Wave Motion最新文献

In the present work, a full nonlinear Timoshenko beam employing nonlinear shape functions is developed. The extended Hamilton principle is employed for deriving the differential equations of motion and the associated boundary conditions. The general form of the boundary conditions is then utilized to determine the static solution of the beam motion. Using this solution for the deformation and rotation of the beam, the nonlinear shape functions of the beam are identified, which leads to the linear and nonlinear mass and stiffness matrices of the Timoshenko beam element. The nonlinear dispersion diagram incorporating the non-linear corrections is obtained using the Linstedt–Poincaré perturbation method. An analysis of the effect of internal transverse shear and bending on the nonlinear dispersion characteristics of wave propagation in two-dimensional periodic network materials made of nonlinear Timoshenko beams is done. The formulated theory shows that the percentage of correction factor of the nonlinear kinematics versus the linear dynamical behavior is inversely proportional to the frequency amplitude. The shear and extension modes are shown to have the higher effect in the non-linear correction term in comparison to the flexural mode.

The existence of surface acoustic waves (SAWs) is studied in laterally periodic superlattices, modelled as an anisotropic elastic half-space with an arbitrary periodic variation of its material properties along the stratification direction (call it X) parallel to the surface. Unlike a homogeneous half-space, such a structure allows for more than one (dispersive) SAW. Specifically, it is shown that any superlattice with a generic shape of periodicity profile admits at most three SAW dispersion branches $\omega \left(K_x\right)$, i.e., at most three different SAW frequencies at any fixed Bloch wavenumber $K_x$. Moreover, the total number of SAWs at fixed $K_x$ in a pair of superlattices with periodicity profiles obtained from one another by the inversion of the axis X cannot exceed three either. At least one SAW branch must exist in one of these two superlattices unless the bulk-wave threshold is the so-called exceptional (i.e., admits surface skimming wave). The SAW branch is unique in the particular case of a superlattice invariant to the inversion $X\to -X$. The above general results are illustrated by the perturbation theory derivations for the weakly modulated superlattices. Explicit leading-order formulas are obtained for the quasi-Rayleigh wave branch evolving from the Rayleigh wave in each of the mutually ”inverse” superlattices and for the quasibulk wave branch evolving from the exceptional bulk-wave threshold in one of the superlattices.

In the current study, the Modified Equal-Width (MEW) equation will be handled numerically by a novel technique using collocation finite element method where cubic Hermite B-splines are used as trial functions. To test the accuracy and efficiency of the method, four different experimental problems; namely, the motion of a single solitary wave, interaction of two solitary waves, interaction of three solitary waves and the birth of solitons with the Maxwellian initial condition will be investigated. In order to verify, the validity and reliability of the proposed method, the newly obtained error norms $L_2$ and $L_{\infty }$ as well as three conservation constants have been compared with some of the other numerical results given in the literature at the same parameters. Furthermore, some wave profiles of the newly obtained numerical results have been given to demonstrate that each test problem exhibits accurate physical simulations. The advantage of the proposed method over other methods is the usage of inner points at Legendre and Chebyshev polynomial roots. This advantage results in better accuracy with less number of elements in spatial direction. The results of the numerical experiments clearly reveal that the presented scheme produces more accurate results even with comparatively coarser grids.

The currents in the ocean have a serious impact on ocean dynamics, since they affect the transport of mass and thus the distribution of salinity, nutrients and pollutants. In many physically important situations the current depends quadratic-ally on the depth. We consider a single layer of fluid and study the propagation of the surface waves in the presence of depth-dependent current with quadratic profile. We select the scale of parameters and quantities, which are typical for the Boussinesq propagation regime (long wave and small amplitude limit) and we also derive the well known KdV model for the surface waves interacting with current.

In this paper, we investigate a fifth-order nonlinear Schrödinger equation, which can be applied to describe the propagation of ultrashort pulses in optical fibers. We provide the eigenfunctions of the Lax pair associated with the elliptic function seed solutions cn and dn. Using the Darboux transformation method, the W-shaped solitons, breathers, periodic solutions and rogue waves on the elliptic functions cn and dn background are obtained, the corresponding dynamical properties and evolutions are illustrated graphically by choosing proper parameters, the variations for amplitudes and periods of these solutions are analyzed. The relationship between parameters and solutions’ structures is discussed. To the best of our knowledge, the W-shaped solitons on the elliptic function background are presented for the first time. The results in this paper might be useful for us to understand some characteristics and relations of breathers and rogue waves on the elliptic functions cn and dn background in various physical equations with higher-order effects.

