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In the regime of above-threshold ionization of gas atom in the field of laser radiation, plasma with photoelectron distribution consisting of peaks at discrete energy values is formed. It is shown that the number of longitudinal waves in such plasma coincides with the number of peaks in the distribution function. When peaks practically don't overlap, the dispersion law of each wave in the region of short waves is determined by electrons from the corresponding peak. In this case the phase and group velocities of the waves are close to the electron velocity, which corresponds to the peak maximum. It is possible to talk about such waves as an electronic sound, since the perturbations of the electron density mainly arise due to pressure perturbations. When the peaks are narrow, but having a finite width, the Cherenkov damping of waves is exponentially small. Numerical calculations the dispersion laws for of the two and four waves in photoionized xenon plasma, in which the electron distribution function consists of two or four narrow peaks are given.

In degenerate quantum plasma the energy behavior of electrostatic modes propagating perpendicular to the external magnetic field is studied by employing the separated spin evolution quantum hydrodynamic (SSE-QHD) model. This model reveals that spin electron cyclotron wave (SECW) appears additionally with the upper hybrid wave (UHW). In case of SECW, the curves for the energy flow speed at different levels of spin polarization effect flip over at a particular value of wave number. The spin polarization effect enhances the energy flow speed before this value of wave number and then suppresses it afterward. The energy flow speed is enhanced by spin polarization effect in the entire range of wave number for the propagation of UHW. The Bohm potential effect drastically increases the energy flow speed at high wave number domain in both the waves. This study may find its applications to understand the energy behavior inspin polarized solid state plasmas

New dependencies of frequency and damping decrement of high-frequency longitudinal waves on the wave vector in photoionized plasma formed by tunnel ionization of atoms in the field of circularly polarized radiation are found. Weakly damped longitudinal waves with a frequency much higher than the electron Langmuir frequency are predicted.

We explore the mode conversion characteristics of electrostatic hybrid surface waves due to the magnetic field orientation in a magnetoplasma slab. We obtain the dispersion relations for the symmetric and anti-symmetric modes of hybrid surface waves for two different magnetic field configurations: parallel and perpendicular. For the parallel magnetic field configuration, we have found that the symmetric mode propagates as upper- and lower-hybrid waves. However, the hybrid characteristics disappear and two non-hybrid waves are produced for the anti-symmetric mode. For the perpendicular magnetic field configuration, however, the anti-symmetric mode propagates as the upper- and lower-hybrid waves and the symmetric mode produces two non-hybrid branches of waves.

Wave coupling exists in asymmetrical structures. In this work, we numerically and experimentally study wave coupling behaviors in one-dimensional origami metamaterials with uncoupled creases. Unlike conventional one-dimensional asymmetrical structures, the coupling characteristics in origami structures can be tuned and rich bandgap regulation can be achieved. By utilizing wave coupling, new flexural-wave bandgaps can be opened and shaped by folding. We propose a coupling ratio to explain different wave coupling phenomena with respect to the folding angles and structure parameters. Based on a proper realization of the creases and with two orthogonal fiber Bragging grating (FBG) displacement sensors, several experiments are performed to support our arguments on wave-coupling behaviors. Through this research, we also provide a systematic analyzing approach and an experimental method in studying coupling waves. Despite focusing on the crease-uncoupled origami structures, the provided multi-dimensional sensing technique enables one to study general origami metamaterials with coupled creases.

Inspired by the recent optical investigations on the hyperbolic metasurfaces, we study the possibility of new quasi-electrostatic waves propagation in such anisotropic 2D materials that do not propagate in the case of a regular 2D medium. The analytical result shows that the dispersion relation of these waves is free from the effects of permittivities of the surrounding media. These new waves may be considered as longitudinal quasi-electrostatic waves.

Flexural vibration suppression in an Euler–Bernoulli beam with attached lateral local resonators (LLR) is studied theoretically and numerically. Hamilton's principle and Bloch's theorem are employed to derive the dispersion relation which reveals that two band gaps are generated. Within both band gaps, the flexural waves are partially transformed into longitudinal waves through a four-link-mechanism and totally blocked. The band gaps can be flexibly tuned by changing the geometry parameter of the four-link-mechanism and the spring constants of the resonators. Frequency response function (FRF) from finite element analysis via commercial software of ANSYS shows large flexural wave attenuation within the band gaps and the effect of damping from the LLR substructures which helps smooth and lower the response peaks at the sacrifice of the band gap effect. The existence of the multi-flexural band gaps can be exploited for the design of flexural vibration control of beams.

