Critical Rays Research Articles

Generalization of P\xf3lya\u2019s Zero Distribution Theory for Exponential Polynomials, and Sharp Results for Asymptotic Growth

An exponential polynomial of order q is an entire function of the form f(z)=P1(z)eQ1(z)+⋯+Pk(z)eQk(z),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} f(z)=P_1(z)e^{Q_1(z)}+\\cdots +P_k(z)e^{Q_k(z)}, \\end{aligned}$$\\end{document}where the coefficients P_j(z),Q_j(z) are polynomials in z such that max{deg(Qj)}=q.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\max \\{\\deg (Q_j)\\}=q. \\end{aligned}$$\\end{document}In 1977 Steinmetz proved that the zeros of f lying outside of finitely many logarithmic strips around so called critical rays have exponent of convergence le q-1. This result does not say anything about the zero distribution of f in each individual logarithmic strip. Here, it is shown that the asymptotic growth of the non-integrated counting function of zeros of f is asymptotically comparable to r^q in each logarithmic strip. The result generalizes the first order results by Pólya and Schwengeler from the 1920’s, and it shows, among other things, that the critical rays of f are precisely the Borel directions of order q of f. The error terms in the asymptotic equations for T(r, f) and N(r, 1/f) originally due to Steinmetz are also improved.

Zero distribution and division results for exponential polynomials

An exponential polynomial of order q is an entire function of the form $$g(z) = {P_1}(z){e^{{Q_1}(z)}} + ...{P_k}(z){e^{{Q_k}(z)}},$$ where the coefficients Pj(z),Qj(z) are polynomials in z such that $$\max \{ deg({Q_j})\} = q.$$ It is known that the majority of the zeros of a given exponential polynomial are in domains surrounding finitely many critical rays. The shape of these domains is refined by showing that in many cases the domains can approach the critical rays asymptotically. Further, it is known that the zeros of an exponential polynomial are always of bounded multiplicity. A new sufficient condition for the majority of zeros to be simple is found. Finally, a division result for a quotient of two exponential polynomials is proved, generalizing a 1929 result by Ritt in the case q = 1 with constant coefficients. Ritt’s result is closely related to Shapiro’s conjecture that has remained open since 1958.

Critical Rays Self-adaptive Particle Filtering SLAM

This paper presents CRSPF-SLAM, a critical rays self-adaptive particle filtering occupancy grid based SLAM system that can operate efficiently with different kinds of odometer in real time, in small and large, indoor and outdoor environments for various platforms. Its basic idea is to eliminate the accumulated error of odometer through scan to map matching based on particle filtering. Through some improvements for the original particle filtering method, the lidar system becomes more robust to conduct accurate localization and mapping. Specifically, in our proposed method, particle filter based on Monte-Carlo algorithm is designed to be out-of-step to the odometer; During the scan matching process, the influence of some critical rays selected through a ray-selection algorithm is enhanced and that of the unreliable rays is weaken or removed; The current optimal match value is regarded as the feedback to reset the particle number and the filtering range; Once the optimal pose and scan are obtained, the previous error scan stored in the map will be removed. It is also introduced in the paper that the method can work effectively with dead reckoning, visual odometry and IMU, respectively. And we have tried to use it on different types of platforms — an indoor service robot, a self-driving car and an off-road vehicle. The experiments in a variety of challenging environments, such as bumpy and characterless area, are conducted and analyzed.

On the behavior of the edge diffracted nonstationary wave in scattering by wedges near the front

We study the behavior of a wave diffracted by a wedge wave near its front in the two-dimensional scattering of an incident plane harmonic wave with a Heaviside type profile. We find the asymptotics of the cylindrical wave diffracted by the edge of a wedge near the front in two cases: near the critical rays and far from the critical rays. The asymptotics turns out to be nonuniform and depends on the magnitude of the wedge. The cases of Dirichlet–Dirichlet, Dirichlet–Neumann and Neumann–Neumann boundary conditions are considered.

