Properties of the zeroth-, first-, and higher-order approximations of attributes of elastic waves in weakly anisotropic media

Use of the perturbation theory in the study of attributes of elastic waves propagating in weakly anisotropic media leads to approximate but transparent and simple formulas, which have many applications in forward and inverse wave modeling. We present and study such formulas. We show that all studied attributes depend on elements of a matrix linearly dependent on parameters of a medium. We study this dependence with the goal to understand which parameters of the medium, and in which combinations, affect individual wave attributes. Alternative auxiliar vector bases, in which the matrix can be specified, are proposed and studied. The vector bases offer alternative specifications of polarization vectors of qS waves. One of the important observations is that the higher-order (n > or = 2) perturbation formulas for qS waves are obtained separately for qS1 and qS2 waves. We also study effects of the use of the perturbation theory on the accuracy of the determination of the acoustical axes in weakly anisotropic media. We show that longitudinal directions in the first-order approximation are identical with actual ones. In singular directions, however, the first-order formulas provide directions, which may deviate from the exact ones, or they may even indicate false singular directions. Again, the above-mentioned matrix depending linearly on the parameters of the medium plays a central role in this study.     

随机扰动对螺旋波动力学的影响研究

心脏中的心肌组织是一种典型的可激发介质, 鉴于心肌细胞分布的离散性, 采用离散可激发介质模型研究了不应态时间随机扰动对螺旋波动力学行为的影响, 在扰动随机出现情况下, 螺旋波的稳定性与受扰元胞的数目和扰动幅度有关, 数值计算结果表明: 在适当的条件下, 可以观察到螺旋波漫游、破碎和消失现象, 并简要分析了产生这些现象的机理. 关键词： 螺旋波 激发介质 随机扰动     

Suppressing arrhythmias in cardiac models using overdrive pacing and calcium channel blockers

Recent findings indicate that ventricular fibrillation might arise from spiral wave chaos. Our objective in this computational study was to investigate wave interactions in excitable media and to explore the feasibility of using overdrive pacing to suppress spiral wave chaos. This work is based on the finding that in excitable media, propagating waves with the highest excitation frequency eventually overtake all other waves. We analyzed the effects of low-amplitude, high-frequency pacing in one-dimensional and two-dimensional networks of coupled, excitable cells governed by the Luo-Rudy model. In the one-dimensional cardiac model, we found narrow high-frequency regions of 1:1 synchronization between the input stimulus and the system's response. The frequencies in this region were higher than the intrinsic spiral wave frequency of cardiac tissue. When we paced the two-dimensional cardiac model with frequencies from this region, we found that spiral wave chaos could, in some cases, be suppressed. When we coupled the overdrive pacing with calcium channel blockers, we found that spiral wave chaos could be suppressed in all cases. These findings suggest that low-amplitude, high-frequency overdrive pacing, in combination with calcium channel inhibitors (e.g., class II or class IV antiarrhythmic drugs), may be useful for eliminating fibrillation. (c) 2002 American Institute of Physics.     

Asymptotic solutions of 2D wave equations with variable velocity and localized right-hand side

In the paper, we consider the Cauchy problem for the inhomogeneous wave equation with variable velocity and with a perturbation in the form of a right-hand side localized in space (near the origin) and in time. In particular, this problem is connected with the question about the creation of tsunami and Rayleigh waves. Using abstract operator theory and in particular Maslov's noncommutative analysis, we show that the solution is separated into two parts: the transient one, which is localized in a neighborhood of the origin and decreases in time and the propagating one, which propagates in space like the wave created by the momentary “equivalent source.” We present several examples covering a wide range of perturbation resulting in rather explicit formulas expressing the solutions it terms of the error function of complex argument.     

The scattering of electromagnetic waves in fractal media

The scattering of electromagnetic waves in fractal media is studied. The fractal dimension is naturally involved in the formulation of two physical problems studied in this paper. The general theory of multiple scattering of electromagnetic wave in fractal media is developed by modifying Twersky's theory. Statistical quantities, such as the average field and average intensity of the multiple scattered wave, are studied for a wave propagating in a fractal medium. The scattering cross section of the medium is deduced. The backscattering of electromagnetic waves is also studied. The results showing the range of dependence of the backscattered signals are in agreement with numerical simulations by Rastogi and Scheucher (1990). It also suggests a method of measuring the fractal dimension of the fractal embedded media using radar sounding. The theory developed in this paper can also be used for problems related to multiple scattering of other kinds of waves, such as acoustic waves, elastic waves etc, in fractal media.     