Studies on electromagnetic surface waves along the surface of a finitely thick plate made from negative-index materials are presented in this paper. Through theoretical analysis we worked out requirements on permeability and permittivity for surface waves to exist. Comparing with the model of a plate with infinite thickness, the one of finite thickness shows released requirements, leading to potential applications in guiding wave and developing radiation sources. With the help of particle-in-cell simulations, we examined theoretical predictions and demonstrated field distributions of surface waves.

Acoustic traveling waves in a gas that saturates a rigid porous medium is investigated under the assumption that the drag experienced by the gas is modeled by Forchheimerʼs law. Exact traveling wave solutions (TWS)s, as well as approximate and asymptotic expressions, are obtained; decay rates are determined; and acceleration wave results are presented. In addition, special cases are considered, critical values of the wave variable and parameters are derived, and comparisons with predictions based on Darcyʼs law are performed. It is shown that, with respect to the Darcy case, most of the metrics that characterize such waveforms exhibit an increase in magnitude under Forchheimerʼs law.

The propagation of coupled forward and backward waves in the system of two tunnel-coupled waveguides, one with negative, and the other with positive refractive index, is considered. General solution of the evolution equations describing the wave propagation is found taking the traveling wave into account. Dependence of wave amplitude and frequency on arbitrary constants is analyzed. We have observed three types of waves in the nonlinear oppositely-directional coupler. Two of them are periodic and solitary waves that are well known in nonlinear optics. We also have found a new type of nonlinear waves for optics of artificial materials that is compacton.

It is shown that the nonlinear interaction between the magnetic field-aligned circularly polarized (CP) dispersive Alfven waves and low-frequency electrostatic perturbations give rise to Alfvenic rogue waves in magnetized plasmas. Our results reveal the left-hand CP Alfvenic wave supports both supersonic and subsonic Alfvenic rogue waves. The latter can propagate for backward wave only. The right-hand CP Alfvenic rouge waves appear on account of the amplitude modulation of the waves by quasi-stationary density modulations. The amplitude of the right-hand CP Alfvenic rogue waves decreases with the increase of the plasma number density, but it increases with the increase of the magnetic field strength. However, the amplitude of the subsonic left-hand CP Alfvenic rogue wave becomes stronger with the excess of the plasma number density, but it shrinks with stronger magnetic field strength. Finally, we briefly discuss the relevance of our investigation to the role of the nonlinear Alfven waves that power the solar wind.

A nonlinear evolution equation for wave propagation in bubbly liquids, taking into account viscosity and heat transfer, has been derived by Kudryashov and Sinelshchikov. In the case of no dissipation the authors have provided analytical solutions representing undistorted waves. These results are cast into a simpler form and studied in more detail. In addition to the wave profiles the corresponding phase curves are presented. Depending on some parameter the solutions represent solitary or periodic waves. Some of the periodic waves exhibit peaks or cusps. From the periodic waves a new type of “meandering” solutions is constructed.

A new method is introduced for studying the propagation of elastic waves in isotropic bodies, based on the Kirchhoff potentials borrowed from electromagnetism. By means of this method we identify and characterize the elastic waves generated in a semi-infinite (half-space) body by the action of an external force localized on, or beneath, the body surface. The method implies coupled integral equations for the wave amplitudes, which we solve for both cases mentioned above. For a force localized on the body surface we identify two transverse waves, corresponding to the two polarizations (normal and parallel to the propagation plane). The longitudinal waves appear as eigenmodes. The waves produced by a force localized beneath the surface are stationary waves along the normal to the surface. We compute the surface displacement in both cases and the force exerted on the surface by a force localized beneath. All these quantities exhibit a characteristic decrease with the distance on the body surface and an oscillatory behaviour. We discuss briefly some possibilities of extending the present method to include the effect of the inhomogeneities on the waves propagation.