A system model for ultrasonic NDT based on the Physical Theory of Diffraction (PTD)

Simulation of ultrasonic Non Destructive Testing (NDT) is helpful for evaluating performances of inspection techniques and requires the modelling of waves scattered by defects. Two classical flaw scattering models have been previously usually employed and evaluated to deal with inspection of planar defects, the Kirchhoff approximation (KA) for simulating reflection and the Geometrical Theory of Diffraction (GTD) for simulating diffraction. Combining them so as to retain advantages of both, the Physical Theory of Diffraction (PTD) initially developed in electromagnetism has been recently extended to elastodynamics. In this paper a PTD-based system model is proposed for simulating the ultrasonic response of crack-like defects. It is also extended to provide good description of regions surrounding critical rays where the shear diffracted waves and head waves interfere. Both numerical and experimental validation of the PTD model is carried out in various practical NDT configurations, such as pulse echo and Time of Flight Diffraction (TOFD), involving both crack tip and corner echoes. Numerical validation involves comparison of this model with KA and GTD as well as the Finite-Element Method (FEM).

Edge Diffraction Coefficients around Critical Rays

The classical GTD (Geometrical Theory of Diffraction) gives a recipe, based on high-frequency asymptotics, for calculating edge diffraction coefficients in the geometrical regions where only diffracted waves propagate. The Uniform GTD extends this recipe to transition zones between irradiated and silent regions, known as penumbra. For many industrial materials, e.g. steels, and frequencies utlized in industrial ultrasonic transducers, that is, around 5 MHz, asymptotics suggested for description of geometrical regions supporting the head waves or transition regions surrounding their boundaries, known as critical rays, prove unsatisfactory. We present a numerical extension of GTD, which is based on a regularized, variable step Simpson's method for evaluating the edge diffraction coefficients in the regions of interference between head waves, diffracted waves and/or reflected waves. In mathematical terms, these are the regions of coalescence of three critical points - a branch point, stationary point and/or pole, respectively. We show that away from the shadow boundaries, near the critical rays the GTD still produces correct values of the edge diffraction coefficients.

Tronquée solutions of the painlevé II equation

We study special solutions of the Painleve II (PII) equation called tronquee solutions, i.e., those having no poles along one or more critical rays in the complex plane. They are parameterized by special monodromy data of the Lax pair equations. The manifold of the monodromy data for a general solution is a twodimensional complex manifold with one- and zero-dimensional singularities, which arise because there is no global parameterization of the manifold. We show that these and only these singularities (together with zeros of the parameterization) are related to the tronquee solutions of the PII equation. As an illustration, we consider the known Hastings-McLeod and Ablowitz-Segur solutions and some other solutions to show that they belong to the class of tronquee solutions and correspond to one or another type of singularity of the monodromy data.

Equiangular spirals provide lossless bends for lightpipes

High efficiency optical throughput is necessary for optical instruments and equipment, including lightpipes: non-imaging optical elements that guide light from one place to another. Typical applications of lightpipes include projector-engine illumination,1 LCD backlight systems,2 and automobile dashboard lighting.3 The shape of these elements vary but the design usually requires bends. Lightpipes can be optimized in a number of ways, e.g., controlling the light angular distribution or spatial distribution. One long-standing issue for lightpipes involves optimizing efficiency by reducing light leakage, especially at bends. Our recent work shows that by using equiangular-shaped spirals, lightpipes can guide light flux with arbitrary bend angles and no leakage, and this design method can also be extended to optical elements that split and mix light.4, 5 A lightpipe uses total internal reflection to guide light. Therefore, the incident angle of a light ray at the lightpipe’s guiding surface should be greater than or equal to the total internal reflection angle, θc , to keep light from leaking out. Hence, the angular distribution of guided rays inside the lightpipe is between -θc to θc . Thus, we can limit our discussion to identify a basic no-loss bent lightpipe geometric unit, which transfers incoming light with an angular distribution between -θc to θc to the other end of a lightpipe without leaks. The most crucial ray to be guided in the bent lightpipe is the ray incident from an inner bend point to the outer surface on the principle section. The key idea of our design is to choose an outer surface that ensures that all critical rays have an incident angle greater than critical angle, θc , so that they can be guided without any light loss. The natural geometric form of the nautilus, a type of mollusk, inspired our choice of shape for our no-loss bent lightpipe (see Figure 1). The shape of the nautilus’s shell is an equiangular spiral, which has a fixed angle between the ray from the center and the surface tangent. As shown in Figure 2, the proposed lossless bent lightpipe contains three parts. First, the straight AB part Figure 1. The general shape of a nautilus is an equiangular spiral.

