Sharp Wave Front Research Articles

Traveling wave solutions of a nonlinear reaction–diffusion–chemotaxis model for bacterial pattern formation

In this paper we consider a nonlinear reaction–diffusion–chemotaxis model for the description of the spatiotemporal evolution of the bacteria of the type Paenibacillus dendritiformis on a thin layer of agar in a Petri dish. We perform a traveling wave analysis for the model equation showing the existence of traveling wave solutions, in particular, the sharp wave front type solutions with minimum speed. Further, we present numerical investigations for a special case. The minimum speed is estimated and the profile of the traveling wave solution is calculated and compared for different numerical methods.

The variant of post-Newtonian mechanics with generalized fractional derivatives

In this article, we investigate mathematically the variant of post-Newtonian mechanics using generalized fractional derivatives. The relativistic-covariant generalization of the classical equations for gravitational field is studied. The equations (i) match the weak Newtonian limit on the moderate scales and (ii) deliver a potential higher than Newtonian on certain large-distance characteristic scales. The perturbation of the gravitational field results in the tiny secular perihelion shift and exhibits some unusual effects on large scales. The general representation of the solution for the fractional wave equation is given in the form of retarded potentials. The solutions for the Riesz wave equation and classical wave equation are clearly distinctive in an important sense. The hypothetical gravitational Riesz wave demonstrates the space diffusion of the wave at the scales of metric constant. The diffusion leads to the blur of the peak and disruption of the sharp wave front. This contrasts with the solution of the D'Alembert classical wave equation, which obeys the Huygens principle and does not diffuse.

Interaction of magnetostatic excitations with 90° domain walls in micrometer-sized permalloy squares

We investigate the magnetic excitations in soft magnetic squares and in particular the role of domain walls in such Landau flux-closure structures. For this we combine synchrotron-based photoemission electron microscopy and simulations using the Landau-Lifshitz-Gilbert equation. We show that the 90\ifmmode^\circ\else\textdegree\fi{} N\'eel walls in such squares act as efficient barriers for magnetic excitations that absorb the incoming energy. Because of the finite elasticity, the absorbed energy leads to oscillations of the domain walls, which subsequently cause emission of sharp wave fronts.

Rotating spirals of curvature flows: a center manifold approach

Rigidly rotating spiral wave patterns of Archimedean shape are a prominent phenomenon in the spatiotemporal evolution of planar reaction diffusion systems. A first analysis by Wiener and Rosenblueth [27] was motivated by electrical waves in heart tissue. For experimental evidence of Archimedean spirals in Belousov– Zhabotinsky systems and other excitable chemical reaction media see, for example, [24–26,5,22,20,21]. See also the recent survey [17] and references therein. Fluid convection experiments, although modeled by the Navier–Stokes system rather than by reaction and diffusion, also exhibit dynamically intricate spiral wave patterns; see [19]. For a survey of mathematical issues related to spiral waves in reaction diffusion systems, see [10]. One mathematical approach, initiated already by Wiener, aims at a geometric description of the spatiotemporal dynamics of the sharp wave fronts that seem to emanate from a highly focused “core” region or “tip” and that take the form of an Archimedeam spiral, far away from the tip. This approach, where the wave front is represented simply by a planar curve z = z(s) ∈ R2, parametrized over arclength s ≥ 0, is sometimes called kinematic theory of spiral waves and has been developed on a mostly formal level. See, for example, [15,14,18,17] and references therein. Typically the dynamics of the curve z(s) is then described by a curvature driven flow

WAVES IN EXCITABLE MEDIA: EFFECTS OF WAVE GEOMETRY

We consider waves in two-dimensional excitable media. The correct form of the eikonal equation (i.e. the formula that predicts propagation speed as a function of curvature) is a question that has surfaced and resurfaced during the last twenty years or so. Good answers have become available in limiting cases and under certain approximations, notably weak curvatures, low frequencies, and sharp wave fronts. The solution is important in some cardiac pathologies, as well as in our basic understanding of excitable-medium mathematics. After a brief review of curvature effects, particularly in heart tissue, we demonstrate how to obtain a drastically corrected formula that is free of restrictions on frequency and sharpness. (In the original limiting cases the result is unchanged.) Our derivation uses a finite-renormalization method. We illustrate the formula in the context of cardiac tissue. For the sake of definiteness we work in terms of FitzHugh–Nagumo-like models.

