Traveling Plane Wave Research Articles

Instability of planar traveling waves in bistable reaction-diffusion systems

This paper is concerned with the stability of a planar traveling wave in a cylindrical domain. The equation describes activator-inhibitor systems in chemistry or biology. The wave has a thin transition layer and is constructed by singular perturbation methods. Let $\varepsilon$ be the width of the layer. We show that, if the cross section of the domain is narrow enough, the traveling wave is asymptotically stable, while it is unstable if the cross section is wide enough by studying the linearized eigenvalue problem. For the latter case, we study the wavelength associated with an eigenvalue with the largest real part, which is called the fastest growing wavelength. We prove that this wavelength is $O(\varepsilon^{1/3})$ as $\varepsilon$ goes to zero mathematically rigorously. This fact shows that, if unstable planar waves are perturbed randomly, this fastest growing wavelength is selectively amplified with as time goes on. For this analysis, we use a new uniform convergence theorem for some inverse operator and carry out the Lyapunov-Schmidt reduction.

A free boundary problem stemmed from combustion theory. Part I: Existence, uniqueness and regularity results

We deal with a free boundary problem, depending on a real parameter λ, in a infinite strip in R 2 , which admits a planar travelling wave solution for every λ∈ R . We prove existence, uniqueness and regularity results for the solutions near the travelling wave.

Standing wave instabilities, breather formation and thermalization in a Hamiltonian anharmonic lattice

Modulational instability of travelling plane waves is often considered as the first step in the formation of intrinsically localized modes (discrete breathers) in anharmonic lattices. Here, we consider an alternative mechanism for breather formation, originating in oscillatory instabilities of spatially periodic or quasiperiodic nonlinear standing waves (SWs). These SWs are constructed for Klein-Gordon or Discrete Nonlinear Schrodinger lattices as exact time periodic and time reversible multibreather solutions from the limit of uncoupled oscillators, and merge into harmonic SWs in the small-amplitude limit. Approaching the linear limit, all SWs with nontrivial wave vectors (0 < Q < π) become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. The dynamics resulting from these instabilities is found to be qualitatively different for wave vectors smaller than or larger than π/2, respectively. In one regime persisting breathers are found, while in the other regime the system thermalizes.

Wave attenuation attributes as flow unit indicators

In this article we present a model-based interpretation algorithm to predict attenuation signatures associated with a reservoir's structure and its intrinsic properties. Scattering losses are caused by multiple reflections associated with the impedance contrast between different reservoir boundaries. Intrinsic losses are due to the fluids in the reservoir, the rock fabric, and the anisotropy due to vertical fractures. Because amplitude data are sensitive to these reservoir characteristics, we calculate the wave attenuation response of the reservoir in the frequency range of surface seismic, VSP, crosswell seismic, and borehole sonic. A model of a plane wave traveling in a poroelastic multilayer earth medium allows us to include anisotropy. The multilayer part of the model simulates elastic scattering; the poroelastic part of the model simulates fluid-flow effects in the reservoir; and the anisotropic part of the model simulates reservoir fractures.

Combustion Fronts in Porous Media

The equations governing combustion in situ in petroleum reservoirs are cast in a way that is appropriate to the study of the time asymptotic fundamental waves: rarefactions and shocks. The physical diffusion terms are maintained so that the internal structure of shocks can be analyzed. In this work, we determine the planar traveling wave solutions for the system of three evolution equations modeling combustion in two phase flow, obtained by neglecting heat losses and volume change effects. The chemical reaction rate is governed by Arrhenius' reaction rate law. Our analysis is intended to capture the main effect of combustion: the reduction of oil viscosity due to temperature increase and its practical result---the increase in oil recovery. It also is intended to resolve the difficulty introduced by the singularity in the Arrhenius reaction rate law. Our analysis shows that the combustion wave is represented by a heteroclinic connection in a nonhyperbolic system of three ordinary differential equations. It follows that numerical simulations need to resolve completely the small parabolic terms in the model to be reliable. The analysis also indicates that the phase diagram should be stable under perturbations of interest. Thus, the general structure of combustion waves in a more realistic model should be similar to the one we found here.