Microwave propagation over the Earth: image inversion

An extremely accurate but simple asymptotic description for the path of a light ray propagating over a curved Earth with steady radial variations in refractive index is derived using simple scaling arguments. It is used to determine effectively exact analytic solutions for the path of rays through refractive-index profiles described in terms of patched quadratics. Such patched quadratics can be used to accurately describe almost all refractive-index profiles of practical interest. The results show that images generated by rays passing through a quadratic refractive-index profile are uniformly magnified in the vertical direction, and magnification and displacement observations can be used to determine the refractive-index profile parameters. For patched quadratics, observations of critical rays can be used to determine the thicknesses of the quadratic layers. An effectively exact solution is also obtained for exponential index profiles and this is used to determine the path of rays through a thin boundary layer attached to the Earth; an inferior mirage situation

The high-frequency asymptotic description of pulses radiated by a circular normal transducer into an elastic half-space

A new method for simulating the propagation of pulses radiated by a circular normal ultrasonic transducer which is directly coupled to a homogeneous and isotropic elastic half-space is proposed. Both nonuniform and uniform high-frequency asymptotics inside geometrical regions as well as boundary layers (penumbra, an axial region, and a vicinity of the critical rays) have been used to describe the transient field by means of harmonic synthesis. The nonuniform asymptotic formulas involving elementary or well-known special functions elucidate the physics of the problem and give explicit dependence of the radiated waves upon the model parameters. The formulas are applicable in the radiating near field which is the near-field with the evanescent wave zone excluded. The code based on the uniform asymptotics has been tested in all regions against an exact numerical solution. It is orders of magnitude faster, but in many realistic cases the accuracy does not suffer. The limits of applicability of the model have been established.

High-frequency asymptotic description of head waves and boundary layers surrounding the critical rays in an elastic half-space

The high-frequency asymptotic description of head waves as well as the field inside boundary layers surrounding the critical rays are obtained for two cases: (a) a point source, and (b) a circular transducer, both acting normally on an isotropic and homogeneous elastic half-space. The edge head waves underneath a circular transducer are described by the asymptotics of a higher order compared to those of direct compressional, edge compressional, and shear waves, but are still discernible in the radiating near zone and thus might be useful in nondestructive evaluation of industrial materials. The asymptotic formulas produced involve in geometrical zones elementary functions and inside boundary layers well-known special functions. Therefore, they allow us to elucidate the physics of the problem and can be used in writing computer codes which simulate the radiating near field of a circular transducer orders of magnitude faster than full numerical schemes. The formulas have been tested against exact integral solutions evaluated numerically.

