motion with continuous eddy on the front face of the ridge followed by upward motion towards the apex of the obstacle. Experimental results also suggested that blockage parameter ξ could be applied to classify various degrees of ISW-ridge interaction, i.e., ξ 〈 0.5 for weak interaction, 0.5 〈 ξ 〈 0.7 for moderate interaction, and 0.7 〈 ξ for wave breaking.

FLEXURAL WAVE DISPERSION IN MULTI-WALLED CARBON NANOTUBES CONVEYING FLUIDS

Motivated by the great potential of carbon nanotubes for developing nanofluidic devices, this paper presents a nonlocal elastic, Timoshenko multi-beam model with the second order of strain gradient taken into consideration and derives the corresponding dispersion relation of flexural wave in multi-walled carbon nanotubes conveying fuids. The study shows that the moving flow reduces the phase velocity of flexural wave of the lowest branch in carbon nanotubes. The phase velocity of flexural wave of the lowest branch decreases with an increase of flow velocity. However, the effects of flow velocity on the other branches of the wave dispersion are not obvious. The effect of microstructure characterized by nonlocal elasticity on the dispersion of flexural wave becomes more and more remarkable with an increase in wave number.

ON FREE WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA

The method of reverberation-ray matrix （MRRM） is extended and modified for the analysis of free wave propagation in anisotropic layered elastic media. A general, numerically stable formulation is established within the state space framework. The compatibility of physical variables in local dual coordinates gives the phase relation, from which exponentially growing functions are excluded. The interface and boundary conditions lead to the scattering relation, which avoids matrix inversion operation. Numerical examples are given to show the high accuracy of the present MRRM.

A novel type of transverse surface wave propagating in a layered structure consisting of a piezoelectric layer attached to an elastic half-space

The existence and propagation of transverse surface waves in piezoelectric coupled solids is investigated, in which perfect bonding between a metal/dielectric substrate and a piezoelectric layer of finite-thickness is assumed. Dis- persion equations relating phase velocity to material con- stants for the existence of various modes are obtained in a simple mathematical form for a piezoelectric material of class 6mm. It is discovered and proved by numerical examples in this paper that a novel Bleustein-Gulyaev （B-G） type of transverse surface wave can exist in such piezoelectric cou- pled solid media when the bulk-shear-wave velocity in the substrate is less than that in the piezoelectric layer but greater than the corresponding B-G wave velocity in the same pie- zoelectric material with an electroded surface. Such a wave does not exist in such layered structures in the absence of pie- zoelectricity. The mode shapes for displacement and electric potential in the piezoelectric layer are obtained and discussed theoretically. The study extends the regime of transverse sur- face waves and may lead to potential applications to surface acoustic wave devices.

Dispersion of Axisymmetric Longitudinal Waves in A Bi-Material Compound Solid Cylinder Made of Viscoelastic Materials

The paper studies the dispersion of axisymmetric longitudinal waves in the bi-material compound circular cylinder made of linear viscoelastic materials.The investigations are carried out within the scope of the piecewise homogeneous body model by utilizing the exact equations of linear viscoelasto-dynamics.The corresponding dispersion equation is derived for an arbitrary type of hereditary operator and the algorithm is developed for its numerical solution.Concrete numerical results are obtained for the case where the relations of the constituents of the cylinder are described through fractional exponential operators.The influence of the viscosity of the materials of the compound cylinder on the wave dispersion is studied through the rheological parameters which indicate the characteristic creep time and long-term values of the elastic constants of these materials.Dispersion curves are presented for certain selected dispersive and non-dispersive attenuation cases under various values of the problem parameters and the influence of the aforementioned rheological parameters on these curves is discussed.As a result of the numerical investigations,in particular,it is established that in the case where the rheological parameters of the components of the compound cylinder are the same,the viscosity of the layers’materials causes the axisymmetric wave propagation velocity to decrease.

Shear Wave Speed Dispersion Characteristics of Seafloor Sediments in the Northern South China Sea

To accurately characterize the shear wave speed dispersion of seafloor sediments in the northern South China Sea,five types of sediments including silty clay,clayey silt,sandy silt,silty sand,and clayey sand were selected,on which the measurements of the shear wave speed at 0.5-2.0 kHz and related physical properties were performed.Results reveal that the shear wave speed of sediments increases as the frequency increases,and the dispersion enhanced in the sediments in the order of silty clay,clayey silt,sandy silt,silty sand,and clayey sand,at a linear change rate of 0.727,0.787,3.32,4.893,and 6.967 m s−1 kHz−1,respectively.Through regression analysis,linear and logarithmic regression equations for the correlation between shear wave speed and frequency were established for each sediment type and the determination coefficients of regression equations indicate that the correlation is closer to a logarithmic relationship.The Grain-Shearing(GS)and Biot-Stoll models were used to calculate the shear wave speed dispersion of the five sediment types,and the comparison between theoretical prediction and measured results of shear wave speeds shows that the GS model can more accurately describe the shear wave speed dispersion characteristics of these sediments in the frequency band of 0.5-2.0 kHz.In the same band,the predictions obtained by using the Biot-Stoll model are significantly different from the measured data.

Wideband dispersion removal and mode separation of Lamb waves based on two-component laser interferometer measurement

Ultrasonic Lamb waves are considered as a sensitive and effective tool for nondestructive testing and evaluation of plate-like or pipe-like structures. The nature of multimode and dispersion causes the wave packets to spread, and the modes overlap in both time and frequency domains as they propagate through the structures. By using a two-component laser interferometer technique, in combination with a priori knowledge of the dispersion characteristics and wave structure information of Lamb wave modes, a two-component signal processing technique is presented for implementing dispersion removal and mode separation simultaneously for two modes mixture signals of Lamb waves. The proposed algorithm is first processed and verified using synthetic Lamb wave signals. Then, the two-component displacements test experiment is conducted using different aluminum plate samples. Moreover, we confirm the effectiveness and robustness of this method.

Numerical simulation for recognition of coalfield fire areas by Rayleigh waves

Effective recognition of a coalfield fire area improves fire-fighting efficiency and helps avoid potential geological hazards. Coalfield fire areas are hard to detect accurately using general geophysical methods. This paper describes simulations of shallow, buried coalfield fires based on real geological conditions. Recognizing the coalfield fire by Rayleigh wave is proposed. Four representative geological models are constructed, namely; the non-burning model, the pseudo-burning model, the real-burning model, and the hidden-burning model. Numerical simulation using these models shows many markedly different characteristics between them in terms of Rayleigh wave dispersion and Eigen displacement. These characteristics, as well as the shear wave velocity obtained by inverting the fundamental dispersion, make it possible to distinguish the type of the coalfield fire area and indentify the real and serious coalfield fire area. The results are very helpful for future application of Rayleigh waves for the detection of coalfield fire area.