Vicinity Of Caustics Research Articles

Mathematical Modeling of the Amplitude, Angular, and Time Characteristics of Short Radio Waves in Slightly Oblique Sounding of the Ionosphere

The method of bicharacteristics is used to study the propagation of decameter radio waves in the ionosphere containing the E, F1, and F2 layers and a traveling ionospheric disturbance (TID) upon slightly oblique sounding with frequency-modulated signals. The propagation of both ordinary and extraordinary waves is considered, the ionograms are compared, and their relative shift is studied. The structure of ionograms in the vicinity of caustics, which appear at the reception point at frequencies reflected in the vicinity of the E layer and TID, is studied. The divergence of the ray flux at the point of radiation reception, which determines a decrease in the amplitude, is calculated.

A model for the investigation of the second-order structure of caustic formations in dispersed flows

The formation of caustics by inertial particles is distinctive of dispersed flows. Their pressureless nature allows crossing trajectories resulting in singularities that cannot be captured accurately by standard Lagrangian approaches due to their fine spatial scale. A promising method for the investigation of caustics is the Osiptsov method or fully Lagrangian approach (FLA). The FLA has the advantage of identifying caustics, but its applicability is hindered by the occurrence of singularities. We present an original robust framework based on the FLA that provides an explicit expression of the dispersed phase structure that does not degenerate in the vicinity of caustics, using a single representative particle. The FLA is extended to account for the Hessian of the dispersed continuum (DC). It demonstrates the integrability of the FLA number density and allows for the calculation of the number density on a given length scale, retaining the functionality of the FLA. Number density models based on the second-order representation of the DC and on the one-dimensional structure of the particle distribution, that account for the anisotropy of the DC on caustics, are derived and applied for analytical flows. The number density is linked to a finite length scale, needed for the introduction of the FLA to spatially filtered flow fields. Finally, the method is used for the calculation of the interparticle separation on caustics. The identification of the structure of caustics presented in this work paves the way to a robust understanding of the mechanisms of particle accumulation.

Measurement of the Gouy phase anomaly for electron waves

We measure the Gouy phase anomaly for matter waves using in-line holography to retrieve the full complex field of an astigmatic electron wave function. Sequential phase shifts of \ensuremath{\pi}/2 rad are observed for electron trajectories along the optic axis that pass through each line-focus caustic of subnanometer transverse width. Our observations demonstrate that anomalous phase shifts of matter waves in the vicinity of caustics can be robustly measured using phase retrieval, extending the current scope of singular electron optics.

Controlling phase space caustics in the semiclassical coherent state propagator

The semiclassical formula for the quantum propagator in the coherent state representation 〈 z ″ | e - i H ˆ T / ℏ | z ′ 〉 is not free from the problem of caustics. These are singular points along the complex classical trajectories specified by z′, z″ and T where the usual quadratic approximation fails, leading to divergences in the semiclassical formula. In this paper, we derive third order approximations for this propagator that remain finite in the vicinity of caustics. We use Maslov’s method and the dual representation proposed in Phys. Rev. Lett. 95, 050405 (2005) to derive uniform, regular and transitional semiclassical approximations for coherent state propagator in systems with two degrees of freedom.

Seismic Wave Field in the Vicinity of Caustics and Higher-Order Travel Time Derivatives

In this paper we consider time-domain asymptotic solutions to the elastodynamic equation. We briefly describe the technique of constructing and analyzing the integral formulas describing seismic wave fields. The formulas are valid at regular points and in the vicinity of caustics.

Numerical Simulation of Sonic Boom Focusing

A numerical method is presented to simulate the focusing of sonic booms. In the vicinity of caustics, the pressure satisfies the nonlinear Tricomi equation. To solve this equation, an iterative algorithm, based on an unsteady version of the equation, is used. The algorithm is a modification of the pseudospectral code used for solving the Zabolotskaya-Khokhlov (KZ) equation. In the linear case, the code is validated by comparison with analytical solutions. For an incoming N wave, the shape of the outgoing wave observed in flight tests is recovered. In the nonlinear case, no analytical solution is known to compare with the numerical output. Validation of the numerical scheme is completed by means of four different tests. First, comparisons with the linear case shows that the numerical solutions behave as expected from the physics. Second, the solution after convergence is proved to be independent of the initial guess. Third, the maximal signal amplitude is proved to converge while increasing the discretization. Finally, the numerical scheme is checked against the nonlinear Guiraud's scaling law. In the last part, an application to the focusing of Concorde sonic boom in acceleration is presented, and ways of reducing focused booms are discussed.