A canonical problem for fluid-solid interfacial wave coupling

Wave-coupling involving defects or obstacles on fluid-solid interfaces is of recurrent interest in geophysics, transducer devices, structural acoustics, the acoustic microscope and related problems in non-destructive testing. Most theoretical analysis to date has, in effect, involved stress or pressure loadings along the interface, or scattering from surface inhomogeneities, that ultimately result in unmixed boundary value problems. A more complicated situation occurs if a displacement is prescribed over a region of the interface, and the rest of the interface is unloaded (or stress/pressure loaded); the resulting boundary value problem is mixed. This occurs if a rigid strip lies along part of the interface and will introduce several complications due to the presence of the edge. For simplicity a lubricated rigid strip is considered, i. e. it is smoothly bonded to the elastic substrate. To consider such mixed problems, e. g. the vibration of a finite rigid strip or diffraction by a finite strip, a canonical semi-infinite problem must be solved. It is the aim of this paper to solve the canonical problem associated with the vibrating strip exactly, and extract the form of the near strip edge stress, and displacement fields, and the far-field directivities associated with the radiated waves. The near strip edge results are checked using an invariant integral based upon a pseudo-energy momentum tensor and gives a useful independent check upon this piece of the analysis. The directivities will be useful in formulating a ray theory approach to finite strip problems. An asymptotic analysis of the solution, in the far field, is performed using a steepest descents approach and the far-field directivities are found explicitly. In the light fluid limit a transition analysis is required to determine uniform asymptotic solutions in an intermediate region over which the leaky Rayleigh wave will have an influence. In a similar manner to related work on the acoustic behaviour of fluid-loaded membranes and plates, there is beam formation in this intermediate region along critical rays.

Non-specular reflection phenomena at a fluid–solid boundary

This paper contains a study of the coupled dynamics of a static, compressible fluid in planar contact with an elastic half-space. The interaction is forced by an incident acoustic bulk wave generated by a uniform, two-dimensional, line source located in the fluid under specified conditions of light fluid loading. An analysis of the exact solution to the corresponding boundary value problem, obtained by Fourier transform methods, is presented and this shows that the ultimate far-field acoustic radiation pattern adopts the usual form of an outgoing cylindrical wave, with an amplitude dependent upon the polar angle θ and decaying algebraically with distance as R -1/2 , where ( R , θ ) are polar coordinates measured from the image source. This is so only for sufficiently large values of R (made precise, in terms of scalings with inverse powers of a small fluid-loading parameter ϵ , in the text) and at lesser distances, a different asymptotic response is found along certain critical directions. This is related to the generation of boundary waves, either of ‘free-mode ’ type or else surface disturbances induced by the transmitted elastic waves in the solid. Whatever their specific nature, the corresponding structure is determined by the coalescence (under certain conditions) of a saddle point with an appropriate singularity in the complex wavenumber plane. Each such occurrence is described in some detail and a suitable transitional solution is derived to provide a full and uniform description valid across these critical rays. In particular, a saddle-point─pole coalescence occurs in the course of the analysis of the free-mode type waves (e.g. the leaky Rayleigh wave and the Schölte wave) and it is noted and demonstrated that standard techniques for dealing with this are inadequate, owing to the presence of a simple zero of the integrand under the same conditions. An alternative method for dealing with this non-uniformity is then derived and the implications of this analysis for the acoustic response are discussed. In particular, it is shown that the far-field disturbance in the fluid along the two critical directions of this nature is always of algebraically small O ( k 0 -1/2 R -1/2 ) amplitude ( k 0 being the acoustic wavenumber), but the precise multiplicative factor changes over a range of distances R = O ( k 0 -1 ϵ -2 ) with smooth transition afforded by a Fresnel-type integral.

AN ANALYTICAL RAYPATH APPROACH TO THE REFRACTION WAVEFRONT METHOD1

AbstractSeismic refraction surveying is still an important tool for determining the geometries and elastic wave propagation velocities of near‐surface layers. Many analytical and graphical methods have been developed over the years for refraction interpretation, and these can be classified into two basic groups. The first group visualizes critically refracted rays converging on a common surface position, while the second group, which includes the wavefront methods, makes use of the critical rays emerging from a common point on the refractor.The method described in this paper is an analytical approach to the wavefront methods. The reverse refracted ray received by a geophone is intersected by the forward refracted rays received by subsequent geophones and a common critical refraction point on the refractor is estimated after a series of comparisons. This process is repeated for each geophone to yield the geometry and the velocity of the refractor. Several interpolations are performed to achieve a better accuracy.Palmer's models are used to test the efficiency of the algorithm. The results are presented together with those of other methods applied to the same models.

