Evolution Of Wavefront Research Articles

Analytical and numerical construction of the optimal outcome function for a class of time-optimal problems

Analytical and numerical algorithms are proposed for constructing the optimal outcome function and its Lebesgue set for the time-optimal control problem with a circular velocity indicatrix. Our approach to the solution of the time-optimal control problem essentially utilizes the specific dynamic properties of the controlled system. The circular vectogram of possible velocities enables us to interpret the cross sections of the controllability set as wavefronts whose source is uniformly distributed on the boundary of the target set. Procedures have been developed for analytical and numerical construction of the evolution of wavefronts based on prior (given the geometry of the target set boundary) identification of their nonsmoothness sets. An essential feature of the construction is the point-to-set distance function. We investigate the differential properties of this function and identify the manifolds on which it loses its smoothness. The proposed wavefront construction algorithms are of independent interest in so far as they enable us to investigate the geometry of the sets and compute their nonconvexity measure. The results are useful not only when studying the evolution of reachability sets of controlled systems, but also for computing the eikonal in geometrical optics and investigating the solutions of the wave equation.

General Relativistic Radiant Shock Waves in the Post-Quasistatic Approximation

An evolution of radiant shock wave front is considered in the framework of a recently presented method to study self-gravitating relativistic spheres, whose rationale becomes intelligible and finds full justification within the context of a suitable definition of the post-quasistatic approximation. The spherical matter configuration is divided into two regions by the shock and each side of the interface having a different equation of state and anisotropic phase. In order to simulate dissipation effects due to the transfer of photons and/or neutrinos within the matter configuration, we introduce the flux factor, the variable Eddington factor and a closure relation between them. As we expected the strong of the shock increases the speed of the fluid to relativistic ones and for some critical values is larger than light speed. In addition, we find that energy conditions are very sensible to the anisotropy, specially the strong energy condition. As a special feature of the model, we find that the contribution of the matter and radiation to the radial pressure are the same order of magnitude as in the mant as in the core, moreover, in the core radiation pressure is larger than matter pressure.

Wave front evolution of negatively refracted waves in a photonic crystal

Using a heterodyne near-field scanning microscope, the authors analyze the phase and amplitude of the electric field of an optical wave across the boundary of positive to negative refraction media. The photonic crystal acts as an extremely anisotropic material with a negative curvature of its dispersion surface whose shape is resolved experimentally. This extreme anisotropy results in the beam having a peculiar phase evolution through propagation that does not occur in isotropic media. A focusing wave is produced by negative refraction, which has diverging wave fronts before the internal focus and converging wave fronts after the focus.

Evolution of a downslope propagating acoustic pulse

During the North Pacific Acoustics Laboratory (NPAL‐04) experiment broadband transmissions (50 Hz bandwidth, 75 Hz center frequency, 27.28 s period length) from a bottom‐mounted source off the island of Kauai were received on a towed array (350 m depth, 200 m aperture) near the island and a large aperture vertical line array (VLA) at a range of 2469.5 km. Previous work [Heaney, J. Acoust. Soc. Am. 117(3), 1635–1642 (2005)] has demonstrated that bottom interaction near the source is a fundamental process in the evolution of the wave‐train. In this paper results from eight receptions at ranges from 2 to 300 km and the reception on the distant VLA is used to examine the wavefront evolution as it propagates into deep water. Results from a local geo‐acoustic inversion will be presented as well as comparison of data with broadband parabolic equation simulations. The initial conclusion is that the first‐order bottom bounce, off the seamount near the source, acts as an image source, contributing to a complex arrival pattern even at great distances.

Chemical lens

Chemical wave propagation in a two dimensional heterogeneous medium consisting of an inner ‘slower’ and an outer ‘faster’ homogeneous medium is studied theoretically and experimentally. The essential features of the evolution of such wave fronts can be derived from the geometrical wave theory based on the Fermat’s principle of minimal propagation time. The interfacial boundary is chosen to realize a perfect image formation. This means that circular fronts starting from a given point, after refraction will also be circular, focusing (travelling inwards and converging) into an other point. This way a chemical lens is constructed and realized experimentally.