Existence of traveling waves propagating without internal reflection in inclined water channels of arbitrary slope is demonstrated. It is shown that traveling non-monochromatic waves exist in both linear and nonlinear shallow water theories in the case of a uniformly inclined channel with a parabolic cross-section. The properties of these waves are studied. It is shown that linear traveling waves should have a sign-variable shape. The amplitude of linear traveling waves in a channel satisfies the same Green's law, which is usually derived from the energy flux conservation for smoothly inhomogeneous media. Amplitudes of nonlinear traveling waves deviate from the linear Green's law, and the behavior of positive and negative amplitudes are different. Negative amplitude grows faster than positive amplitude in shallow water. The phase of nonlinear waves (travel time) is described well by the linear WKB approach. It is shown that nonlinear traveling waves of any amplitude always break near the shoreline if the boundary condition of the full absorption is applied.

We have numerically calculated chaotic waves of the focusing nonlinear Schrr̈odinger equation (NLSE), starting with a plane wave modulated by relatively weak random waves. We show that the peaks with highest amplitude of the resulting wave composition (rogue waves) can be described in terms of exact solutions of the NLSE in the form of the collision of Akhmediev breathers. © 2009 Elsevier B.V. All rights reserved., The work is supported by the Australian Research Council (Discovery Project scheme DP0985394). J.M.S.C. acknowledges support from the M.E.y C. under contract FIS2006-03376.

The phonon–polariton behaviors of two-dimensional piezoelectric phononic crystals (PPCs) were studied using the plane wave expansion method. The governing equations combine Maxwell's equations and Newton's equations of motion. A mode-repulsion can be formed by strong coupling between electromagnetic (EM) waves and elastic waves in the vicinity of the center of the first Brillouin zone for PPC that comprises piezoelectric material and with opposite polarization in different periodically organized areas. Take a 2D ZnO PPC as a numerical example, it was decoupled into two independent groups. One refers to the mixed mode of the in-plane elastic waves and the transverse-magnetic (TM) mode EM waves. The other group refers to the mixed mode of the out-of-plane elastic waves and the transverse-electric (TE) mode EM waves. Coupling repulsion is also observed in these two groups.

By using of the multiple scattering methods, we study the negative refraction imaging effect of solid acoustic waves by two-dimensional three-component phononic crystals composed of coated solid inclusions placed in solid matrix. We show that localized resonance mechanism brings on a group of flat single-mode bands in low-frequency region, which provides two equivalent frequency surfaces (EFS) close to circular. The two constant frequency surfaces correspond to two Bloch modes, a right-handed and a left-handed, whose leading mode are respectively transverse (T) and longitudinal (L) modes. The negative refraction behaviors of the two kinds of modes have been demonstrated by simulation of a Gaussian beam through a finite system. High-quality far-field imaging by a planar lens for transverse or longitudinal waves has been realized separately. This three-component phononic crystal may thus serve as a mode selector in negative refraction imaging of solid acoustic waves.

A new type of wave solutions, called as multiple-mode waves, which can be expressed in the superposition forms of more than two types of single-mode waves of Vakhnenko equation have been investigated in this Letter. A new general method for obtaining the multiple-mode waves is proposed, based on which four cases of the possible forms of wave solutions with two-mode have been derived. The explicit expressions of the two-mode waves as well as the existence conditions have been presented, which may be the nonlinear combinations between periodic waves, solitons, compactons, etc., with different wave speeds, respectively. It is pointed out that more complicated multiple-mode waves with more than three single-mode waves can be derived accordingly, which can be used to reveal the evolution of interactions between different types of waves, especially between various solitons.

Influence of quantum effects on the internal waves and the Rayleigh-Taylor instability in plasma is investigated. It is shown that quantum pressure always stabilizes the RT instability. The problem is solved both in the limit of short-wavelength perturbations and exactly for density profiles with layers of exponential stratification. In the case of stable stratification, quantum pressure modifies the dispersion relation of the inertial waves. Because of the quantum effects, the internal waves may propagate in the transverse direction, which was impossible in the classical case. A specific form of pure quantum internal waves is obtained, which do not require any external gravitational field., 9 pages, 2 figures

We analyze theoretically the structure of the field created in a semi-infinite polycrystal by an acoustic wave, coming from an isotropic homogeneous medium and incident normally onto its surface. The elastic anisotropy of the polycrystal is supposed to be small, and the perturbation theory is applied. It is shown that the effective medium approach is not valid. In addition to the transmitted wave propagating in the polycrystal with an effective sound speed, there is one more bulk wave, whose amplitude decreases at a distance of the order of the mean size of the grain from the interface. The structure of the reflected wave is the same as when reflecting from an isotropic solid. However, the relation between the amplitudes of reflected and transmitted waves differs from that in an isotropic solid.

