Infinitesimal Waves Research Articles

Generalization of nonlinear Murnaghan elastic model for viscoelastic materials

This paper presents a generalization of Murnaghan elastic material to viscoelastic behavior using the Green–Rivlin multiple-integral approach. In the linear limit, the model coincides with the generalized Maxwell model. To create a nonlinear generalization, all possible second-order corrections were included in the constitutive equations written in the internal strains representation. Using this approach, we obtained expressions for the time- and frequency-dependent nonlinear dynamic moduli. We applied the developed nonlinear viscoelastic model to the description of infinitesimal strain waves superposed on a finite prestrain. Furthermore, we considered the generation of higher harmonics by the nonlinear interaction of two strain waves, which we showed can provide a method for measuring all viscoelastic constants of the developed model.

Partial dissipation and sub-shock

To study the dissipation property of a physical system one first considers infinitesimal waves for the analysis of weakly nonlinear phenomena. For some physical systems, the dissipation is partial and there is the appearance of sub-shocks in a strong traveling trajectory. The phenomenon of partial dissipation can occur for systems of hyperbolic balance laws and also for viscous conservation laws in continuum physics. We illustrate the phenomenon for a simple relaxation model and for the Navier-Stokes equations for compressible media. The admissibility criteria and the formation of sub-shocks are studied through the zero viscosity limit.

New solutions for interfacial capillary waves of permanent form

Progressive capillary waves on the interface between two homogeneous fluids confined in a channel with rigid walls parallel to the undisturbed interface are investigated. This problem is formulated as a system of integrodifferential equations that can be solved numerically via a boundary integral equation method coupled with series expansions of the unknown functions. With this highly accurate scheme and numerical continuation, we explore the global bifurcation of periodic travelling waves. It is found that there are two types of limiting profile, self-intersecting and boundary-touching, which appear either along a primary branch bifurcating from infinitesimal periodic waves or on an isolated branch existing above a certain finite amplitude. For particular sets of parameters, these two types of bifurcation curves can intersect, which can be viewed as a secondary bifurcation phenomenon occurring on the primary branch. Based on asymptotic and numerical analyses of the almost limiting waves, it is found that the boundary-touching solutions feature a circular geometry, i.e. the interface is pieced together by circular arcs of the same radius. A theoretical investigation yields the necessary conditions for the existence of these extreme waves, whereby we can predict the limiting configurations for most parameter sets. The comparisons between theoretical predictions and numerical results show good agreement.

Hybrid Zakharov-kinetic simulation of nonlinear stimulated Raman scattering

We present a novel 2D reduced numerical model for stimulated Raman scattering (SRS) in laser fusion plasmas in which envelope equations for the electromagnetic fields are coupled to a hybrid description of the electron species. Specifically, the electron distribution is split between a bulk part described by a Zakharov-like linear model and a kinetic tail discretized using a particle-in-cell-like (PIC) scheme. By avoiding to sample the bulk-electron distribution, this approach greatly reduces the numerical cost of SRS simulations compared with PIC codes, while still being able to describe the nonlinear evolution of the electron tail and trapping-related kinetic phenomena. First, our model is shown to reproduce accurately the linear Landau damping of an infinitesimal electron plasma wave (EPW) whose phase velocity falls into the tail of the electron distribution. Then, applying it to the simulation of the trapped-particle modulational instability of a large-amplitude EPW, results comparable to those of previously published 2D Vlasov simulations are obtained. Finally, we simulate the excitation of kinetic backward SRS from a single strong laser speckle (λ=0.527 μm, I=1016 W cm−2) in an underdense (ne=0.036 nc) plasma, which drives an EPW with wavenumber kλD≈0.34. The model predictions fairly agree with the results of a PIC simulation regarding the kinetic saturation mechanisms (i.e., trapped-particle instabilities), and with experimental data and Vlasov simulations related to the frequency shift of nonlinear EPWs. For this SRS simulation, we estimate that our hybrid model is over an order of magnitude less costly than an equivalent PIC simulation due to the lower particle count.

