Form Of Traveling Waves Research Articles

Interaction of harmonic plane waves with a mobile elastic phase boundary in anti-plane shear

This paper is concerned with the effect of sustained infinitesimal harmonic plane wave excitation of a phase boundary in a non-linearly elastic material that is subject to anti-plane shear deformation. The phase boundary is capable of motion that is here described by a harmonic travelling waveform. The reflected wave is also a harmonic plane wave, however the transmitted wave may be either in the form of a harmonic plane wave or a harmonic surface wave. The phase boundary motion is determined on the basis of a standard kinetic relation that involves a single mobility parameter. This gives phase boundary motion that is synchronized with the incident wave for the case of a transmitted plane wave, but is not synchronized with the incident plane wave for the case of a transmitted surface wave. A certain fraction of the energy provided by the incident wave is dissipated by phase boundary motion in a fashion that can be explicitly quantified. Special incident angles can suppress the reflected wave, suppress the transmitted wave or cause the dissipation to vanish.

Extremal properties of random trees.

We investigate extremal statistical properties such as the maximal and the minimal heights of randomly generated binary trees. By analyzing the master evolution equations we show that the cumulative distribution of extremal heights approaches a traveling wave form. The wave front in the minimal case is governed by the small-extremal-height tail of the distribution, and conversely, the front in the maximal case is governed by the large-extremal-height tail of the distribution. We determine several statistical characteristics of the extremal height distribution analytically. In particular, the expected minimal and maximal heights grow logarithmically with the tree size, N, h(min) approximately v(min) ln N, and h(max) approximately v(max) ln N, with v(min)=0.373365ellipsis and v(max)=4.31107ellipsis, respectively. Corrections to this asymptotic behavior are of order O(ln ln N).

Traveling waves, front selection, and exact nontrivial exponents in a random fragmentation problem.

We study a random bisection problem where an interval of length x is cut into two random fragments at the first stage, then each of these two fragments is cut further, etc. We compute the probability P(n)(x) that at the nth stage, each of 2(n) fragments is shorter than 1. We show that P(n)(x) approaches a traveling wave form, and the front position x(n) increases as x(n) approximately n(beta)rho(n) for large n with rho = 1.261 076ellipsis and beta = 0.453 025ellipsis. We also solve the m-section problem where each interval is broken into m fragments and show that rho(m) approximately m/(lnm) and beta(m) approximately 3/(2lnm) for large m. Our approach establishes an intriguing connection between extreme value statistics and traveling wave propagation in the context of the fragmentation problem.

Optical gap solitons: Past, present, and future; theory and experiments.

Optical gap solitons refer to nonlinear waves propagating in optical fibers whose linear refractive index has a periodic variation. Stationary gap solitons came to light first in 1987 [Chen and Mills, Phys. Rev. Lett. 58, 160 (1987)]; two years later, they re-emerge in Christodoulides and Joseph [Phys. Rev. Lett. 62, 1746 (1989)] and are first extended to a more general traveling wave form in Aceves and Wabnitz [Phys. Lett. A 141, 37 (1989)]. But it was not until seven years later, that the first experimental demonstration [Eggleton et al., Phys. Rev. Lett. 76, 1627 (1996); J. Opt. Soc. Am. B 14, 2980 (1997)] was reported. Since then, there has been an increase in the study of the dynamics and applications of such solitons. This paper is a brief survey of some of the ongoing and future research on optical gap solitons. (c) 2000 American Institute of Physics.

Global Travelling Waves in Reaction–Convection–Diffusion Equations

We study the existence and properties of solutions in travelling wave form, u(x, t)=φ(x−st), defined for every z=x−st∈R, for the reaction-convection- diffusion equationut=a(um)xx+b(un)x+kupforx∈R,t>0,with a, m, n>0; b, k, p∈R. In the reaction case k>0 we prove that there exist travelling waves vanishing for z→∞ if and only if b>0 and min{1, n}⩽m+p2⩽max{1, n},p⩽max{1, n}.Moreover, if m+p≠2n, there exists a minimal velocity s*(a, b, k, m, n, p)>0, for which there are travelling waves only with s⩾s*, while in the case m+p=2n there are travelling waves only when 4amk⩽b2n and for every velocity s⩾0. Some properties of the function s* are established. All the waves are decreasing in their support and waves having bounded support from the right exist if and only if m>min{1, n}. Also, the absorption case k<0 is treated, where we find that, for different values of the parameters, there exists a unique travelling wave for every velocity s∈R, but for some case where only negative velocities exist. The cases b=0 or k=0 are well known in the literature.