White matter tractography by anisotropic wavefront evolution and diffusion tensor imaging

Determination of axonal pathways provides an invaluable means to study the connectivity of the human brain and its functional network. Diffusion tensor imaging (DTI) is unique in its ability to capture the restricted diffusion of water molecules which can be used to infer the directionality of tissue components. In this paper, we introduce a white matter tractography method based on anisotropic wavefront propagation in diffusion tensor images. A front propagates in the white matter with a speed profile governed by the isocontour of the diffusion tensor ellipsoid. By using the ellipsoid, we avoid possible misclassification of the principal eigenvector in oblate regions. The wavefront evolution is described by an anisotropic version of the static Hamilton–Jacobi equation, which is solved by a sweeping method in order to obtain correct arrival times. Pathways of connection are determined by tracing minimum-cost trajectories using the characteristic vector field of the resulting partial differential equation. A validity index is described to rate the goodness of the resulting pathways with respect to the directionality of the tensor field. Connectivity results using normal human DTI brain images are illustrated and discussed. We also compared our method with a similar level set-based tractography technique, and found that the anisotropic evolution increased the validity index of the obtained pathways by 18%.

Wavefront and ray-density plots using seventh-order matrices

The optimization of an optical system benefits greatly from a study of its aberrations and an identification of each of its elements’ contribution to the overall aberration figures. The matrix formalism developed by the author was the object of a previous paper and allows the expression of image-space coordinates as high-order polynomials of object-space coordinates. In this paper we approach the question of aberrations, both through the evaluation of the wavefront evolution along the system and its departure from the ideal spherical shape and the use of ray density plots. Using seventh-order matrix modelling, we can calculate the optical path between any two points of a ray as it travels along the optical system and we define the wavefront as the locus of the points with common optical path length; the results are presented on the form of traces of the wavefront on the tangential plane, although the formalism would also permit sagittal plane plots. Ray density plots are obtained by actual derivation of the seventh-order polynomials.

Sound and vorticity interactions: transmission and scattering

We review several aspects of the propagation of sound in vortical flows. We restrict ourselves to isothermal, humidity-free flows at low Mach number M. Since vorticity plays a major role in vortex-flow interactions we focus on vortical flows. We consider two main canonical situations. The first concerns the transmission of sound. We analyze the evolution of acoustic wavefronts as they propagate across a single vortex. The second situation addresses the scattering of sound waves by nonstationary vortices. We study the evolution of the acoustic pressure emitted in the far field, at an angle with the initial direction of propagation. In this geometry one performs direct spectroscopy of the flow vorticity field. In each case, we review theoretical results and compare with experimental measurements and numerical simulations when available. We also briefly report how the following new acoustic techniques have recently been used to study complex or turbulent flows: time-resolved acoustic spectroscopy, speckle interferometry and Lagrangian particle tracking.

Wave front evolution in strongly heterogeneous layered media using the fast marching method

SUMMARY The fast marching method (FMM) is a grid based numerical scheme for tracking the evolution of monotonically advancing interfaces via finite-difference solution of the eikonal equation. Like many other grid based techniques, FMM is only capable of finding the first-arriving phase in continuous media; however, it distinguishes itself by combining both unconditional stability and rapid computation, making it a truly practical scheme for velocity fields of arbitrary complexity. The aim of this paper is to investigate the potential of FMM for finding later arriving phases in layered media. In particular, we focus on reflections from smooth subhorizontal interfaces that separate regions of continuous velocity variation. The method we adopt for calculating reflected phases involves two stages: the first stage initializes FMM at the source and tracks the incident wave front to all points on the reflector surface; the second stage tracks the reflected wave front by reinitializing FMM from the interface point with minimum traveltime. Layer velocities are described by a regular grid of velocity nodes and layer boundaries are described by a set of interface nodes that may be irregularly distributed. A triangulation routine is used to locally suture interface nodes to neighbouring velocity nodes in order to facilitate the tracking of wave fronts to and from the reflector. A number of synthetic tests are carried out to assess the accuracy, speed and robustness of the new scheme. These include comparisons with analytic solutions and with solutions obtained from a shooting method of ray tracing. The convergence of traveltimes as grid spacing is reduced is also examined. Results from these tests indicate that wave fronts can be accurately tracked with minimal computational effort, even in the presence of complex velocity fields and layer boundaries with high curvature. Incident wave fronts containing gradient discontinuities or shocks also pose no difficulty. Further development of the wave front reinitialization scheme should allow other later arrivals such as multiples to be successfully located.