The scattering of electromagnetic waves in fractal media

Abstract

The scattering of electromagnetic waves in fractal media is studied. The fractal dimension is naturally involved in the formulation of two physical problems studied in this paper. The general theory of multiple scattering of electromagnetic wave in fractal media is developed by modifying Twersky's theory. Statistical quantities, such as the average field and average intensity of the multiple scattered wave, are studied for a wave propagating in a fractal medium. The scattering cross section of the medium is deduced. The backscattering of electromagnetic waves is also studied. The results showing the range of dependence of the backscattered signals are in agreement with numerical simulations by Rastogi and Scheucher (1990). It also suggests a method of measuring the fractal dimension of the fractal embedded media using radar sounding. The theory developed in this paper can also be used for problems related to multiple scattering of other kinds of waves, such as acoustic waves, elastic waves etc, in fractal media.     

Transverse coupling of chemical waves

The transverse coupling of chemical waves is investigated using a model scheme for excitable media. Chemical waves supported on the surfaces of a semipermeable membrane couple via diffusion through the membrane, resulting in new types of spatiotemporal behavior. The model studies show that spontaneous wave sources may develop from interacting planar waves, giving rise to a complex sequence of patterns accessible only by perturbation. Coupled circular waves result in the spontaneous formation of spiral waves, which subsequently develop patterns in distinct domains with characteristic features. The long time entrainment behavior of coupled spiral waves reveals regions of 1:2 phase locking.     

Properties of surface waves in granular media under gravity

Acoustical waves propagating along the free surface of granular media under gravity are investigated in the framework of elasticity theory. The influence of stress on a surface wave is analyzed. The results have shown that two types of surface waves, namely sagittal and transverse modes exist depending on initial stress states, which may have some influence on the dispersion relations of surface waves, but the influence is not great. Considering that the present experimental accuracy is far from distinguishing this detail, the validity of elasticity theory on the surface waves propagating in granular media can still be maintained.     

Involutes: the geometry of chemical waves rotating in annular membranes

According to earlier theories certain parts of a chemical wave front propagating in a 2-D excitable medium with a convex obstacle should be involutes of that obstacle. The present paper discusses a special case where self-sustained chemical waves are rotating around a central obstacle in an annular 2-D excitable region. A simple geometrical model of wave propagation based on the Fermat principle (minimum propagation time) is suggested. Applying this model it is shown that the wave fronts in the case of an annular excitable region should be purely involutes of the central obstacle in the asymptotic state. This theory is supported by experiments in a novel membrane reactor where a catalyst of the Belousov-Zhabotinsky reaction is fixed on a porous membrane combined with a gel medium. Involutes of circular and triangular obstacles are observed experimentally. Deviations from the ideal involute geometry are explained by inhomogeneities in the membrane. (c) 1995 American Institute of Physics.     

A singular perturbation theory for the FitzHugh-Nagumo nerve conduction equation

A singular perturbation theory is applied to the FitzHugh-Nagumo nerve conduction equation with cubic nonlinearity to obtain jacobian elliptic-function travelling-wave profiles. Starting with zeroth-order elliptic-function solutions, we obtain explicit first-order correction to the propagating waves and pulses.     

Organizing Filament of Small Amplitude Scroll Waves

We theoretically analyze the organizing filament of small amplitude scroll waves in general excitable media by perturbation method and explicitly give the expressions of coefficients in Keener theory.In particular for the excitable media with equal diffusion,we obtain a close system for the motion of the filament.With an example of the Oregonator Model,our results are in good agreement with those simulated by Windree.     

Nonlinear dynamics of self-sustained supersonic reaction waves: Fickett's detonation analogue

The present study investigates the spatiotemporal variability in the dynamics of self-sustained supersonic reaction waves propagating through an excitable medium. The model is an extension of Fickett's detonation model with a state-dependent energy addition term. Stable and pulsating supersonic waves are predicted. With increasing sensitivity of the reaction rate, the reaction wave transits from steady propagation to stable limit cycles and eventually to chaos through the classical Feigenbaum route. The physical pulsation mechanism is explained by the coherence between internal wave motion and energy release. The results obtained clarify the physical origin of detonation wave instability in chemical detonations previously observed experimentally.     