Shell function method for computing the acoustic pressure field in arbitrary inhomogeneous media

The shell function method is presented as a new technique to evaluate the acoustic pressure field produced by a practical source of finite area, in a medium having an arbitrary spatially varying sound speed in three dimensions. Instead of initially subdividing the source surface area into many area elements, the shell function method determines the surface element of source area associated with the ray received at the observation point for a specified travel time. Special attention is given to the deficiency of conventional ray tracing for the case of a caustic, where an infinite pressure is predicted. On the other hand, the shell function method takes into account both the convergent state of a sound beam and the corresponding surface element of the source area. As a consequence, the sound pressure at and in the vicinity of caustics can be given an appropriate value directly. Numerical examples are included to show the computational procedure of the shell function method and the ensuing results.

Maslov theory for surface wave propagation on a laterally heterogeneous earth

SUMMARY The usual JWKB ray-theoretical description of Love and Rayleigh surface wave propagation on a smooth, laterally heterogeneous earth model breaks down in the vicinity of caustics, near the source and its antipode. In this paper we use Maslov theory to obtain a representation of the wavefield that is valid everywhere, even in the presence of caustics. The surface wave trajectories lie on a 3-D manifold in 4-D phase space (θ, φ, kθ, kφ), where θ is the colatitude, φ is the longitude, and kθ and kφ are the covariant components of the wave vector. There are no caustics in phase space; it is only when the rays are projected onto configuration space (θ, φ), the mixed spaces (kθ, φ) and (θ, kφ), or momentum space (kθ, kφ), that caustics occur. The essential strategy is to employ a mixed-space or momentum-space representation in the vicinity of configuration-space caustics, where the (θ, φ) representation fails. By this means we obtain a uniformly valid Green's tensor and an explicit asymptotic expression for the surface wave response to a moment tensor source.

Application of ray theory with beam displacement to three-dimensional sound propagation in a wedge-shaped ocean

Ray theory with beam displacement (RTBD) has been developed to evaluate the acoustic field in three dimensions in a wedge-shaped ocean with a lossy fluid bottom. An algorithm for tracing eigenrays in three dimensions is described. A procedure for determining the location of caustics in three dimensions is also presented. It is shown that the density of caustics reduces with increasing bottom attenuation. Sharp spurious peaks appear in the acoustic field in the vicinity of caustics. These caustic-related peaks can be readily identified and smoothed out, and no further corrections would be necessary in most situations of practical interest. Agreement between the smoothed RTBD solution and the ‘‘exact’’ two-way coupled-mode solution is quite good for the two-dimensional problem of up-slope propagation in the ASA benchmark wedge with a penetrable lossy bottom. Numerical results for the three-dimensional problem are presented for several bottom slopes and directions of propagation.

Wavefields near transverse cusp caustics produced by reflecting ultrasonic transients and tone bursts from curved surfaces

Abstract : Ultrasonic wavefields reflected from curved surfaces were studied in the vicinity of caustics, Acoustical and optical transverse cusp diffraction catastrophes produced by reflections from a curved metal surface in water were imaged by displaying the amplitude or intensity in an observation plane transverse to the general direction of propagation. The optical image was used to locate the cusp point in the observation plane. Acoustical diffraction patterns for sine waves, described by the Pearcey function, were calculated with the parameters determined by the experimental setup leaving no adjustable scaling parameters. The calculated and experimental acoustical diffraction patterns showed good agreement near the cusp point. The acoustical diffraction pattern showed the expected mirror symmetry about an axis. The pattern was shown to scale properly with frequency. The transverse cusp caustic separates space into a region with three rays and a region with one ray. Inside the caustic there are three rays, on the curve two of the rays merge and disappear leaving one ray outside. Transient signals reflected from curved surfaces exhibited the merging and disappearance of rays on the caustic. Relative arrival times for signals in calculated and recorded time traces agree well. The relation to the wavefront parameters of the temporal orientation of the travel time surface is discussed. The general shape of the travel time surface is that of the swallow tail caustic surface.