The evolution of wave patterns of a generalized KdV equation is investigated in this Letter. For the topological vector filed, a singular line can be observed which separates the phase trajectories on the phase plane into two parts. All possible waves including compactons, solitary waves, smooth periodic waves and non-smooth periodic waves with peaks as well as the existence conditions have been presented. Specially, a new type of waves called as kink-compactons which combine both half structures of compacton and kink-wave can be found.

Long dynamo waves are a characteristic feature of interface dynamo models with spatially localized α and Ω effects. The evolution of such waves is described by the modified Korteweg–de Vries equation. Solutions to this equation take the form of solitary waves, breathers, and snoidal and cnoidal waves, and represent nonlinear waves of magnetic activity that migrate towards the equator, as observed on the Sun. Averaging techniques extend the theory to longer times and relate the amplitude of these waves to the dynamo number.

We derive a normal form of nonlinear equations for short equatorial waves considered in the framework of the rotating shallow water model. We show dynamical splitting of equatorial Rossby and inertia-gravity waves. We derive an effective Hamiltonian for the short inertia-gravity waves and consider their kinetics using the weak turbulence approach. Stationary power-law energy spectra are obtained. They have different slopes for eastward and westward propagating waves due to the fact that resonant triads of inertia-gravity waves exist only in specific regions of the phase-space.

A study of transverse acceleration wave phenomena in the context of nonlinear elasticity is carried out. Using singular surface theory, exact amplitude expressions for acceleration waves arising from a sinusoidal boundary condition (BC) are obtained and their temporal evolution and propagation speeds determined. It is shown that amplitude blow up is possible only if the medium ahead is in motion, otherwise a constant amplitude wave occurs. Analytical results are illustrated numerically using simple computational methods/tools. Lastly, erroneous results that were reported in a recently published related study are noted and/or corrected.

We consider the turbulence of potential deep-water surface gravity waves, which have dispersion law Ω k = g k (g is the gravity constant, k=| k | ). It is well known that there are no three wave resonance interaction for such waves, and the lowest-order resonance involves four waves. We show the importance of almost resonance three wave interactions; in particular, they determine the life-span of these waves, provided the range of scales is large enough.

Existence of finite-amplitude inertia–gravity waves propagating without change of form is proved in the two-layer rotating shallow water model. We find exact solutions of the full equations of motion corresponding to such waves. The form of the nonlinear waves depends on the parameters of the model (density and height ratios of the layers) and on the phase speed. Three distinct families of waves having different forms and responding differently to the increase of amplitude are identified.

On a basis of the relativistic Vlasov–Maxwell equations for an electron unmagnetized collisionless plasma, a three-dimensional analysis of the nonlinear mixing of electromagnetic and longitudinal waves is performed. Using a conducting relativistic plasma model two waves generate a nonlinear current which is the source of a longitudinal plasma wave described by a nonlinear differential equation and by a nonlinear conductivity tensor. A dielectric relativistic plasma model is used to explain the nonlinear excitation of a transverse wave, as an effect of the nonlinear polarization, and described by a nonlinear differential equation and by a nonlinear susceptibility tensor.

Nonlinear resonant wave phenomena are treated. It is assumed that near and at the resonance first-order linear terms in perturbed wave equations annihilate each other. As a result, the perturbed wave equations reduce to basic highly nonlinear ordinary differential equation or the basic algebraic equation for traveling waves. These equations determine the evolution of smooth waves into shock waves. Then the jump curls and eventually breaks, nucleating drops, bubbles or vortices.

Parametric decay of an obliquely (to the magnetic field) propagating electron plasma wave to an obliquely propagating electron plasma sideband and a lower hybrid wave is studied. This decay process is then considered as a cascade channel for electron plasma waves to broaden their downshifted frequency spectra. The dependence of the threshold pump field on the number of cascade is determined.

