Traveling Plane Wave Research Articles

Traveling waves in a nonlinear medium with cubic nonlinearity

Unlike linear nondispersive media, which allow propagation of wave packets of arbitrary forms, nonlinear media admit only certain profiles of traveling waves. Here we examine media with Duffing oscillators, i.e., with bound electrons for which an equilibrium disturbance causes forces proportional to the first and third powers of deviation. We show that the linearly polarized traveling plane waves such media can transmit have profiles modulated as Jacobi elliptic functions. When discussing propagation across an interface between different media, only incidence from the side of the linear medium is considered. Even in this case, to launch a traveling wave in the nonlinear medium, a severe restriction must be imposed on the incident wave’s amplitude.

Axial radiation force of a Bessel beam on a sphere and direction reversal of the force

An expression is derived for the radiation force on a sphere placed on the axis of an ideal acoustic Bessel beam propagating in an inviscid fluid. The expression uses the partial-wave coefficients found in the analysis of the scattering when the sphere is placed in a plane wave traveling in the same external fluid. The Bessel beam is characterized by the cone angle beta of its plane wave components where beta=0 gives the limiting case of an ordinary plane wave. Examples are found for fluid spheres where the radiation force reverses in direction so the force is opposite the direction of the beam propagation. Negative axial forces are found to be correlated with conditions giving reduced backscattering by the beam. This condition may also be helpful in the design of acoustic tweezers for biophysical applications. Other potential applications include the manipulation of objects in microgravity. Islands in the (ka, beta) parameter plane having a negative radiation force are calculated for the case of a hexane drop in water. Here k is the wave number and a is the drop radius. Low frequency approximations to the radiation force are noted for rigid, fluid, and elastic solid spheres in an inviscid fluid.

On the evolution of two-dimensional patterns in an inviscid liquid sheet

We perform an analysis of the pattern formation for a moving sheet of inviscid fluid. The sheet, which is assumed to have an infinite horizontal extent, moves at some prescribed velocity into a passive surrounding gas. The sheet’s thickness is assumed much smaller than the horizontal scale of the fluid motion. By considering a system that is symmetric with respect to the horizontal planes, long scale asymptotics are used to reduce the full governing equations in three dimensions to a set of three coupled nonlinear partial differential equations for the horizontal components of the velocity field and the height of the interface profile. The interfacial conditions consisting of the kinematic and normal stress balance are incorporated into these evolution equations. Investigations are carried out as function of the sole dimensionless parameter, namely the Weber number. A small amplitude stability analysis around the planar gas–liquid interface reveals that wave patterns in the form of traveling plane waves occur subcritically, and are therefore unstable. The reduced evolution equations are solved numerically for fixed values of the Weber number. Since the reduced system of equations is homogeneous, the wave motion is generated by initial conditions. Five initial conditions have been imposed: one-dimensional rolls, two-dimensional squares, two-dimensional hexagons, two-dimensional ridges, and smooth peaks. The ensuing evolution of the liquid sheet’s shape and corresponding flow fields are described by illustrations of the changes in the sheet’s morphology with time.

Instability criteria and pattern formation in the complex Ginzburg-Landau equation with higher-order terms

We study the modulational instability and spatial pattern formation in extended media, taking the one-dimensional complex Ginzburg-Landau equation with higher-order terms as a perturbation of the nonlinear Schrödinger equation as a model. By stability analysis for the original partial differential equation, we derive its stability condition as well as the threshold for amplitude perturbations and we show how nonlinear higher-order terms qualitatively change the behavior of the system. The analytical results are found to be in agreement with numerical findings. Modulational instability mediates pattern formation through the lattice. The main feature of the traveling plane waves is its disintegration in pulse train during the propagation through the system.