Symmetry reductions, generalized solutions and dynamics of wave profiles for the (2+1)-dimensional system of Broer–Kaup–Kupershmidt (BKK) equations

In this article, a (2+1)-dimensional system of Broer–Kaup–Kupershmidt (BKK) equations, which describes the non-linear, and long gravity waves in a dispersive system, is investigated by applying the method of Lie group of invariance. Applying the Lie group technique, the infinitesimals, vector fields, commutator table, and adjoint table are constructed for the BKK system. Furthermore, a one-dimensional optimal system of subalgebras is determined with the help of the adjoint transformation matrix; thereafter, the BKK system of equations is reduced into many systems of ordinary differential equations (ODEs) with respect to the similarity variables obtained through the symmetry reduction. These systems of ODEs are solved under some parametric constraints to obtain the exact closed form solutions. The obtained results are interpreted physically via graphical representation. Thus, dark–bright solitary waves, multi-peak mixed waves, breather type waves, periodic waves, multi-peakon solitons, and multi-solitons profiles of the obtained solutions are presented to make this research physically significant. A comparison is also presented between the solutions obtained in this article and the solutions reported by Kassem and Rashed in Kassem and Rashed (2019). The solutions obtained in this article are more general, and completely different in view of solutions obtained by Kassem and Rashed. The resulting solutions are found to be useful to understand the dynamics of BKK model and represent the authenticity as well as the effectiveness of the Lie group method. Therefore, the obtained solutions, and their dynamical wave structures are quite significant for understanding the propagation of the long gravity waves in a dispersive system. To obtain the infinitesimal generators and wave profiles of obtained solutions, symbolic computation is performed in the software package MAPLE and MATHEMATICA.

Progressive flexural–gravity waves with constant vorticity

Abstract

Nonlinear wave propagation and dynamic reconfiguration in two-dimensional lattices with bistable elements

This paper examines the propagation of elastic waves of large amplitudes in two-dimensional square lattices that include an alternating pattern of linear springs and nonlinear bistable springs. Because of the presence of bistable springs, these lattices have multiple stable configurations. In this paper, a theoretical model that takes into account the non-linearity of the bistable springs while neglecting geometric nonlinearities is used to study non-linear wave propagation in these lattices. Results from numerical simulations demonstrate that, for a lattice that is initially in a deformed stable equilibrium configuration, stimuli of large amplitudes are able to cause a reconfiguration of the lattice to a configuration of lower potential energy. Even for initial stable configurations that exhibit omnidirectional or directional bandgaps for infinitesimal wave propagation, waves of large amplitudes can propagate in the lattice due to its reconfiguration; however, due to the nonlinearity of bistable springs, the transmitted response tends to have multiple spectral peaks or be broadband for narrow-band stimuli of large amplitude. Simulations are used to study how the reconfiguration of the lattice depends on the amplitude and duration of the stimulus, on damping, and on the energy barrier between the two stable equilibria of the bistable springs. These numerical results suggest that these multistable lattices can serve as reconfigurable wave guides for which the amplitude of the stimulus offers opportunities for enhanced tunability.

Wave-propagation in an incompressible hollow elastic cylinder with residual stress

A study is presented to observe the effect of residual stress on waves in an incompressible, hyper-elastic, thick and hollow cylinder of infinite length. The problem is based on the non-linear theory of infinitesimal deformations occurring after a finite deformation. A prototype model of strain energy function is used which adequately includes the effects of residual stress and deformation. The expressions for internal pressure and the axial load are calculated and graphical illustrations are presented. Analysis of infinitesimal wave propagation is carried for the axisymmetric case in the considered cylinder. Numerical solution is obtained in the undeformed configuration and analyzed for the two-point boundary-value problem. Dispersion curves are plotted for varying choice of parameters.