D13 共鳴管を使った熱音響スターリングエンジン

A thermoacoustic Stirling Engine with a high efficiency needs both the traveling wave phase and high acoustic impedance. However, the acoustic impedance of a freely traveling plane wave is fixed at the ρδ, resulting in significant viscous energy losses due to high acoustic velocities. So we focus on a position in a resonance tube, where both a high acoustic impedance and a traveling wave phase are formed. We put a regenerator stacked with many screen meshes into this paticular position and externally imposed steep temperature gradients along the regenerator. We succeed in the amplification and attenuation of sound intensity depending on the sign of the temperature gradient.

Spatially extended sound equalization in rectangular rooms.

The results of a theoretical study on global sound equalization in rectangular rooms at low frequencies are presented. The zone where sound equalization can be obtained is a continuous three-dimensional region that occupies almost the complete volume of the room. It is proved that the equalization of broadband signals can be achieved by the simulation of a traveling plane wave using FIR filters. The optimal solution has been calculated following the traditional least-squares approximation, where a modeling delay has been applied to minimize reverberation. An advantage of the method is that the sound field can be estimated with sensors placed in the limits of the equalization zone. As a consequence, a free space for the listeners can be obtained.

Helical localized wave solutions of the scalar wave equation

A right-handed helical nonorthogonal coordinate system is used to determine helical localized wave solutions of the homogeneous scalar wave equation. Introducing the characteristic variables in the helical system, i.e., u = zeta - ct and v = zeta + ct, where zeta is the coordinate along the helical axis, we can use the bidirectional traveling plane wave representation and obtain sets of elementary bidirectional helical solutions to the wave equation. Not only are these sets bidirectional, i.e., based on a product of plane waves, but they may also be broken up into right-handed and left-handed solutions. The elementary helical solutions may in turn be used to create general superpositions, both Fourier and bidirectional, from which new solutions to the wave equation may be synthesized. These new solutions, based on the helical bidirectional superposition, are members of the class of localized waves. Examples of these new solutions are a helical fundamental Gaussian focus wave mode, a helical Bessel-Gauss pulse, and a helical acoustic directed energy pulse train. Some of these solutions have the interesting feature that their shape and localization properties depend not only on the wave number governing propagation along the longitudinal axis but also on the normalized helical pitch.

Linear stability of propagating phase boundaries in capillary fluids

Dynamical phase changes occurring along planar traveling waves in capillary fluids are considered. Their multi-dimensional stability, which is encoded in the spectrum of a third order variational differential operator, is addressed by two approaches. On one hand, an original energy method is used to show that these diffuse interfaces are weakly stable. More precisely, the spectrum of the involved operator is found to coincide with the imaginary axis. On the other hand, an Evans function technique reveals a bifurcation phenomenon about zero, the interpretation of which still needs some clarification. It is the counterpart of the surface waves exhibited in an earlier work for the corresponding sharp interfaces.

A critical case of stability in a free boundary problem

We study the stability of the planar travelling wave solution to a free boundary problem for the heat equation in the whole \( \Bbb R^2 \) . We turn the problem into a fully nonlinear parabolic system and establish a stability result which is the proper generalization of the one-dimensional case. The curvature terms contribute a gradient squared corresponding to critical growth. The latter is eliminated by means of the Hopf-Cole transformation.

Chaotic dynamics of coupled transverse-longitudinal plasma oscillations in magnetized plasmas.

The propagation of intense electromagnetic waves in cold magnetized plasma is tackled through a relativistic hydrodynamic approach. The analysis of coupled transverse-longitudinal plasma oscillations is performed for traveling plane waves. When these waves propagate perpendicularly to a static magnetic field, the model is describable in terms of a nonlinear dynamical system with 2 degrees of freedom. A constant of motion is obtained and the powerful classical mechanics methods can be used. A new class of solutions, i.e., the chaotic solutions, is discovered by the Poincaré surface of sections. As a result, coupled transverse-longitudinal plasma oscillations become aperiodically modulated.