Resonance quasi-levels in symmetrical double-barrier structures have been calculated and compared with bound states in quantum well. The relation between the resonance quasi-levels and bound states in corresponding quantum well has been discussed, and the discrepancy between them decreases with the increasing barrier width. The coupling effect between transverse and longitudinal wave vector was evaluated systematically. With the transverse wave vector increasing the quasi-levels depart form the values without considering coupling effect, and vanish at the higher value of longitudinal wave vector. Furthermore, we found that the coupling effect has more influence on the higher energy levels. In the end the visualization of coupling effect was plotted in 3D curves.

We investigate asymptotic equations describing small amplitude surface elastic waves in the half-plane (Rayleigh waves). For hyperelastic materials such model equations are Hamiltonian systems, and are seen to lead to the formation of singularities in the surface elastic displacement. At the time of singularity formation the Fourier spectra of the solutions exhibit power law decay, and the observed exponents suggest the existence of both differentiable and non-differentiable singular profiles.

We extend the quasi-continuum method of approximation to find waves of general speed in a diatomic lattice. Our results include as a special case the one speed which previous methods have found; we find the shape of nonlinear waves of arbitrary velocity up to some maximum speed.

We study the limiting behavior of smooth periodic waves for the Degasperis–Procesi equation as the parameters trend to some special values. Using an improved qualitative method, the existence of smooth periodic waves is considered. For some limiting values of parameters of smooth periodic waves, one obtains peaked solitary waves.

The multi-order exact solutions of the one-dimensional complex Ginzburg–Landau equation are obtained by the use of the wave packet theory. In these solutions, the zeroth-order exact solution is a plane wave; the first-order exact solutions are shock waves between the amplitude and the shift of phase, spiral waves between the shift of phase and distance or between the shift of phase and the amplitude; the second-order exact solutions are two periodic waves with certain conditions and their limits – the shock waves and solitary waves can also be found.

We prove that the solitary waves of a model for nonlinear dispersive waves in cylindrical compressible hyperelastic rods are orbitally stable. This establishes that the shape of the wave is stable.

Parametric surface waves in water or liquefiable layers are described by a wave-type equation. Approximate solutions of this equation are presented. These solutions describe unfamiliar waves which can have properties of both standing waves and travelling waves. They are a new kind of waves which are not d’Alembert-type waves. Analytic expressions for different periodical wave patterns in the x–t plane are found. Theoretical results are compared with recent experimental data. It is noted that the solutions may be valid for different wave fields of Nature. According to the theory an infinite quantity of localised modes may be generated in the fields due to any periodical parametric excitation. Sometimes these modes are indistinguishable from particles. Thus the equation which is also valid for springs and lattices has a rich variety of nonlinear solutions. This variety is huge in comparison with a variety of solutions of ordinary differential equations.

We reveal a mechanism of resonant excitation of non-dispersive coastal Kelvin waves by pairs of incident/reflected inertia–gravity waves in the rotating stratified fluid. In the simplest rotating shallow water model on the semi-infinite plane we show that the mechanism works for a continuum of incoming waves, and thus should be ubiquitous in the ocean. Subsequent slow evolution of thus excited Kelvin waves is governed by harmonically forced simple-wave equation and leads to nontrivial transport and mixing properties.

In the trans-resonant frequency band a balance between the nonlinear, spatial, dissipative and dispersive effects varies together with the excited frequency. Therefore nonlinear resonant waves may be excited in spherical resonators which it is difficult to classify as well known soliton- or conoidal- or shock- or breather-type waves. It follows from the presented theory that spatial and dissipative effects are very important for spherical waves. These effects can prevent the formation of diverging and converging spherical discontinuous shock waves. Oscillating localized waves focusing at the origin are responsible for the sonoluminescence at the exact resonance.

The time evolution of three-dimensional hydromagnetic waves in a nonideal magnetohydrodynamic (MHD) plasma is investigated. Two types of exact time-dependent wave solutions of the nonlinear resistive and dissipative MHD equations are found. One solution shows a damped Alfven wave and the other one shows a magnetically compressional wave with the magnetic and velocity fields having different damping times. It is shown that in the asymptotic limit, t → ∞, the plasma relaxes toward the equilibrium state with no magnetic field and constant (for the former solution) or sheared (for the latter solution) velocity.

The thin-flux-tube approximation developed by Zhugzhda [Phys. Plasmas 3 (1996) 10] allows one to describe analytically the weakly non-linear dynamics of long-wavelength sausage body magnetosonic waves in magnetic flux tubes. A Kortewegde Vries equation is derived for slow body waves. The weakly non-linear sausage body waves appear as narrowings of the flux tube, with rarefaction, cooling, and acceleration of a plasma flow due to the squeezing of the tube. The “breaking” of the non-linear slow body waves is discussed.