Discrete nonlinear Schrödinger approximation of a mixed Klein–Gordon/Fermi–Pasta–Ulam chain: Modulational instability and a statistical condition for creation of thermodynamic breathers

We analyze certain aspects of the classical dynamics of a one-dimensional discrete nonlinear Schrödinger model with inter-site as well as on-site nonlinearities. The equation is derived from a mixed Klein–Gordon/Fermi–Pasta–Ulam chain of anharmonic oscillators coupled with anharmonic inter-site potentials, and approximates the slow dynamics of the fundamental harmonic of discrete small-amplitude modulational waves. We give explicit analytical conditions for modulational instability of travelling plane waves, and find in particular that sufficiently strong inter-site nonlinearities may change the nature of the instabilities from long-wavelength to short-wavelength perturbations. Further, we describe thermodynamic properties of the model using the grand-canonical ensemble to account for two conserved quantities: norm and Hamiltonian. The available phase space is divided into two separated parts with qualitatively different properties in thermal equilibrium: one part corresponding to a normal thermalized state with exponentially small probabilities for large-amplitude excitations, and another part typically associated with the formation of high-amplitude localized excitations, interacting with an infinite-temperature phonon bath. A modulationally unstable travelling wave may exhibit a transition from one region to the other as its amplitude is varied, and thus modulational instability is not a sufficient criterion for the creation of persistent localized modes in thermal equilibrium. For pure on-site nonlinearities the created localized excitations are typically pinned to particular lattice sites, while for significant inter-site nonlinearities they become mobile, in agreement with well-known properties of localized excitations in Fermi–Pasta–Ulam-type chains.

Combustion waves in a model with chain branching reaction

In this paper the steady planar travelling waves in the adiabatic model with two-step chain branching reaction mechanism are investigated numerically. The properties of these solutions are demonstrated to have similarities with the properties of non-adiabatic combustion waves that is, there is a residual amount of fuel left behind the travelling waves and the solutions can exhibit extinction. It is also shown that the model possesses a new type multiple travelling wave solutions (which we call wave trains) with complex structure of the profiles and varying speeds

Plane wave decomposition in the unit disc: Convergence estimates and computational aspects

This paper deals with the numerical simulation of time-harmonic wave fields using progressive plane waves. It is shown that a plane wave travelling in arbitrary direction can be numerically recovered with an accuracy of the order of the machine precision with a collocation formulation and the square root of the machine precision with a least-square formulation. However, strongly evanescent and nearly singular wave fields cannot be properly recovered with standard double-precision floating-point arithmetic. Some of the ideas are applied to the elastic wave equation and a simple optimization algorithm is proposed to find a good compromise between the accuracy and the number of plane waves.

Modeling an emittance-dominated elliptical sheet beam with a [formula omitted]-dimensional particle-in-cell code

Modeling a 3-dimensional (3-D) elliptical beam with a 2 1 2 -D particle-in-cell (PIC) code requires a reduction in the beam parameters. The 2 1 2 -D PIC code can only model the center slice of the sheet beam, but that can still provide useful information about the beam transport and distribution evolution, even if the beam is emittance dominated. The reduction of beam parameters and resulting interpretation of the simulation is straightforward, but not trivial. In this paper, we describe the beam parameter reduction and emittance issues related to the initial beam distribution. As a numerical example, we use the case of a sheet beam designed for use with a planar traveling-wave amplifier for high power generator for RF ranging from 95 to 300 GHz [Carlsten et al., IEEE Trans. Plasma Sci. 33 (2005) 85]. These numerical techniques also apply to modeling high-energy elliptical bunches in RF accelerators.

Bifurcation of a helical wave from a traveling wave

Spin mode instabilities have been observed in solid-state combustion reactions and in propagating fronts of polymerization among others, that is, when some conditions are changed, a planar traveling wave may lose its stability and there appears a planar wave or a helical wave propagating in the form of spiral encircling the cylindrical sample with several reaction spots. In the present paper, we study the existing condition of stable helical waves and the transition process of wave patterns from a static mode to pulsating and/or helical modes. By using a full system of reaction-diffusion equations, we clarify that a stable helical wave can bifurcate directly from a planar traveling wave and that helical waves with different numbers of reaction spots can coexist stably.

Theory of gaseous detonations.