T Traveling Slow Waves of Neural Activity: A Novel Form of Network Activity in Developing Neocortex

Spontaneous neuronal firing during development has the potential to shape many aspects of neuronal wiring throughout the brain. Bursts of electrical activity coordinated among large numbers of neurons, occurring during a brief developmental window, have been described in many regions of the CNS, including retina, hippocampus, and spinal cord, but evidence for this type of activity in developing neocortex has so far been lacking. To identify conditions that may give rise to patterned spontaneous electrical activity in developing neocortex, cholinergic agonists were applied to immature rat cortical slices while large-scale activity was imaged optically with fura-2 AM. Here I show that activation of muscarinic acetylcholine receptors results in waves of correlated neural activity. Waves recruit large numbers of neurons, are slowly propagating, regenerative events involving depolarization and associated calcium transients, and advance for many millimeters as a sharp wave front perpendicular to the pial surface, at speeds ranging between 50 and 300 m/sec. The expression of waves is restricted temporally to a brief period in postnatal development, until postnatal day 6, and spatially to some neocortical areas. The ability of isolated neocortical networks to generate large-scale patterned activity endogenously during a period of massive neurite extension and synaptogenesis raises the possibility that at least in some cortical areas these processes might be influenced by patterned neuronal firing generated independently of thalamocortical input.

Fluidlike surface-wave behavior of a polymer powder layer atop a dielectric sheet triggered through corona charging-discharging

When a film of oil is separated from a ground plate by a slab of insulating dielectric, a scheme of corona precharging followed by passive discharging leads, under favorable conditions, to the creation of a class of circularly expanding wave on oil surface (Lee and Phan [1]). In this experimental work, oil film was substituted by a mm-thick layer of polymer powder ranging from commercially available copier toner to those used in electrostatic coating applications for which grain size is typically below 10 gm. Point-plane corona precharging of up to 30 kV (dc, both polarities) was applied. When passive wire grounding was introduced suddenly, an analogous circular expanding wave was observed on the surface of the powder layer starting at a finite radius. The very sharp wavefront leaves behind a domain of quiescence wherein optical reflectivity appeared radically different from that of the region yet to be affected. Velocity decay set in after some propagation. This report details the apparent wave characteristics, along with the nature of pulsings in the accompanying current. A comparison between this powder-supported wave and a corresponding oil wave is made.

The generalized lagging response in small-scale and high-rate heating

The generalized lagging behavior in solids under high-rate heating is derived by precise correlation with the hyperbolic two-step model. The ballistic behavior of heat transport in the electron gas is found to be captured by the second-order effect of the phase lag of the heat flux vector. In contrast to the parabolic two-step model, the ballistic behavior results in a sharp wavefront in the history of heat propagation. The analytical expression for the thermal wave speed is derived. In comparison with the classical diffusion and the wave models, the two phase lags in the lagging response result in a much deeper thermal penetration depth and much higher temperature in the heat-affected zone.

Near-inertial motions in the coastal ocean

Internal-inertial waves are frequently observed in the upper ocean and below the thermocline. A three-dimensional general circulation model with turbulence-closure mixed layer is used to study the generation and propagation of near-inertial motion below the mixed layer. In particular, the problem of effect of a coastal wall on the wind induced inertial-internal wave field is re-examined using a fully nonlinear model. Responding to a wind pulse, a sharp wavefront propagates offshore. After the wavefront passage, strong near-inertial internal waves, marked by the tilting velocity isolines and the interface oscillations, are generated. The predicted near-inertial motion is consistent with the wave dispersion relation. Downward energy propagation occurs after the wavefront passage, and both kinetic and potential energy are strongly modified. After several inertial periods, the kinetic energy in the upper layer can be completely removed. The theoretical results, which are supported by observations, indicate that internal-inertial wave are important for mixing in the upper coastal ocean.