Instabilities in a Two-Dimensional Combustion Model with Free Boundary

We prove instability of the planar travelling wave solution in a two-dimensional free boundary problem modelling the propagation of near- equidiffusional premixed flames in the whole plane. We reduce the problem to a fixed boundary fully nonlinear parabolic system. The spectrum of the linearized operator contains an interval [0,ω c ], ω c > 0, so we cannot construct backward solutions. We use an argument about stability of dynamical systems in Banach spaces in order to prove pointwise instability of the moving front.

A three-dimensional scattering model for fading channels in land mobile environment

Clarke's (1968) scattering model, one of the most widely accepted channel models for the land mobile environment, is a two-dimensional (2-D) model because of the assumption of horizontal traveling plane waves. By introducing a nonzero elevation angle of the arriving wave, a three-dimensional (3-D) model is more general and accurate, especially for the urban environment. Though the concept of the 3-D model has been proposed already, there is still a lack of satisfying results regarding the distribution of the elevation angle, from both theoretical analysis and field measurement, and the power spectral density (PSD) of the received signal. In this paper, a family of functions with two parameters, {m,n}, where m and n are positive integers, for both the symmetrical and asymmetrical probability density function (PDF) of the elevation angle (EA), is proposed. Among these functions, those with odd m and n lead to analytical solutions of the PSD of the received signal in addition to satisfying other requirements for a PDF of the EA previously proposed in literature. The PSDs in closed form associated with m and n equal to one and three are derived in particular, and the autocorrelation functions are obtained numerically. Since a family of functions rather than a single function is proposed for the PDF of the EA, it provides certain flexibility in application and covers a wide range of environments. Another contribution of this paper is a new expression which directly relates the PDF of the EA in the 3-D model to the PSD of the received signal.

Velocity and Stability of Solitary Planar Travelling Wave Solutions of Intracellular [Ca]2+

Normal cardiac muscle contraction occurs in response to a rapid rise followed by a slower decay in intracellular calcium concentration. When cardiac muscle cells are loaded with calcium, an intracellular store releases calcium into the cytosol by the process of calcium-induced calcium release (CICR). This release contributes to the rise in intracellular calcium which in turn triggers contraction. We use two qualitative piecewise linear reaction-diffusion models of this behaviour to investigate the speed, stability and waveform of plane waves using singular perturbation techniques.Copyright 1999 Society for Mathematical BiologyTo whom correspondence should be addressed.

Polarization properties for the electromagnetic field in an unboundedn-type semiconductor medium

In this paper, we derive and discuss the polarization properties of a plane wave traveling in an unbounded n-type semiconductor medium. Starting from the Maxwell's equations and the constitutive relations, we first compute the formal expression of the electromagnetic field associated with the two eigenmodes of propagation sustained in the n-type semiconductor medium. After that, we determine the conditions that the constitutive parameters must satisfy in order to have linear and circular polarization, and discuss possible applications of the n-type semiconductor medium. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 17: 332–335, 1998.

Acoustic power and heat fluxes in the thermoacoustic effect due to a travelling plane wave

The thermoacoustic effect in a single plate due to a travelling plane wave is analyzed with respect to the conditions for the refrigerator or prime mover cycle to occur. The relation between the conduction of heat along a normal to the plate and the absorbed or generated acoustic power is also shown.

Multidimensional stability of planar travelling waves

The multidimensional stability of planar travelling waves for systems of reaction-diffusion equations is considered in the case that the diffusion matrix is the identity. It is shown that if the wave is exponentially orbitally stable in one space dimension, then it is stable for x ∈ R n , n ≥ 2 x\in \mathbf {R}^n,\,n\ge 2 . Furthermore, it is shown that the perturbation of the wave decays like t − ( n − 1 ) / 4 t^{-(n-1)/4} as t → ∞ t\to \infty . The result is proved via an application of linear semigroup theory.