High-Frequency Wave Propagation in Random Media—a Unified Approach

High-frequency wave propagation in a random medium over long propagation distances, where the fluctuations in the wave field are not small and caustic formation is likely, is studied. Techniques based on the geometrical optics approximation or ray theory as well as those based on the parabolic wave equation are examined. It is shown that for small fluctuations in the refractive index of order $\sigma ( {\sigma \ll 1} )$ and long propagation distances of order $\sigma ^{ - 2/3} $, that both these methods are equivalent at high frequencies, at least away from caustics. Since previous theories have identified this $\sigma ( \sigma ^{ - 2/3} )$ distance scale as that on which random caustics first appear, we also derive a regularization of the ray approximation, which is called the beam method, and which is uniformly valid in the vicinity of caustics. On the $\sigma ^{ - 2/3} $ scale, the random ray process converges to a diffusion Markov process, and equations can be derived for the joint distribution of an arbitrary number of ray positions and ray tube areas. It is thus possible to compute arbitrary moments of the wave field for both the rays and the beams. It is shown that the parabolic wave equation can also be scaled for a Markovian limit for propagation distances of order $\sigma ^{ - 2/ 3} $ from which we derive the parabolic moment equations. This is the first derivation of the moment equations that gives the distance scale on which they are valid. The first and second moments can be computed explicitly for both the rays and beams. The results agree with the parabolic equation. The fourth moment of the field at four distant points as computed using ray theory is seen to be singular when all the points coalesce; i.e., the computed scintillation index diverges. The correct result is obtained using the beam method and yields \[ \left\langle {| v |^4 } \right\rangle = \log ( \omega \sigma ^{2/3} )f( t ) + O( 1 ). \] Here t is the scaled propagation distance and $\omega $ is the frequency. The function $f( t )$ is universal depending on the statistics of the medium only through a single distance scale parameter and can be related to the probability of caustic formation.

Hybrid integration technique for efficient and accurate computation of diffraction integrals

Numerical evaluation of diffraction integrals is often time consuming because the integrand is a rapidly varying function. Therefore much effort has been expended in developing asymptotic techniques. However, these techniques are invalid in many circumstances, for example, in the vicinity of caustics and cusps of geometrical or diffracted rays. When asymptotic theories fail, efficient computation can be obtained by using a hybrid integration technique, which combines numerical integration with asymptotic methods. The accuracy, applicability, and computational savings of this hybrid technique are examined.

Hybrid (ray)-(parabolic equation) analysis of propagation in ocean acoustic guiding environments

Acoustic ray theory provides an effective means for predicting propagation in general environments. Ray theory fails in the vicinity of caustics and must there be augmented by uniform asymptotic treatment. However, even such corrections are inadequate at long ranges in a ducting channel, where the caustics due to multiple reflected and/or refracted rays accumulate at locations near the duct axis. To repair this deficiency of ray acoustics, the caustic forming ray spectra have previously been replaced by guided modes in a self-consistent ray-mode scheme. It is now shown that the troublesome ray spectra at long ranges can be replaced as well by a narrow-angle parabolic equation (PE) approximation to the full pressure field. Taken together, the hybrid ray-PE form can be regarded either as correcting via PE the failure of ray acoustics near the guiding axis or as removing the narrow-angle restriction from PE by filling the wide angle spectra with rays. The theory is developed first for a range-independent surface duct, and then for a range-dependent guiding-to-antiguiding transition. Unlike the adiabatic modes, the numerical PE algorithm can accommodate changes from guiding to antiguiding. The analogous problem of propagation along a circular concave boundary and along a boundary with concave–convex transition is treated briefly in the Appendix.

Caustics in a simpleSU(2) lattice gauge theory model

This paper describes the beginning of an attempt to study the weak coupling limit of lattice gauge theories, using the Hamiltonian formulation and the semiclassical approximation. The topics are caustics and the behavior of the ground state wave function in the vicinity of caustics. For two very simpleSU(2) models (one plaquette, two plaquettes) we demonstrate the existence of caustics, determine their locations and study the peaking behavior of the ground state wave function on them in the limitg2→0.