A Fast Method to Simulate Travelling Waves in Nonhomogeneous Chemical or Biological Media

Waves in excitable media can be treated by a simple geometric theory. The propagation velocity is assumed known and evolution of wave fronts is determined by elementary physical principles (Fermat's principle, Huygens' principle). Based on this geometric theory a fast computational method is developed. By this method the distorting effect of the spatial grid is avoided. The method is applied to the cases when a circular obstacle is surrounded by a homogeneous and heterogeneous medium, respectively. The numerical simulations show that the method is convenient, fast and reliable.

Equations of laser‐induced plasma shock wave motion in air

AbstractSimple equations of laser‐induced plasma shock wave motion in air are studied, based on the point blast theory. The theoretical models of the shock wave front evolution characterize two stages of shock wave propagation: the initial formation stage and the free attenuation stage. Due considerations of the boundary conditions, the validity range of the attenuation equation is larger than the one of the t2/5 law of the self‐similar solution. Furthermore, the theoretical results are in good agreement with the experimental data and nuclear explosion results. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 38: 75–79, 2003

Excitation fronts in a spatially modulated light-sensitive Belousov-Zhabotinsky system.

The evolution of excitation wave fronts in a spatially modulated light-sensitive Belousov-Zhabotinsky system is investigated experimentally and theoretically. The excitation wave propagates in a thin, quasi-two-dimensional reaction layer, which is illuminated through a periodical gray level mask. The light-induced differences in excitability and velocity give rise to a temporal and spatial modulation of the initially flat fronts. The experimental front evolution is described in the framework of a kinematical theory as developed earlier for nonuniformly curved systems.

Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations

The fundamental electromagnetic properties of left-handed materials (LHMs) are reviewed and verified by finite-element method full-wave analysis using rectangular waveguide structures loaded by a LHM and adopting an effective medium approach. The negative phase velocity, positive intrinsic impedance, and modified boundary conditions at an interface with a right-handed medium are verified by loading a waveguide section with a LHM that has edges perpendicular to the waveguide axis. In addition, the negative angle of refraction is demonstrated by loading the junction of a T-junction waveguide with a LHM having one edge 45° with respect to the waveguide axis. These properties are shown by the evolution of wave fronts in the LHM and by analysis of the S-parameters of the waveguide structures.

Dynamic photoelastic study of the transient stress field in solids during shock wave lithotripsy.

Photoelastic and shadowgraph imaging techniques were used to visualize the propagation and evolution of stress waves, and the resultant transient stress fields in solids during shock wave lithotripsy. In parallel, theoretical analysis of the wavefront evolution inside the solids was performed using a ray-tracing method. Excellent agreement between the theoretical prediction and experimental results was observed. Both the sample size and geometry were found to have a significant influence on the wave evolution and associated stress field produced inside the solid. In particular, characteristic patterns of spalling damage (i.e., transverse and longitudinal crack formation) were observed using plaster-of-Paris cylindrical phantoms of rectangular and circular cross sections. It was found that the leading tensile pulse of the reflected longitudinal wave is responsible for the initiation of microcracks in regions inside the phantom where high tensile stresses are produced. In addition, the transmitted shear wave was found to play a critical role in facilitating the extension and propagation of the microcrack.