极化电场对可激发介质中螺旋波的控制

螺旋波在不同的物理、化学和生物系统中普遍存在.周期外场,比如极化电场,尤其是具有旋转对称性的圆极化电场可对螺旋波动力学产生重要影响.本文综述了极化电场对可激发介质中螺旋波的控制,包括共振漂移、同步、手征对称性破缺、多臂螺旋波的稳定、次激发介质中的螺旋波、三维回卷波湍流态的控制、心脏组织中螺旋波的去钉扎、心脏组织中螺旋波湍流态的控制等.     

Third-order Stokes wave solutions for interfacial internalwaves in three-layer dendity-stratified fluid

<正>Interfacial internal waves in a three-layer density-stratified fluid are investigated using a singular perturbation method,and third-order asymptotic solutions of the velocity potentials and third-order Stokes wave solutions of the associated elevations of the interfacial waves are presented based on the small amplitude wave theory.As expected,the third-order solutions describe the third-order nonlinear modification and the third-order nonlinear interactions between the interfacial waves.The wave velocity depends on not only the wave number and the depth of each layer but also on the wave amplitude.     

Selection of spiral waves in excitable media with a phase wave at the wave back

Universal relationships between the medium excitability and the angular velocity and the core radius of rigidly rotating spiral waves in excitable media are derived for situations where the wave front is a trigger wave and the wave back is a phase wave. Two universal limits restricting the region of existence of spiral waves in the parameter space are demonstrated. The predictions of the free-boundary approach are in good quantitative agreement with results from numerical reaction-diffusion simulations performed on the Kessler-Levine model.     

Using weak impulses to suppress traveling waves in excitable media

Here we propose mechanisms for suppressing non-steady-state motions--propagating pulses, spiral waves, spiral-wave chaos--in excitable media. Our approach is based on two points: (1) excitable media are multistable; and (2) traveling waves in excitable media can be separated into fast and slow motions, which can be considered independently. We show that weak impulses can be used to change the values of the slow variable at the front and back of a traveling wave, which leads to wave front and wave back velocities that are different from each other. This effect can destabilize the traveling wave, resulting in a transition to the rest state.     

Pursuit-evasion predator-prey waves in two spatial dimensions

We consider a spatially distributed population dynamics model with excitable predator-prey kinetics, where species propagate in space due to their taxis with respect to each other's gradient in addition to, or instead of, their diffusive spread. Earlier, we have described new phenomena in this model in one spatial dimension, not found in analogous systems without taxis: reflecting and self-splitting waves. Here we identify new phenomena in two spatial dimensions: unusual patterns of meander of spirals, partial reflection of waves, swelling wave tips, attachment of free wave ends to wave backs, and as a result, a novel mechanism of self-supporting complicated spatiotemporal activity, unknown in reaction-diffusion population models.     

Theory of periodic and solitary space charge waves in extrinsic semiconductors

We present a theory of the existence and stability of traveling periodic and solitary space charge wave solutions to a standard rate equation model of electrical conduction in extrinsic semiconductors which includes effects of field-dependent impurity impact ionization. A nondimensional set of equations is presented in which the small parameter β = (dielectric relaxation time) / (characteristic impurity time) 1 plays a crucial role for our singular perturbation analysis. For a narrow range of wave velocities a phase plane analysis gives a set of limit cycle orbits corresponding to periodic traveling waves. while for a unique value of wave velocity we find a homoclinic orbit corresponding to a moving solitary space charge wave of the type experimentally observed in p-type germanium. A linear stability analysis reveals all waves to be unstable under current bias on the infinite one-dimensional line. Finally, we conjecture that solitary waves may be stable in samples of finite length under voltage bias.     

Topological Constraints on Scroll and Spiral Waves in Excitable Media

A conservation equation for topological charges of phase singularities （scroll and spiral waves） in excitable media is given. It provides some topological properties of scroll （spiral） waves： for example, the topological charge of the generated or annihilated spiral pair must be opposite. Additionally, we obtain another equation on scroll waves, which shows that singular filaments of scroll waves occur on a set of one-dimensional curves which may be either closed loops or infinite lines.