Travelling wave behaviour for a Porous-Fisher equation

We study the propagation properties of the reaction-diffusion equation of Fisher typewith p&lt;1&lt;m. Taking into account that solutions of the Cauchy problem are nonunique if m+p⩾2, we prove that the minimal solutions in this case tend to propagate with certain minimal speed c∗(m, p). More precisely, if we translate any solution with a velocity ∣c∣&lt;c∗, we get the limit in time value one, and if the initial value u(·, 0) vanishes, say, for x⩾0, the minimal solution translated with velocity c&gt;c∗ tends to zero. Also, an interface appears for the minimal solutions, whose asymptotic velocity is c∗. This behaviour depends upon the existence of special solutions of travelling wave form. Travelling waves have been widely studied for diffusion equations related with the above. We characterize here the minimal velocity c∗ for which travelling waves exist, as an analytic function of the parameters m and p, for every m+p⩾2, by viewing it as an anomalous exponent. Some local properties of the minimal solutions and their interfaces in the case m+p⩾2 are also proved.

The evolution of travelling waves in reaction-diffusion equations with monotone decreasing diffusivity. II. Abruptly vanishing diffusivity

In this paper we continue our study of some of the qualitative features of chemical polymerization processes by considering a reaction-diffusion equation for the chemical concentration in which the diffusivity vanishes abruptly at a finite concentration. The effect of this diffusivity cut-off is to create two distinct process zones; in one there is both reaction and diffusion and in the other there is only reaction. These zones are separated by an interface across which there is a jump in concentration gradient. Our analysis is focused on both the initial development of this interface and the large time evolution of the system into a travelling wave form. Some distinct differences from our previous analysis of smoothly vanishing diffusivity are found.

Approximate traveling wave solution for impulsively perturbed circular membrane

It is instructive to transform the normal modes solution for a circular membrane (radius R) suddenly displaced ξ0 at the boundary into an approximate purely periodic traveling waveform ξ=ξ0Σ ×An√r/R cos[(n −1/4)π(r/R ± ct/R) − π/4], where c=wave speed (although strictly good for n≫1, r/R≪1). The series evidently has period T=8R/c. Individual mode periods T/3,T/7,T/11,... account for the notable characteristic symmetry of the waveform about the center of the periods T and asymmetry about the quarter points. The −π/2 focal phase shift (finite Hilbert transform) and the simple sign reversal (±π shift) at the boundary are readily deduced. The result also applies to ideal supersonic jets, ξimplies pressure perturbation, ctimplies (axial distance) √M2−1. For cell length (axial period) one may take sM = 8R√M2−1 for the series, or sP =8/3R√M2−1 for the first term. The two‐dimensional case is unambiguous sM=4(semiwidth) √M2−1. Of course, the exact solution is not periodic at all, the major discrepancy being in fir...

Wave-number spectrum and normal mode solutions for sound scattering from internally loaded cylindrical shells

In a companion paper [L. B. Felsen and Y. P. Guo, J. Acoust. Soc. Am. 94, ⧠⧠ (1993)], two-dimensional acoustic wave scattering from a thin elastic cylindrical shell with internal loading has been expressed in terms of a hybrid ray-mode algorithm that combines ray-type traveling wave fields in the fluid with resonant modes in the structure. The algorithm is based on decomposition of the overall problem into a background problem for the empty shell, on which the internal loads are accounted for by induced equivalent forces at the load attachment points. In the numerical implementation, it was found convenient to express the induced forces due to the internal loading and consequent modal resonances in harmonic series form. This use of the angular harmonics solution for the coupling forces is not only computationally efficient, but also reveals the mechanisms that are responsible for many features observed in the far field. Nevertheless, it is of interest to formulate and solve the problem entirely in a traveling-wave-based representation, which is the counterpoise to the previous solution strategy based entirely on angular harmonics [Y. P. Guo, ‘‘Sound scattering from an internally loaded cylindrical shell,’’ J. Acoust. Soc. Am. 91, 926–938 (1992)]. The traveling wave form isolates individual scattering events and charts the evolution of resonances by successive multiple interactions. Poisson summation is shown to transform one solution into the other, thereby exhibiting for each wave constituent the traveling versus standing wave alternatives.

Pattern of Travelling Wave Form in Non-even Coupled Map Lattice Models

In the research of high dimension dynamical system, the coupled map lattice models (CML) have been used to study the spatio-temporal complexity recently. Since the research system itself is chaotic, we can consider the influence of low dimension chaos on high

The simultaneous process of filament winding and curing for polymer composites

The simultaneous process of filament winding and curing a composite cylindrical shell is considered. From the macrokinetic point of view it is characterized by the small thickness of the relative effective reaction zone. The thermochemical conditions in manufacturing are governed by the following process variables: speed of winding, initial conversion and temperature of the filament bundle wound, and the heating applied. A thermochemical model is developed for simulation of the temperature and conversion distributions during the process. An approximate solution of the thermochemical model in the form of the constant traveling wave is derived for constant values of process variables and a thermally isolated mandrel. A non-uniform cure stress model of the process is developed for the case of using elastic orthotropic constitutive relations for a composite material. It is found that the thickness of the relative effective reaction zone drastically influences the cure stresses. The dependence of the stress level on the process variables is analyzed.

Stability of steady combustion of low-gas composition with heat and mass transfer through the lateral surface of a cylindrical sample

The mathematical model of low-gas condensed systems permits the investigation of a wide range of frontal processes of exothermal conversion of porous condensed compositions with partially gasifying reaction products. However, this model, like the phenomenon itself, is complex, nonlinear, and multiparametric. Lacking the means to obtain an analytical solution in traveling-wave form and establish the limits of its stability, the influence of various determining factors on the limit of stability has been investigated in the present work. This provides a qualitative picture of the influence of the LGC parameters on the stability of steady combustion and permits rational control of the combustion conditions.

Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion

We consider the evolution of a gasless combustion front. We derive coupled complex Ginzburg-Landau type equations for the amplitudes of waves along the front as functions of slow temporal and spatial variables. The equations are written in characteristic variables and involve averaged terms which reflect the fact that in the slowest time scale, the effect on one wave, of a second wave traveling with the group velocity in the opposite direction on the intermediate time scale, enters only through its average. Solutions of the amplitude equations in the form of traveling, standing, and quasiperiodic waves are found, and regions of stability for these solutions are determined. In particular we find that the traveling and quasiperiodic (including standing) waves are not stable simultaneously. Finally, we observe that the stability analysis for coupled complex Ginzburg-Landau equations with averaged terms differs from that for coupled complex Ginzburg-Landau equations with the averages removed.

Self-Excited Oscillations of a Circular Disk Rotating in Air.

Self-excited oscillations are often observed in a circular disk when it is rotated in air. Theoretical as well as experimental studies are conducted on these oscillations. As a preparatory study, free vibrations of the disk are discussed theoretically without taking into account the effect of air. Then, utilizing the results of this preparatory study, and taking into account the effect of air, free vibrations of the disk are studied. It is shown that self-excited oscillations can occur at a rotating speed higher than a certain value in the form of abackward traveling wave. Experiments confirm the theoretical analysis.

Self-Excited Oscillations of a Circular Disk Rotating in Air.

Self-excited oscillations are often observed in a circular disk when it is rotated in air. Theoretical as well as experimental studies are conducted on these oscillations. As a preparatory study, free vibrations of the disk are discussed theoretically without taking into account the effect of air. Then, utilizing the results of this preparatory study, and taking into account the effect of air, free vibrations of the disk are studied. It is shown that self-excited oscillations can occur at a rotating speed higher than a certain value in the form of abackward traveling wave. Experiments confirm the theoretical analysis.

Joint inversion of travel times and waveforms by means of annealing algorithms. Application to a seismic refraction experiment

An algorithm of annealing is applied to a joint inversion of travel times and waveforms belonging to a synthetically generated seismic refraction experiment. The medium (crust and upper mantle) is modelled by a set of plane stratified layers and a halfspace. The obtained structure (elastic parameters and depth of layers) shows that, in spite of contaminating seismic noise and poor knowledge of the seismic source, annealing methods are a good tool in these kinds of inversion problems. We think that many characteristics of the annealing process described here could be used with real data and more sophisticated media for the crust and upper mantle of the earth than the present example.

Traveling waves and chaos in thermosolutal convection.

Numerical experiments on two-dimensional thermosolutal convection reveal oscillations in the form of traveling, standing, modulated, and chaotic waves. Transitions between these wave forms and steady convection are investigated and compared with theory. Such rich nonlinear behavior is possible in fluid layers of wide horizontal extent, and provides an explanation for waves observed in recent laboratory experiments with binary fluid mixtures.

Electromagnetic Field-to-Wire Coupling in the SHF Frequency Range And Beyond

Methods have been developed for the treatment of field-to-wire coupling at superhigh frequencies (SHF's) and beyond. In this region, transmission-line lengths and wire separations become very large electrically, and field-to-wire coupling problems become intractable. Several types of transmission lines have been examined in the SHF range, including uniform and nonuniform transmission lines, coaxial transmission lines, and multiconductor transmission lines. Current distributions are found to be predominantly of a standing-wave or traveling-wave form. The higher order modes are not significant for those cases examined. Bounds may be obtained for induced currents in the field-to-wire problem by considering the transmission line as a receiving antenna. The maximum value of induced current under conditions of maximum susceptibility is shown to vary little with frequency.

Long-time response of a finite cantilever plate to antisymmetric dynamic surface loading

This paper treats the transient response of a finite, isotropic, homogenous, elastic cantilever plate in a state of plane strain to an antisymmetric surface line load, which is assumed to be a step function in time. This analysis is based on the method given by J. Miklowitz for solving non-separable waveguide problems by using a double Laplace transform and an entirety condition on the solution. The corner stress singularities are considered in the evaluation of the stress distribution at the fixed end. Then, the near field solution is found by means of asymptotic expansion. Moreover, the transverse displacement along the plate is obtained in the travelling wave form as well as in the vibrational one, which is evaluated numerically and discussed. It is concluded that the engineering methods in which a “dynamic load factor” in used in conjunction with the static solution tend to underestimate the values of the deflections beyond the point of load up to the free end of the cantilever plate.

Nonlinear electrostatic surface waves in a semi-infinite plasma

The nonlinear equations which govern the filamentation of a large-amplitude high-frequency electrostatic surface wave are studied considering solutions in the form of stationary traveling waves. It is shown that solitary surface waves exist at certain amplitude and phase conditions.