On the Wave Theory in Heat Conduction

This work contains three major components: a thorough review on the research emphasizing engineering applications of the thermal wave theory, special features in thermal wave propagation, and the thermal wave model in relation to the microscopic two-step model. For the sake of convenience, the research works are classified according to their individual emphases. Special features in thermal wave propagation include the sharp wavefront and rate effects, the thermal shock phenomenon, the thermal resonance phenomenon, and reflections and refractions of thermal waves across a material interface. By employing the dual-phase-lag concept, we show that the energy equation may be reduced to that governing the heat transport through the metal lattice in the microscopic two-step model. The dual-phase-lag concept can thus capture the microscopic mechanisms in some limiting cases.

Numerical simulation of the dynamics of solids mixing in fluidized beds

In this paper, the dynamics of solids mixing in a fluidized bed with horizontal tubes subjected to impulse and pulse input of tracer is simulated using the ‘Cinematic’ model. The dynamic mass balances are also solved using Laplace transforms followed by numerical inversion and a time-domain comparison is presented from the time of injection of the tracer until the steady is attained. It is demonstrated that due to the presence of sharp wave fronts, the numerical inversion procedure does not result in complete and accurate solutions. In general, the inversion is very slow and the accuracy of the results is restricted to a maximum of three significant digits. The numerical inversion required the computations to be performed individually for the three phases while the ‘Cinematic’ model calculates complete profiles in one sweep. In most trials, the numerical inversion procedure required a large number of terms to achieve convergence. Also, performing an important check such as the conservation of the tracer solids in the bed proved to be a very tedious task with Laplace transforms while the ‘Cinematic’ model did not demand additional efforts in performing the same check. Since many countercurrent systems can be represented by a system of hyperbolic partial differential equations similar to the one presented in this paper, the method of solution given in this paper can be regarded as applicable to other countercurrent systems as well.

A model for aggregation-dispersion dynamics of a population

An important feature that distinguishes the movement of living systems from the random motion of inorganic material is a delicate balance between spreading and concentrating. This movement is based on the kind of interactions which a bacterial colony may establish during migration. Namely, the antagonistic effects of dispersal which take place preferentially down the population gradient and the tendency in grouping together. In this work a model is proposed which considers these effects. The phase plane analysis and the numerical calculations reveal the existence of stable sharp wave front solutions. The speed of the wave front is modulated by the compromise between the tendencies of spreading and aggregating. The results obtained were compared with experimental observations in cultures of Escherichia coli and Streptococcus faecalis. The agreement between both types of results supports the hypothesis on which the model was based.

Conflicting density dependent dynamics of a bacterial population

We consider how the antagonistic effects of dispersal, which takes place preferentially down a population gradient, and the tendency to group together governs the dynamics of dispersal of a population. We propose a model which considers these effects. The phase plane analysis and the numerical calculations reveal the existence of stable sharp wave front solutions.

Solving the Hodgkin–Huxley Equations by a Random Walk Method

A numerical method is described for solving the Hodgkin–Huxley cable equations, set up as an initial boundary value problem. The method uses a gradient random walk combined with creation and destruction of the diffusing elements. It is designed to be efficient in the presence of sharp wavefronts. Details of implementation are given with particular emphasis on satisfying the boundary conditions without introducing excessive noise.

Tunneling by an electron packet with an initially sharp wavefront

We have studied the time behavior of electron wavepackets traversing one-dimensional potentials. These packets are described as plane wave states that have been cut off to give a sharp initial wavefront. We find that the shift, or delay time, of the main part of the pulse is comparable to that of a Gaussian wavepacket with the same momentum, and that both kinds of pulses have the same broad-pulse (sharp-momentum) limit. This is shown to hold for a general class of potentials. We show explicitly that the steepest descents calculation described by Stevens does not lead to a finite tunneling velocity. An exact expression is given for the packet transmitted through a delta-function barrier, which suggests a new interpretation of the tunneling velocity that has been obtained in other calculations.