Static and dynamic effective moduli of elastic-perfectly plastic granular aggregates under normal compression

Based on an existing simplified theoretical model for the normal contact interaction between two elastic-perfectly plastic spherical particles, we derived explicit expressions for the static and dynamic normal and dynamic tangential contact stiffnesses of elastic-perfectly plastic two-particle combination at pre-yield, yield, and post-yield conditions of normal loading. We used “static stiffness” or “loading stiffness” to refer to the slope of the force-displacement curve during monotonically increasing load. The “dynamic stiffness” or “unloading stiffness” refers to the stiffness that controls the speed of infinitesimal strain elastic waves propagating through the contacts. The static and dynamic contact stiffnesses are compared with numerical modeling of a two-sphere combination using the finite-element method. Furthermore, we used the explicit expressions for contact stiffnesses with the commonly used statistical averaging scheme to derive the static and dynamic effective bulk and shear moduli of a dry, random packing of identical elastic-perfectly plastic spherical particles. Elastic contact/mechanics-based effective medium models are unable to model the growth of contact area between inelastic (e.g., plastic) particles under normal force, which results in inaccurate predictions of contact stiffnesses and effective moduli. Once the particle reaches the limit of elasticity with onset of plastic deformation (yielding), further loading of two elastic-perfectly plastic spherical particles leads to a larger contact area than for two elastic particles under the same normal loading. As a result, after yielding, the dynamic effective moduli become stiffer than the corresponding moduli in the elastic case, whereas the static effective moduli remain constant, rather than increasing as in the elastic case.

Miracles, misconceptions and scotomas in the theory of solitary waves

ABSTRACTIn an age of billion dollar particle accelerators and Mars rovers, it is surprising that solitary waves were first discovered by a man on horseback with no tools but his own eyes. A century and a half later, more complicated patterns of ridges, so-called hyperelliptic two-polycnoidal waves, were discovered in the ocean during a beach vacation. The inverse scattering method, which solves nonlinear partial differential equations through a sequence of solving purely linear equations, is a blend of quantum theory and hydrodynamics that arose from informal, unstructured conversations (i.e. goofing off) among a group of postdocs from different disciplines who were randomly assigned to the same office. The cnoidal wave in the lemniscate case is well-approximated by a nonlinear solitary wave and equally well approximated by a linear sine wave. It is always and exactly the superposition of solitary waves even in the limit in which it is an infinitesimal sine wave. The history and science of solitary waves has the disorienting quality of an M. C. Escher drawing. Here, we cannot give an understanding of these deep subjects in so brief an article; rather we strive to unveil the beauty and unexpectedness of these topics to give the reader a reason to pursue these in the much more comprehensive reviews and books we cite. Further, we stress the “scotomas” (blind spots), misconceptions and surprises, the sociology and epistemology of science. It is true that failed theories, scotomas, serendipity and cognitive saltation (progress in jumps) is characteristic of science. It is also true that the invention of the train was the invention of the train wreck. Engineering learns from each disaster and science should do the same. The highly nonlinear history of nonlinear waves is reported not to disrespect the past but to replace scientific fatalism with a constructive wariness. We are not smarter or more enlightened than Scott Russell or Stokes or Landau, but we can learn from their scotomas and misconceptions as much as from their triumphs.