A new kind of solitary waves is found coexisting with bistable steady waves in driven drift-wave system. The energies of these steady waves distribute in the gap between that of the bistable steady waves. The gap state is a combined result of the carrier steady wave and the soliton-like structure of perturbation wave trapped by the carrier. The effect of trapped structure is to reduce the ‘potential well’ of carrier wave.

We develop a theory for surface electromagnetic waves at the boundary of a layer-periodical structure and a vacuum. The excitation of these waves by a charged particle, moving in vacuum and specularly reflected by the boundary, is investigated. The radiation field in vacuum is obtained. This field consists of fields of electromagnetic waves of a different physical nature: the surface cylindrical wave and the volume spherical wave, whose powers are of the same order as the power of the transition radiation.

A number of novel phenomena arising in systems with O(2) symmetry are described. These include pulsing traveling waves with either a periodically or quasiperiodically modulated phase velocity, and heteroclinic waves which alter their appearance with time, taking successively the form of pure traveling waves, standing waves and steady states. Chaotic waves can also be present. These states result from the codimension-two interaction between a reflection-breaking steady state bifurcation and a reflection-preserving Hopf bifurcation from a circle of nontrivial equilibria. The resulting waveforms are illustrated using a model of magnetoconvection with periodic boundary conditions.

The equation for the spectral intensity of small scale drift eigenmodes (cylindrical waves) for a radially inhomogeneous plasma is obtained. The induced scattering of these waves by bare ions is found to suppress the short waves with k⊥ϱs621 and to provide the growth of the long waves with k⊥ϱs

The influence of nonlinearity on surface elastic waves is studied in the framework of a simple lattice model for cubic crystals including nonlinear interactions between the first, second, and third neighbors. It is shown that in the model nonlinear effects are of the same order as the dispersion ones and they may change the penetration length of surface waves. In particular, nonlinearity may support surface waves not existing in linear theory. Analytical results are obtained in two limit cases: deeply penetrating nonlinear surface waves and nonlinear waves strongly localized at the crystal surface. It is predicted that surface waves may be totally localized at the surface layer when the nonlinearity is of the order of a certain critical value.

The coupling due to the van der Waals and the dissipative thermal interactions of surface waves propagating along a thin vacuum gap between nonpiezoelectric crystals is investigated. The propagation of the Rayleigh and the shear horizontal surface waves in coupled massive crystals and the slow bending waves in coupled thin films is considered. It is shown that the van der Waals attractive forces cause a negative shift of the velocity and the existence of a terminal point k 0 for the propagation of the long-wavelength Rayleigh and bending waves, in which the frequency of the coupled waves goes to zero. The anomalous absorption of the grazing bulk shear acoustic waves scattered from the gap is predicted.

The roots k perpendicular-to versus omega and k parallel-to of the electrostatic dispersion relation in a magnetized Maxwellian plasma in the frequency range that covers the first few ion-cyclotron harmonics, are followed when the external magnetic field is decreased down to arbitrarily low values and compared with the roots of the non-magnetic dispersion relation (the ion-acoustic roots). It is found that the "forward" branch of the magnetic case tends to the less damped ion-acoustic root and that the whole series of backward Bernstein waves become, in the limit, the second, third, etc., less damped waves in the ion-acoustic series.

In the context of one-dimensional nonlinear waves in plasmas, we derive an integrable model of coupled nonlinear equations for the description of the interaction of the Langmuir waves with the acoustic waves through the ponderomotive force. Our model includes nonlinearity and dispersion of the acoustic wave and accounts for a resonant scattering of the electrostatic wave. The integrability of the model allows us to understand the instability of the Langmuir wave as (i) a mutual trapping of the acoustic wave and the scattered Langmuir wave, (ii) an asymptotic time evolution towards the formation of a local singularity of the Langmuir wave (collapsing).

We present a cylindrically symmetric solution of the gravitational field equations which describes the propagation of rotating gravitational waves in empty space. The radiation field can be represented as a superposition of incoming and outgoing waves. The singularities of the rotating waves are examined and their significance for the speed-of-light catastrophe is discussed.