Dynamic photoelastic study of the transient stress fields in solids during shock wave lithotripsy

Photoelastic and shadowgraph imaging techniques were used to visualize shock wave propagation, evolution, and the resultant transient stress fields in solids during shock wave lithotripsy. In parallel, theoretical analysis of the wavefront evolution inside the solids was performed using the ray-tracing method. Excellent agreement between the theoretical prediction and experimental results was observed. Moreover, the effects of sample size and shape on wave evolution and associated stress fields induced inside the solids were evaluated, both theoretically and experimentally. Finally, stone fragmentation tests were carried out using stone phantoms of different geometry and size, and the characteristics of the damage patterns were documented. By correlating the stone fragmentation patterns with the behavior of different wave components induced in the target stones, it was found that while the damage cracks near the posterior surface of the sample were initiated by the reflected longitudinal tensile wave, the transmitted shear wave plays a critical role in crack extension. [Work supported by NIH.]

Stable singularities of wave-fronts in general relativity

A wave-front in a space-time M is a family of null geodesics orthogonal to a smooth spacelike two-surface in M; it is of some interest to know how a wave-front can fail to be a smoothly immersed surface in M. In this paper we see that the space of null geodesics N of M, considered as a contact manifold, provides a natural setting for an efficient study of the stable singularities arising in the time evolution of wave-fronts.

Refraction of active waves in reaction—diffusion media

The refraction of active waves is analyzed for a stable—metastable reaction—diffusion system consisting of two regions with different diffusion coefficients. The equations governing the evolution of wavefronts are derived by means of an asymptotic perturbation method for boundary layers. These equations describe non-stationary refraction near the steady state regime. It is shown that the dynamics of wavefronts separates into that in the region near the boundary and that far from the boundary.

High-frequency nonlinear acoustic beams and wave packets

Gaussian beams and wave packets are formally equivalent to high-frequency solutions of the linear wave equation with complex-valued phase functions. The theory of weakly nonlinear high-frequency waves is extended in this paper to allow for complex phase solutions. The procedure is similar to nonlinear geometrical optics and the nonlinearity causes the phase to vary, leading to the development of weak shocks. However, unlike standard geometrical optics that describes the evolution of wave fronts, complex phase solutions correspond to Gaussian decay away from a central ray, and the associated curvatures are complex valued. Geometrical singularities due to caustics and foci do not occur, and the only singularities in the theory are purely nonlinear. The theory is developed specifically for nonlinear acoustic waves in a homogeneous inviscid fluid. The time taken for nonlinear singularities to develop is compared for a real spherical wave front and a Gaussian beam with Gaussian radius equal to the real radius of curvature of the wave front. The time to blowup for the Gaussian beam is intermediate between the shortest possible, which is for the converging wave front, and the longest possible, for the diverging front.

Cross‐borehole simulation of wave propagation

Synthetic seismograms are used to study elastic wave propagation in multi‐layered media for cross‐borehole geometries. The calculations are done using the discrete wavenumber method. We compute the wavefield for a series of receiver arrays located at various offsets to follow the evolution of the wavefronts and the distribution of the seismic energy in space. The results show the complexity of the wavefield at large offsets. Trapped waves and conical waves perturb the identification of direct and primary reflected phases. The display of polarization diagrams of the guided waves shows elliptical prograde motions. The source position plays an important role in the energy distribution within the medium. In order to study this dependency, we compare the cases of sources located in relatively low and high‐velocity layers. In the former case, most of the energy is trapped within the source layer and in the neighboring region and propagates horizontally. The S‐converted waves generated at the interfaces bounding the source layer have larger amplitude than the direct P‐wave. When the source is placed in a relatively high‐velocity layer the energy of the P‐wavefront spreads rapidly throughout the medium as the offset increases.

Wavefront propagation in a class of random microstructures—1. bilinear elastic grains

We show that a Markov property of disturbance propagation forms the basis for study of wavefront evolution in one-dimensional piecewise constant microstructures with randomness in constitutive moduli and grain lengths. A new general method of solution to transient wave problems in such microstructures combining the method of characteristics and Markov diffusion processes is developed. In this paper we give a diffusion approximation for the case of linear elastic grains, and use it to determine rules of propagation of wavefronts in the case of bilinear elastic grains. Both, soft and hard constitutive laws are investigated.