ONE-DIMENSIONAL QUASI-STATIC CONSOLIDATION MODEL CONSIDERING INERTIA OF FLUID PHASE

Terzaghi’s one-dimensional classic consolidation theory ignores inertia of pore water in saturated soils, and has outstanding differences in its derivation and formulation during various publication periods. This leads to a strange phenomenon that considerable misunderstandings and confusion of it still prevail in the current literature. Within the previous framework of large strain dynamic consolidation theory, a one-dimensional infinitesimal strain consolidation wave model is obtained to consider the effect of inertia of pore water with necessary simplification and hypotheses. The present consolidation wave model is characterized by velocity dispersion and dissipative attenuation. The method of separation of variables is used to obtain an analytical solution for a consolidation wave model under the condition of one-way drainage and instantaneous loading. A numerical case study shows that behaviors of consolidation wave are actually controlled by the value of a dimensionless number D c . For cases of higher values of D c , jump and fluctuation of dimensionless excess pore water pressure between positive and negative values are prone to occur, in contrast with the cases of lower values of D c that result into special phenomena including the Mandel-Cryer effect observed in laboratory tests. The inherent ambiguities about Terzaghi’s classic theory models proposed in the early stage and the later stage respectively are investigated to draw a conclusion that the early Terzaghi’s (1923,1925) consolidation model can be interpreted as a large strain model with respect to the general soil coordinates contrasted with an infinitesimal strain model with respect to the solid-phase volume coordinates. Accordingly, the present consolidation wave model can be extended to various formulations based on the corresponding coordinate properties. Consolidation wave theory is of significance to probe into an innovative uncertainty principle which shows that for scale model testing, the observed consolidation wave response of undisturbed soil samples cannot equal the response of the same soil in practical conditions. Therefore, it is advisable to pay attention to the size effect of soil samples in consolidation study from the perspective of microscopic soil mechanics. The theoretical precision of evaluated excess pore water pressure by classic Terzaghi’s consolidation model varies with the value of dimensionless quantity D c , which is a vital parameter in the proposed consolidation wave theory.

Nonlinear phase-resolved reconstruction of irregular water waves

We develop and validate a high-order reconstruction (HOR) method for the phase-resolved reconstruction of a nonlinear wave field given a set of wave measurements. HOR optimizes the amplitude and phase of $L$ free wave components of the wave field, accounting for nonlinear wave interactions up to order $M$ in the evolution, to obtain a wave field that minimizes the reconstruction error between the reconstructed wave field and the given measurements. For a given reconstruction tolerance, $L$ and $M$ are provided in the HOR scheme itself. To demonstrate the validity and efficacy of HOR, we perform extensive tests of general two- and three-dimensional wave fields specified by theoretical Stokes waves, nonlinear simulations and physical wave fields in tank experiments which we conduct. The necessary $L$, for general broad-banded wave fields, is shown to be substantially less than the free and locked modes needed for the nonlinear evolution. We find that, even for relatively small wave steepness, the inclusion of high-order effects in HOR is important for prediction of wave kinematics not in the measurements. For all the cases we consider, HOR converges to the underlying wave field within a nonlinear spatial-temporal predictable zone ${\mathcal{P}}_{NL}$ which depends on the measurements and wave nonlinearity. For infinitesimal waves, ${\mathcal{P}}_{NL}$ matches the linear predictable zone ${\mathcal{P}}_{L}$, verifying the analytic solution presented in Qi et al. (Wave Motion, vol. 77, 2018, pp. 195–213). With increasing wave nonlinearity, we find that ${\mathcal{P}}_{NL}$ contains and is generally greater than ${\mathcal{P}}_{L}$. Thus ${\mathcal{P}}_{L}$ provides a (conservative) estimate of ${\mathcal{P}}_{NL}$ when the underlying wave field is not known.

Tuning Elastic Waves in Soft Phononic Crystal Cylinders Via Large Deformation and Electromechanical Coupling

Soft electroactive materials can undergo large deformation subjected to either mechanical or electrical stimulus, and hence, they can be excellent candidates for designing extremely flexible and adaptive structures and devices. This paper proposes a simple one-dimensional soft phononic crystal (PC) cylinder made of dielectric elastomer (DE) to show how large deformation and electric field can be used jointly to tune the longitudinal waves propagating in the PC. A series of soft electrodes, which are mechanically negligible, are placed periodically along the DE cylinder, and hence, the material can be regarded as uniform in the undeformed state. This is also the case for the uniformly prestretched state induced by a static axial force only. The effective periodicity of the structure is then achieved through two loading paths, i.e., by maintaining the longitudinal stretch and applying an electric voltage over any two neighboring electrodes or by holding the axial force and applying the voltage. All physical field variables for both configurations can be determined exactly based on the nonlinear theory of electroelasticity. An infinitesimal wave motion is further superimposed on the predeformed configurations, and the corresponding dispersion equations are derived analytically by invoking the linearized theory for incremental motions. Numerical examples are finally considered to show the tunability of wave propagation behavior in the soft PC cylinder. The outstanding performance regarding the band gap (BG) property of the proposed soft dielectric PC is clearly demonstrated by comparing with the conventional design adopting the hard piezoelectric material. One particular point that should be emphasized is that soft dielectric PCs are susceptible to various kinds of failure (buckling, electromechanical instability (EMI), electric breakdown (EB), etc.), imposing corresponding limits on the external stimuli. This has been carefully examined for the present soft PC cylinder such that the applied electric voltage is always assumed to be less than the critical voltage except for one case, in which we illustrate that the snap-through instability of the axially free PC cylinder made of a generalized Gent material may be used to efficiently trigger a sharp transition in the BGs.