Corrections for critical rays in 2-D seismic modelling

The usual high-frequency (ω) ray method of modelling reflections from smooth boundaries does not account for interference arising near critical angles of incidence. When the reflector is effectively convex towards the incident-wave side, the ray method fails (i) in a transition zone of width O(ω-1/2) separating regions of partial and total ray reflection, (ii) for the transmission in a zone along the reflector having thickness O(ω-1/2) and where a whispering-gallery is formed by multiply-reflected turning waves, and (iii) for turning rays refracted back into the first medium at distances from the critical point less than O(mω-1/4), where m is the number of turns. The interference is a local effect and approximate analytic waveforms can be found for laterally varying media. The 2-D acoustic case is described here. The fields near the point of critical incidence and about the critically reflected ray are found in a way similar to the grazing ray solution described in an earlier paper. However, through the critical ray problem the first two terms of the asymptotic expansion must be considered. The whispering-gallery is obtained by exploiting work on modes by Buldyrev, Lewis, Ludwig and others. The modal solution is then matched with the solution near the critical point to determine the initial modal amplitudes. The individual ray contributions separate from the modes as the waves progress along the boundary. The modal form of solution is needed, even if one wishes to demonstrate only the matching with ray theory for the primary refracted/turning wave near the critical point. This is because the transition zone for the refraction is much wider than that for the reflected wave. Though it is more complicated than the earlier grazing ray solution, the critical-ray diffracted wavefield is still continued way from the boundary using ray coordinates. Hence, it is not difficult to provide ray tracing programs with corrections to ray synthetics. Some preliminary numerical examples are presented to indicate the waveforms and potential significance of these corrections. They are small in comparison with the total reflection, but they are very significant in comparison with the weaker refracted waves. Even where ray theory is (just) valid, for practical reasons one may wish to avoid tracing multiple reflections close to a numerically specified boundary. The boundary-layer formulae provide a more ‘natural’ solution to both the theoretical and numerical problems for the refracted part of the field.

A note on the location of complex zeros of solutions of linear differential equations

For second-order equations W′′ + A( z) w = 0, where A( z) is a polynomial, there is a classical result which states that there are (deg A) + 2 critical rays, arg z = φj such that for any ∊ > 0, all but finitely many zeros of any solution f (z) ≢ 0 lie in the union of the ∊-sectors . In this paper, we determine when an analogous result holds for higher-order equations.

Critical visibility and outward rays

The notion ofclear visibility has appeared recently in the literature concerning the geometry of starshaped sets. A somewhat converse notion, that ofcritical visibility, is explored here. As a by-product it is proved that the points of local nonconvexity of a closed connected set S determine the nontrivial portions of the boundary of the star of a point in S. Paragraph 4 is devoted to the study of the concepts ofoutward ray through a boundary point andinner stem of that point. A new characterization of the convex kernel of a starshaped set results from this study, and Krasnoselsky-type theorems are obtained in the planar case.

A note on the location of complex zeros of solutions of linear differential equations

For second-order equations W′′ + A( z) w = 0, where A( z) is a polynomial, there is a classical result which states that there are (deg A) + 2 critical rays, arg z = φj such that for any ∊ > 0, all but finitely many zeros of any solution f (z) ≢ 0 lie in the union of the ∊-sectors . In this paper, we determine when an analogous result holds for higher-order equations.

A note on the zeros of solutions [wacute]+P(z)w= 0 where P is a polynomial

For the class of equations described in the title, there is a classical result which states that there are (deg P) + 2 critical rays, arg z = θj, such that for any e > 0, all but finitely many zeros of any solution f(z) ≢ 0 must lie in the union of the sectors |arg z - θj| < e. We prove that any infinite set of zeros in such a sector (for sufficiently small e) must actually approach a definite ray, which is a translate ofthe critical ray (and which can be explicitely calculated). In addition, we estimate the rate at which the zeros approach the ray, thus obtaining new information on the exact location of the zeros. The class of equations treated here contains equations which arise in applications in other areas, such as Airy's equation and Titchmarsh's equation