Causality, caustics, and the structure of transient wave fields

Transient acoustic wave fields in the vicinity of caustics in both two and three space dimensions are considered. In the vicinity of some possible caustics, the time history of the wave field due to an impulsive source includes a precursor. This noncausal behavior is reconciled with the physical principle of causality by requiring that such caustics be a part of certain higher dimensional caustics. With this constraint all noncausal arrivals are preceded by a causal arrival and the total time history is causal. This constraint is used to derive some simple general properties of transient wave fields in the short wave limit, e.g., the number of arrivals must be an odd number.

A parabolic equation for the combined refraction–diffraction of Stokes waves by mildly varying topography

A parabolic equation governing the leading-order amplitude for a forward-scattered Stokes wave is derived using a multiple-scale perturbation method, and the connection between the linearized version and a previously derived approximation of the linear mild slope equation is investigated. Two examples are studied numerically for the situation where linear refraction theory leads to caustics, and the nonlinear model is shown to predict the development of wave-jump conditions and significant reductions in amplitude in the vicinity of caustics.

Synthetic body wave seismograms for laterally varying layered structures by the Gaussian beam method

Summary. Several approaches to computing body wave seismograms in 2–D and 3–D laterally inhomogeneous layered structures are suggested. They are based on the Gaussian beam method, which has been recently applied to the evaluation of time-harmonic high-frequency wavefields in inhomogeneous media. Three variants are discussed in some detail: the spectral method, the convolutory method and the wave-packet method. The most promising seems to be the wave-packet approach. In this approach, the wavefield, generated by a source, is expanded into a system of wave packets, which propagate along rays from the source in all directions. The wave packets change their properties due to diffusion, spreading, reflections/transmissions, etc. The resulting seismogram at any point of the medium is then obtained as a superposition of those packets which propagate close to the point. The final expressions in all the three methods are regular even in regions, in which the ray method fails, e.g. in the vicinity of caustics, in the critical region, at boundaries between shadow and illuminated regions, etc. Moreover, they are not as sensitive to the minor details of the medium as the ray method and, what is more, they remove the time-consuming two-point ray tracing from computations. Numerical examples of synthetic seismograms computed by the wave-packet approach are presented.

Application of the WKBJ Green’s function to acoustic propagation in horizontally stratified oceans

The Green’s function (response to an impulsive point source) for ocean acoustic propagation is found using the WKBJ seismogram technique [Chapman, Geophys. J. R. Astron. Soc. 54, 481–518 (1978)]. It is shown that the WKBJ Green’s function reduces to geometric ray theory; in the vicinity of caustics it gives the well-known Airy function range dependence of each harmonic component of the acoustic field. Arrival patterns constructed using the WKBJ Green’s function are in excellent agreement with a solution to the parabolic equation (PE) except when the source and/or receiver depth is near the turning depth of a geometric ray. Agreement with a measured arrival pattern is also found to be good.

Boundaries of geometrical optics applicability and related problems

On the basis of the Huygens‐Fresnel principle as well as the concept of a Fresnel volume (the part of space essential for diffraction), the following criteria of the ray method applicability are suggested. (1) Parameters of the medium involved and characteristics of the wave (amplitude and phase gradient) should vary just slightly over the cross section of the Fresnel volume. (2) The phase difference of two rays arriving at the same point of space should exceed π. These criteria may be evaluated from the conditions of applicability of the stationary phase method. The effectiveness of the suggested criteria is demonstrated by considering various problems allowing either rigorous or approximate solutions. Moreover, one can formulate generalized criteria for the case of space‐time geometrical optics and for geometrical theory of diffraction as well. And finally, some related problems are considered, the most important one being estimation of the wave field intensity in the vicinity of caustics in terms of the energy conservation law in a ray tube corresponding to the caustical zone width.

Numerical calculation of the intensity distribution in sound channels using coherence theory

We have demonstrated that coherence theory can be applied to the practical computation problem of the propagation of a beam in an infinite half‐space with stratified but arbitrarily varying speed of sound variations. The intensity distributions found in this manner are the same as those of an equivalent point source in regions removed from the caustics of the geometrical ray family, but the intensity formula is continuous everywhere in the field and may be used in the vicinity of caustics. The computer time and storage requirements are equivalent to those of a ray program.