Spin waves in rings of classical magnetic dipoles

We theoretically and numerically investigate spin waves that occur in systems of classical magnetic dipoles that are arranged at the vertices of a regular polygon and interact solely via their magnetic fields. There are certain limiting cases that can be analyzed in detail. One case is that of spin waves as infinitesimal excitations from the system’s ground state, where the dispersion relation can be determined analytically. The frequencies of these infinitesimal spin waves are compared with the peaks of the Fourier transform of the thermal expectation value of the autocorrelation function calculated by Monte Carlo simulations. In the special case of vanishing wave number an exact solution of the equations of motion is possible describing synchronized oscillations with finite amplitudes. Finally, the limiting case of a dipole chain with is investigated and completely solved.

On mechanical waves and Doppler shifts from moving boundaries

We investigate the propagation of infinitesimal harmonic mechanical waves emitted from a boundary with variable velocity and arriving at a stationary observer. In the classical Doppler effect, Xs(t)=vt is the location of the source with constant velocity v. In the present work, however, we consider a source co‐located with a moving boundary x=Xs(t), where Xs(t) can have an arbitrary functional form. For ‘slowly moving’ boundaries (i.e., ones for which the timescale set by the mechanical motion is large in comparison to the inverse of the frequency of the emitted wave), we present a multiple‐scale asymptotic analysis of the moving boundary problem for the linear wave equation. We obtain a closed‐form leading‐order (with respect to the latter small parameter) solution and show that the variable velocity of the boundary results not only in frequency modulation but also in amplitude modulation of the received signal. Consequently, our results extend the applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (i) for non‐uniform boundary motion can be inserted in place of the constant velocity v in the classical Doppler formula and (ii) that the non‐uniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated. Copyright © 2017 John Wiley & Sons, Ltd.

Stability and dynamics of two-dimensional fully nonlinear gravity–capillary solitary waves in deep water

The stability and dynamics of two-dimensional gravity–capillary solitary waves in deep water within the fully nonlinear water-wave equations are numerically studied. It is well known that there are two families of symmetric gravity–capillary solitary waves – depression waves and elevation waves – bifurcating from infinitesimal periodic waves at the minimum of the phase speed. The stability of both branches was previously examined by Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) by means of a numerical spectral analysis. Their results show that the depression solitary waves with single-valued profiles are stable, while the elevation branch experiences a stability exchange at a turning point on the speed–amplitude curve. In the present paper, we provide numerical evidence that the depression solitary waves with an overhanging structure are also stable. On the other hand, Dias et al. (Eur. J. Mech. B, vol. 15, 1996, pp. 17–36) numerically traced the elevation branch and discovered that its speed–amplitude bifurcation curve features a ‘snake-like’ behaviour with many turning points, whereas Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) only considered the stability exchange near the first turning point. Our results reveal that the stability exchange occurs again near the second turning point. A branch of asymmetric solitary waves is also considered and found to be unstable, even when the wave profile consists of a depression wave and a stable elevation one. The excitation of stable gravity–capillary solitary waves is carried out via direct numerical simulations. In particular, the stable elevation waves, which feature two troughs connected by a small dimple, can be excited by moving two fully localised, well-separated pressures on the free surface with the speed slightly below the phase speed minimum and removing the pressures simultaneously after a period of time.