The objective of the present paper is to review some developments that have occurred in detonation theory over the last ten years. They concern nonlinear dynamics of detonation fronts, namely patterns of pulsating and/or cellular fronts, selection of the cell size, dynamical self-quenching, direct (blast) or spontaneous initiation, and transition from deflagration to detonation. These phenomena are all well documented by experiments since the sixties but remained unexplained until recently. In the first part of the paper, the patterns of cellular detonations are described by an asymptotic solution to nonlinear hyperbolic equations (reactive Euler equations) in the form of unsteady (sometime chaotic) and multidimensional traveling-waves. In the second part, turning points of quasi-steady solutions are shown to correspond to critical conditions of fully unsteady problems, either for (direct or spontaneous) initiation or for spontaneous failure (self-quenching). Physical insights are tentatively presented rather than technical aspects. The challenge is to identify the physical mechanisms with their relevant parameters, and more specifically to explain how the length-scales involved in detonation dynamics are larger by two order of magnitude (at least) than the length-scale involved in the steady planar traveling-wave solution (detonation thickness).

Transmission of dipole radiation through interfaces and the phenomenon of anti-critical angles.

Radiation emitted by an electric dipole consists of traveling and evanescent plane waves. Usually, only the traveling waves are observable by a measurement in the far field, since the evanescent waves die out over a length of approximately a wavelength from the source. We show that when the radiation is passed through an interface with a medium with an index of refraction larger than the index of refraction of the embedding medium of the dipole, a portion of the evanescent waves are converted into traveling waves, and they become observable in the far field. The same conclusion holds when the waves pass through a layer of finite thickness. Waves that are transmitted under an angle larger than the so-called anti-critical angle theta (1) ac are shown to originate in evanescent dipole waves. In this fashion, part of the evanescent spectrum of the radiation becomes amenable to observation in the far field. We also show that in many situations the power in the far field coming from evanescent waves greatly exceeds the power originating in traveling waves.

Stability in a two-dimensional combustion model

We study the stability of a planar travelling wave in the two-dimensional NEF-combustion model when the reduced Lewis number is equal to zero. The functional analytic setting consists of spaces

A new method to predict the evolution of the power spectral density for a finite-amplitude sound wave.

A method to predict the effect of nonlinearity on the power spectral density of a plane wave traveling in a thermoviscous fluid is presented. As opposed to time-domain methods, the method presented here is based directly on the power spectral density of the signal, not the signal itself. The Burgers equation is employed for the mathematical description of the combined effects of nonlinearity and dissipation. The Burgers equation is transformed into an infinite set of linear equations that describe the evolution of the joint moments of the signal. A method for solving this system of equations is presented. Only a finite number of equations is appropriately selected and solved by numerical means. For the method to be applied all appropriate joint moments must be known at the source. If the source condition has Gaussian characteristics (it is a Gaussian noise signal or a Gaussian stationary and ergodic stochastic process), then all the joint moments can be computed from the power spectral density of the signal at the source. Numerical results from the presented method are shown to be in good agreement with known analytical solutions in the preshock region for two benchmark cases: (i) sinusoidal source signal and (ii) a Gaussian stochastic process as the source condition.

Propagation of a planar flame front studied by molecular dynamics

In this paper we study the propagation of a planar traveling wave in a nonisothermal autocatalytic reaction diffusion system. This is done by performing a series of molecular dynamics simulations for different densities and heat of reactions. In the special case where the reaction is isothermal, it is shown that the molecular dynamics simulation qualitatively resembles the approximative macroscopic description that is based on the reaction diffusion equation. Simulations for different densities and heat releases further show that the speed of propagation is not simply the minimum speed of the corresponding macroscopic equation and that the speed is always constant even though the Lewis number is greater than unity.