Kelvin wave fronts, Rossby solitary waves and the nonlinear spin‐up of the equatorial oceans

This paper examines the spin‐up from rest of an equatorial ocean with meridional boundaries following the sudden onset of spatially uniform winds by using a nonlinear, one‐active‐layer, reduced‐gravity shallow‐water model. It is shown that the results obtained can differ considerably from corresponding integrations with a linear model and that this occurs within realistic parameter ranges. In the westward wind case a Kelvin wave front can develop, across which there is a discontinuity in the zonal current and in the depth of the model pycnocline. A vortex pair resembling a Rossby solitary wave as described theoretically by Boyd (1980b) is generated by the impact of this front on the eastern boundary. In the eastward wind case a series of equatorial Rossby solitary waves is generated at the eastern boundary. An extension of Boyd's theory is given that allows for a background zonal current and is used to account for the westward propagation speed of these waves. An important effect of nonlinearity in this case is to prevent equatorial waves from being able to bring to rest the equatorial (Yoshida) jet that is established during the first stage of the integration. The model response is shown to depend on the values of only two parameters, one of which (X) measures the nonlinearity and the other (δ) the ratio of the equatorial Rossby radius to the longitudinal basin width L. The former parameter varies as the product of L and the surface wind forcing and as the inverse square of the linear gravity wave speed. It is known that there is a threshold value of X (Xcrit) for the sharp Kelvin wave front to occur. It is shown that the results presented here for a nonlinear, one‐active‐layer, reduced‐gravity shallow‐water model are of relevance to a continuously stratified ocean model, provided there is a thermocline‐trapped mode that is dominantly excited by wind forcing, such as the second baroclinic mode in the equatorial oceans.

EXPERIMENTS STUDY ON SOIL-PILE DYNAMICS USING ELECTROMAGNETIC INDUCTION TYPE SHOCK WAVE SOURCE

In order to clarify soil-pile dynamics, a new experimental method using an electromagnetic induction type shock wave source was developed. In the experiment, a ground model was made of water-cured urethane polymer, and a pile of urethane was buried in this gel-like ground. An impulse with duration time of 0.5ms was given to the pile head by the shock wave source. This device consists of an aluminium circular plate and a flat solenoid coil close to this plate. A capacitor discharge through the solenoid serves as the energy source. Using this device makes it possible not only to estimate exact response characteristis of soil-pile systems pretty well but also to follow sharp wave fronts by photoelastic experiments.

ON THE NUMERICAL SOLUTION OF HYPERBOLIC HEAT CONDUCTION

Non-Fourier heat conduction is governed by the hyperbolic heat conduction equation, which involves the wave nature of thermal energy transport. In such cases, energy propagates in the medium as a wave with sharp discontinuities at the wave front. Difficulties encountered in the numerical solution of such problems include, among others, numerical oscillations and the representation of sharp discontinuities with good resolution at the wave front. In this work it is shown that a numerical technique based on MacCormack's predictor-corrector scheme can be used to handle discontinuities at the wave front with high resolution and little oscillation. Numerical predictions are compared with the exact analytic solutions for a wide variety of strict test conditions. For all the cases considered, the present numerical scheme remains stable and produces high resolution at the sharp wave front.

Formation of the Wavefront Image of Progressive Acoustic Wave by Self-Imaging Optical Method with a Diode-Laser Strobe

Theory holds that when a progressive acoustic wave not as intense as Raman-Nath parameter v≤1 is illuminated by an optical plane wave, its strain wavefront image appears alone in a limited region of the diffractied field. This imaging is based on the generation and superposition of the diffracted lights due to optical phase modulation by the acoustic wave. The experiment revealed that distinct and sharp wavefront fringes of the wave were formed in such a diffraction zone and these were helpful in our understanding of the vibrational mechanism of an energy-trapping transducer.

Fast axonal transport in the c-fibers of the garfish olfactory nerve.

AbstractThe application of aqueous solutions of tritiated L‐Leucine to the olfactory epithelium of the longnosed garfish (Lepisosteus osseus) results in a rapid sorption of the labeled precursor and the subsequent establishment along the nerve of a profile of labeled macromolecular constituents. This profile consists of a sharp wavefront, a peak and a saddle region. At 23°C, the wavefront enters the nerve 1.8 hr after the isotope application and moves away from the cell bodies at an average rate of 222 ± 2.4 mm/day. As the labeled material is propelled along the nerve fiber, its distribution undergoes several characteristic profile changes. The peak broadens and decreases in height and the slope of the wavefront becomes gradually less steep, while the area of the profile increases. These systematic, time‐dependent changes suggest a redistribution of some of the labeled axonal material during transport.