Analytical models of stationary nonlinear gravitational waves

Euler’s equations with standard boundary conditions for the problem of potential surface waves of an arbitrary amplitude in a homogeneous liquid layer with a flat bottom are converted into the new system, including integral and differential equations for the of the potential and its time derivative near the surface. The basic formula of the theory of infinitesimal waves, paired Korteweg-de Vries (KdV) and Kadomtsev− Petviashvili (KP) equations, the envelope Zakharov−Shabat soliton follows from the system in limiting case. The resulting generalized equation, unlike traditional KdFand KP-equations is suitable for the description of waves on the surface of the initially quiescent fluid. A new exact solutions for gravity waves in a deep water, expressed in terms of complex Lambert’s functions are constructed.

Transonic gliding edge dislocations/slip pulse near and on an interface/fault

Three problems are solved in this paper that are related to transonic earthquakes. (1) The shear shock Mach wave emanating from a transonic gliding edge dislocation which impacts an interface. (2) A transonic edge dislocation gliding parallel and near to an interface. (3) The transonic edge dislocation gliding on the interface itself. The dislocation is in uniform or quasi‐uniform motion. The interface separates isotropic elastic material of slightly different properties. The first and last problems are essentially the same. The problems are solved with the building blocks of Mach waves, with logarithmic and arctan distributions of infinitesimal shock waves and with subsonic (with respect to longitudinal wave velocities) glide edge and climb edge dislocations. The need for the smeared infinitesimal shock wave distributions is an interesting feature of the solutions. Distributions of infinitesimal smeared Mach waves will exist wherever physical discontinuities exist near a transonic dislocation. A slip pulse of infinitesimal moving edge dislocations can be self sharpening because of attractive forces between like sign dislocations in certain velocity ranges.

Spectral Density Analysis for Wave Characteristics in Pohang New Harbor

The Pohang New Harbor (PNH), located at the Yongil bay in the northeastern part of Pohang city, South Korea, has experienced extreme wave hazards of about 3.0–5.0 m in elevation due to the seasonal swell from the far ocean. In this paper, both analytical and numerical studies are performed to investigate the wave-induced oscillations in an arbitrary shaped harbor with corner point consideration. By taking the consideration of the actual topography and bathymetry data, the boundary of PNH is constructed. Our theoretical model is based on the assumptions of inviscid, irrotational fluid, infinitesimal wave amplitude, and finally, the Helmholtz equation and its Weber’s solution. The numerical simulations are conducted to analyze the spectral density of the standing waves in PNH at eight respective synthetic record points. The simulation results are validated with real-time measurement data, which is obtained by wave heights and tide gauges at the specified record points within and outside the PNH. To improve the harbor’s design, a tactic such as building the breakwater at the entrance of the harbor is implemented and then spectral density is estimated in the modified geometry of the PNH. The consequential effects are proposed at the same time, suggesting the feasibility of the improvement measures.

Waves of intermediate length through an array of vertical cylinders

We report a semi-analytical theory of wave propagation through a vegetated water. Our aim is to construct a mathematical model for waves propagating through a lattice-like array of vertical cylinders, where the macro-scale variation of waves is derived from the dynamics in the micro-scale cells. Assuming infinitesimal waves, periodic lattice configuration, and strong contrast between the lattice spacing and the typical wavelength, the perturbation theory of homogenization (multiple scales) is used to derive the effective equations governing the macro-scale wave dynamics. The constitutive coefficients are computed from the solution of micro-scale boundary-value problem for a finite number of unit cells. Eddy viscosity in a unit cell is determined by balancing the time-averaged rate of dissipation and the rate of work done by wave force on the forest at a finite number of macro stations. While the spirit is similar to RANS scheme, less computational effort is needed. Using one fitting parameter, the theory is used to simulate three existing experiments with encouraging results. Limitations of the present theory are also pointed out.