Differential flow instability in the Ginzburg-Landau and Swift-Hohenberg approximations

The effects of a differential flow of the components of a reaction-diffusion system which is close to its stability boundary are described within the long wavelength approximation. In the vicinity of the Hopf bifurcation the system's evolution is governed by a complex Ginzburg-Landau equation modified by a purely imaginary convective term. If the system is near the zero real eigenvalue bifurcation, the governing equation is a modified Swift-Hohenberg equation. In both cases the homogeneous, stable reference steady state may be destabilized by the differential flow. In the Ginzburg-Landau equation, the destabilization occurs as long as the flow velocity exceeds some critical value vcr, which tends to zero as the system approaches the Hopf bifurcation. In the modified Swift-Hohenberg equation, the flow has either a destabilizing or stabilizing effect, depending on the sign of one of the system parameters. Destabilization occurs when the flow velocity exceeds some threshold; however in this case, the threshold remains finite even at the bifurcation point. In both Ginzburg-Landau and Swift-Hohenberg equations the differential flow instability produces traveling plane waves. The stability analysis shows that once a periodic plane wave is established, its spatial period remains unchanged over a finite range of the flow velocity and changes in discrete steps — the phenomenon of ‘wavenumber locking’. ‘Wavenumber locking’ is verified in numerical experiments with the Ginzburg-Landau equation. Near Hopf bifurcation, the Benjamin-Feir instability may occur. In this case irregular traveling waves are found, but a regular component of the wave pattern survives. Depending on a parameter, the differential flow either promotes or deters the Benjamin-Feir instability. As a result, the increasing flow may swithc the periodic wave pattern into a irregular state or, conversely, may stabilize the previously induced irregular pattern and produce periodic waves.

Direct numerical simulation of acoustic/shear flow interactions in two-dimensional ducts

The interaction between an imposed monochromatic, time-dependent acoustic disturbance and a steady mean shear flow in a two-dimensional duct is studied using direct numerical simulation. Unlike previously reported numerical results, the long-time acoustic response is captured through implementation of accurate nonreflecting boundary conditions. Below a certain cutoff frequency, the acoustic field is a nearly planar traveling wave propagating axially along the duct. Above this cutoff frequency, oblique waves are generated due to both acoustic refraction and cross-stream dependent source oscillation, leading to an alternating pattern of higher acoustic pressures at the wall and the centerline, downstream of the disturbance source. At resonant conditions, the growth of the oblique wave, which is nearly transverse, dominates the axial wave, leading to a phase change of 180 deg after their interaction. The thickness of the acoustic boundary layer and its response to imposed disturbances are also in good agreement with theory. These results are consistent with previously published theoretical predictions and represent their first numerical verification. Nomenclature #00 = reference speed of sound, also reference velocity Cp = constant pressure specific heat, temperature independent et = internal energy per unit volume h = duct half-width i, j,k = coordinate directions in Cartesian tensor notation L = computational length of duct M = mean flow Mach number Pr = Prandtl number p = pressure qj = heat flux vector component in the jth direction Re = acoustic Reynolds number in the calculation Rec = duct Reynolds number T = temperature t = time variable uk -• velocity component in the &th coordinate direction

Transverse instability of plane wavetrains in gas-fluidized beds

Using a two-fluid model of gas-fluidized beds, it is shown that periodic plane voidage waves travelling against gravity are unstable to perturbations with large transverse wavelength. This secondary instability sets in at arbitrarily small amplitudes of the plane wave and correspondingly small transverse wavenumbers of the two-dimensional perturbation. More precisely, if the bed is wide enough to accommodate sufficiently long horizontal waves, then the plane wave becomes unstable as soon as its amplitude has grown to the order of the square of the transverse wavenumber. The instability can be stationary or oscillatory in nature and has its origin in the interaction between the plane wave and four least-stable modes with small transverse wavenumber. Two of them represent a pair of bubble-like ‘mixed modes‘; the other two are initially, i. e. at the onset of the primary wave, pure transverse modes, one consisting only of an initially pure vertical velocity perturbation of the state of uniform fluidization. Depending on a relation between the eigenvalues of the least-stable modes at the primary bifurcation point, either one of these can be the dominant mode, which becomes (most) unstable along the growing vertically travelling plane wave. While the transverse modes gain longitudinal structure during this process, the mixed modes obtain a vertical component of the vertically averaged velocity as well, so that it appears that the secondary instability described here is a variant of Batchelor &amp; Nitsche's (1991) ‘overturning’ instability found recently for unbounded stratified fluids, see also Batchelor (1993).