A multi-channel correlation method detects traveling γ-waves in monkey visual cortex

Correlations among simultaneously recorded signals are mostly analyzed pairwise and include temporal averaging. However, pairwise methods are not suitable for characterizing relationships among multiple channels for signals which vary temporally in an unpredictable way. Here we develop a time-resolved spatio-temporal correlation (STC) measure among simultaneously recorded signals. We demonstrate the capabilities of the method with artificial data sets and with multiple-channel recordings from striate cortex of awake monkeys. We concentrate on correlations in the γ-frequency range (γ: 30–90 Hz) because they were prominent in the analyzed recordings and gained high interest in the recent years due to their assumed role in associative processing, including perceptual binding. Former analyses of γ-activities in visual cortex, using pairwise correlation methods, mostly revealed zero-delay correlation, indicating synchrony. In cat and monkey visual cortex this γ-synchrony is restricted to 1.5–3.0 mm (half-height decline). However, our spatio-temporal correlation (STC)-method demonstrates for striate cortex from awake monkeys that γ-synchrony is a local phenomenon of more global traveling plane waves that appear stimulus-induced at randomly varying orientations. These γ-waves are coupled over much larger cortical distances (≈7 mm half-height decline) than the γ-synchrony ranges obtained by pairwise correlation analyses from the same data. Our STC-method therefore suggests that the previously reported results of short-range and zero-delay correlations were often due to temporal averaging of traveling γ-waves.

Spatial separation of the traveling and evanescent parts of dipole radiation.

Electric dipole radiation consists of traveling and evanescent plane waves. When radiation is detected in the far field, only the traveling waves will contribute to the intensity distribution, as the evanescent waves decay exponentially. We propose a method to spatially separate the traveling and evanescent waves before detection. It is shown that when the radiation passes through an interface, evanescent waves can be converted into traveling waves and can subsequently be observed in the far field. Let the radiation be observed under angle theta(t) with the normal. Then there exists an angle theta(ac) such that for 0 < or = theta(t) < theta(ac) all intensity originates in traveling waves, whereas for theta(ac) < theta(t) < pi/2 only evanescent waves contribute. It is shown that with this technique and under the appropriate conditions almost all far-field power can be provided by evanescent waves.

Explicit and exact travelling plane wave solutions of the (2+1)-dimensional Boussinesq equation

The deformation mapping method is applied to solve a system of (2+1)-dimensional Boussinesq equations. Many types of explicit and exact travelling plane wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and others exact solutions, are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and the cubic nonlinear Klein-Gordon equation.

Stability in a two-dimensional free boundary combustion model

We prove rigorously the stability of the planar travelling wave solution to a free boundary system in R 2 , arising as a model in combustion theory, for appropriate values of the reduced Lewis number. After changements of coordinates and unknowns, the problem is reduced to the stability of the null solution of a fixed boundary parabolic system. The difficulties consist in the fact that the final system is fully nonlinear, and the spectrum of the linearized system contains the halfline (−∞,0]. They are overcome setting the problem in a suitably weighted Hölder space.

Discrete nonlinear Schrödinger equation with defects.

We investigate the dynamical properties of the one-dimensional discrete nonlinear Schrödinger equation (DNLS) with periodic boundary conditions and with an arbitrary distribution of on-site defects. We study the propagation of a traveling plane wave with momentum k: the dynamics in Fourier space mainly involves two localized states with momenta +/-k (corresponding to a transmitted and a reflected wave). Within a two-mode ansatz in Fourier space, the dynamics of the system maps on a nonrigid pendulum Hamiltonian. The several analytically predicted (and numerically confirmed) regimes include states with a vanishing time average of the rotational states (implying complete reflections and refocusing of the incident wave), oscillations around fixed points (corresponding to quasi-stationary states), and, above a critical value of the nonlinearity, self-trapped states (with the wave traveling almost undisturbed through the impurity). We generalize this treatment to the case of several traveling waves and time-dependent defects. The validity of the two-mode ansatz and the continuum limit of the DNLS are discussed.

Wave front tracing and asymptotic stability of planar travelling waves for a two-dimensional shallow river model

The propagation of surface water waves in a frictional channel with a uniformly inclined bed is governed by a two-dimensional shallow river model. In this paper, we consider the time-asymptotic stability of weak planar travelling waves for a two-dimensional shallow river model with Darcy's law. We derive an effective parabolic equation to analyze the wave front motion. By employing weighted energy estimates, we show that weak planar travelling waves are time-asymptotically stable under sufficiently small perturbations.