This paper derives a semi-discrete conservative difference scheme for the rotation-two-component Camassa–Holm system based on its Hamiltonian invariants. Mass, momentum and energy are preserved for the semi-discrete scheme. Furthermore, a fully discrete finite difference scheme is proposed without destroying any one of the conservative laws. Combining a nonlinear iteration with a threshold strategy, the accuracy of the scheme is guaranteed. Meanwhile, this scheme captures the formation and propagation of solitary wave solutions in long time behavior under smooth/nonsmooth initial data. Remarkably, a new type of asymmetric wave breaking phenomenon is revealed in the case of the nonzero rotational parameter.

The diffraction of a time-harmonic plane wave on collinear finite defects in a square lattice is studied. This problem is reduced to a matrix Wiener–Hopf equation. This work adapts the recently developed iterative Wiener–Hopf method to this situation. The method was motivated by wave scattering in continuous media but it is shown here that it can also be employed in a discrete lattice setting. The numerical results are validated against a different method using discrete Green’s functions. Unlike the latter approach, the complexity of the present algorithm is shown to be virtually independent of the length of the cracks.

Long crested waves above monotone bathymetries with different steepness are shown to be related by a time scaling. The scaling is explicitly present in the usual WKB approximation and more generally in the Hamiltonian potential theory for incompressible, irrotational inviscid fluid motion (Zakharov, 1968). The scaling uses the depth instead of the spatial distance as position marker, which is a canonical transformation in the action functional. This implies that waves above different slopes are related by a simple space-time scaling. At depths before near-coastal effects of run-up become relevant, the scaling property is valuable for understanding the wave propagation and may reduce laboratory experiments. Taking into account non-Hamiltonian coastal effects of breaking and coastal run-up, nonlinear simulations show correlations above 0.8 for waves above different slopes until a typical depth for many offshore activities of 15 m (Forrsitall, 2004). Numerical simulations with second- and third order nonlinearity are performed with HAWASSI software (Kurnia and Van Groesen, 2014), a variant of a higher order spectral method (Dommermuth and Yue, 1987; West et al., 1987). An example of the scaling is also shown to be present for an air-water CFD potential simulation (Aggarwal et al., 2020).

In this paper, we investigate the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which is widely used in fluid mechanics and theoretical physics. The $N$-soliton solution is obtained via the Hirota’s bilinear method. The Wronskian solution is derived by using the Wronskian technique for the bilinear form. Through the exchange formula, we deduce the bilinear Bäcklund transformation consisting of four equations and six parameters. In order to consider the quasi-periodic wave having complex structure, one-, two- and three-periodic waves are investigated systemically by combining the Hirota’s bilinear method with Riemann theta function. Furthermore, the corresponding graphs of periodic wave are presented by considering the geometric properties between the characteristic lines. The propagation characteristics of periodic waves are investigated by virtue of the characteristic lines. Finally, the asymptotic relationships between quasi-periodic wave solutions and soliton solutions are established theoretically under a condition of the small amplitude limit. The analytical method used in this paper can be applied in other integrable systems.

Ultrasonic testing techniques such as guided wave-based structural health monitoring aim to evaluate the integrity of a material with sensors and actuators that operate in situ, i.e. while the material is in use. Since ultrasonic wave propagation is sensitive to environmental conditions such as pre-deformation of the structure, the design and performance evaluation of monitoring systems in this context is a complicated task that requires quantitative data and the associated modeling effort. In our work, we propose a set of numerical tools to solve the problem of mechanical wave propagation in materials subjected to pre-deformation. This type of configuration is usually treated in the domain of acoustoelasticity. A relevant modeling approach is to consider two different problems: a quasi-static nonlinear problem for the large displacement field of the structure and a linearized time-domain wave propagation problem. After carefully reviewing the modeling ingredients to represent the configurations of interest, we propose an original combination of numerical tools that leads to a computationally efficient algorithm. More specifically, we use 3D shell elements for the quasi-static nonlinear problem and the time-domain spectral finite element method to numerically solve the wave propagation problem. Our approach can represent any type of material constitutive law, geometry or mechanical solicitation. We present realistic numerical results on 3D cases related to the monitoring of both isotropic and anisotropic materials, illustrating the genericity and efficiency of our method. We also validate our approach by comparing it to experimental data from the literature.