Scalar Wave Field Research Articles

Time-dependent plane-wave spectrum representations for radiation from volume source distributions

A new time-domain spectral theory for radiation from a time-dependent source distribution, is presented. The full spectral representation is based on a Radon transform of the source distribution in the four-dimensional space-time domain and consists of time-dependent plane waves that propagate in all space directions and with all (spectral) propagation speeds vκ. This operation, termed the slant stack transform, involves projection of the time-dependent source distribution along planes normal to the spectral propagation direction and stacking them with a progressive delay corresponding to the spectral propagation speed vκ along this direction. Outside the source domain, this three-fold representation may be contracted into a two-fold representation consisting of time-dependent plane waves that satisfy the spectral constraint vκ=c with c being the medium velocity. In the two-fold representation, however, the complete spectral representation involves both propagating time-dependent plane waves and evanescent time-dependent plane waves. We explore the separate role of these spectral constituents in establishing the causal field, and determine the space-time regions where the field is described only by the propagating spectrum. The spectral theory is presented here for scalar wave fields, but it may readily be extended to vector electromagnetic fields.

Path-linking interpretation of Kirchhoff diffraction

The Kirchhoff-diffraction integral is often used to describe the (scalar) wave field from a monochromatic point source in the presence of ‘opaque’ screens. Despite criticisms that can be made of its ‘derivation’, the Kirchhoff field is an exact solution of the wave equation, and exactly obeys definite, though unusual, boundary conditions (Kottler 1923, 1965). Here, the path-integral picture of wave fields is used to interpret the Kirchhoff-diffraction field in terms of all conceivable propagation paths, whether or not they pass through the opaque screens. Specifically, it is noted that the Kirchhoff field equals Ʃ(1 ─ m )ψ m , where the sum is over all integers m , and ψ m is the wave field due to all paths from the source to the field point for which the number of outward screen crossings minus the number of backwards screen crossings is m . Expressed more topologically, m is the total linking number of a path, when closed by any unobstructed path, with the screen edge lines. Other models of diffraction by screens are compared with Kirchhoff diffraction in the path interpretation.

A new representation of a scalar wave field diffracted by solid obstacles that satisfies both the wave equation and hard or soft boundary conditions

A new diffraction principle formulated by assuming a hypothetical observer at a point of interest in the sound field and considering the wave propagation in a space seen by the observer virtually is proposed in this paper. This new approach allows one to grasp a simple concept of diffraction by solid obstacles and to formulate the representation of diffracted waves that satisfies both the wave equation and hard or soft boundary conditions. Thus the diffraction problem for the scalar wave equation is solved in principle by this approach. Heuristic derivation of the new principle of diffraction, followed by the theoretical analysis, is presented and the meaning and limitation of the conventional representations of diffracted waves are analyzed in the light of the new principle.

Sound and light propagation in a weakly inhomogeneous atmosphere†

Abstract Small-scale inhomogeneities caused by atmospheric turbulence have a considerable effect on sound and light propagation, producing the fluctuations of these wave fields. V A Krasilnikov [1, 2] performed experiments on phase and amplitude variations of a sound wave propagating through the atmosphere. Fluctuations of light-wave parameters occur, for example, in the well known phenomenon of star scintillation, apparently strongly connected with turbulent irregularities of the atmospheric temperature field [3, 4]‡ ‡ The theoretical analysis of turbulent fluctuations of the temperature field and some corresponding experimental data can be found in [5–7]. . Some calculations of phase (arrival angle) and amplitude fluctuations for a wave propagating through a turbulent medium are described by Krasilnikov [1, 3, 8]. All these and similar calculations are based on the geometrical optics (acoustics) approximation, which may be the reason for disagreement between calculation and experimental data in some cases. Thus, for example, amplitude fluctuations in the geometrical approximation turn out to be proportional to the distance of propagation through a turbulent medium to the power of 3/2. However, observations usually show much slower fluctuation growth. This paper represents an attempt to consider the problem of amplitude and phase variations for a scalar wave field in terms of more general equations including some diffraction effects. Incidentally, the range of validity of geometrical optics theory becomes clear.

Solution of the inverse-scattering problem for grazing incidence upon a rough surface

The inverse-scattering problem for a scalar wave field incident at grazing angles upon a one-dimensional moderately rough surface is considered. The problem is solved directly by treating the scattering integral as an integral equation in the unknown field derivative at the surface and coupling this to a simple equation relating the derivative to the surface itself. An algorithm is described for the solution of this system, and results in which complicated rough surfaces are accurately reconstructed are presented.

Direct solution of the inverse problem for rough surface scattering at grazing incidence

Considers the inverse scattering problem for a scalar wavefield incident at grazing angles on a one-dimensional rough surface. The problem is formulated first as a pair of coupled integral equations in two unknown functions, knowledge of which immediately yields the surface. A method is described for the direct approximate solution of this system. Preliminary results are presented in groups of complicated rough surfaces which are closely recaptured in all details except for scale.

Gravitational and quantum effects in rotating cosmological models

Specific effects of the dynamics of (spinor and scalar) wave fields are considered in rotating uniform Godel-type cosmological models. It is shown that the gravitational interaction of the spinor field can be reduced to the interaction between its pseudovector current and the angular velocity of space-time rotation and is similar to its interaction with the pseudotrace of the space-time twisting. The mean values of energy-momentum tensor of the quantized scalar field in vacuum are calculated in rotating cosmological models and the difference between these values and their mean counterparts in vacuum is determined for Friedman's nonrotating cosmological models.

Semiclassical theory of spin-orbit coupling.

Newly developed methods for finding the semiclassical or WKB eigenvalues and eigenfunctions for vector wave fields are applied to spinning particles moving in central potentials subject to spin-orbit forces. The new methods are like the familiar Bohr-Sommerfeld or Einstein-Brillouin-Keller quantization methods for scalar wave fields, but involve additional issues relating to Berry's phase, gauge structures, and monopolelike singularities. All these extra issues occur in the asymptotics of spin-orbit coupling. The role of angular momentum and conservation laws is particularly interesting, due to the role played by gauge fields in the classical-quantum correspondence. The work presented is a contribution to ongoing semiclassical studies of shell structure in nuclei and other fermion systems such as metal clusters, as well as an example of the general methods of semiclassical vector wave quantization.

Numerical Solution of the Second Moment Equation for Waves in an Inhomogeneous Waveguide

An accurate numerical method for the solution of the second moment equation is presented. This may be applied to scalar wavefields in a wide variety of scattering regimes and inhomogeneous waveguides, for many of which explicit analytical solutions cannot be obtained. The method is applied here to the solution of a waveguide similar to the acoustic near-surface ocean channel. For the case of a Gaussian beam in a parabolic channel the method yields the asymptotic behaviour of focusing with increasing scattering, expressed as functional laws.

Multilinear array manifold interpolation

Two algorithmic solutions for interpolating between array manifold grid points are presented. The algorithms are for sensor arrays in vector or in scalar wavefields. The algorithms make successful use of the condition that the reciprocal DOA (direction-of-arrival) spectrum is a multilinear function in the small, i.e. over the region of adjacent grid points. Since polarization is a linear parameter subspace, a DOA spectrum can be computed without recourse to the unknown polarizations. Then, vector interpolation is replaced by bivector interpolation to address the radio wavefield case. The approach distinguishes between sensor arrays in scalar wavefields (e.g. acoustic) and those in vector wavefields (e.g. electromagnetic). Since both storage and computational load vary with the reciprocal of the grid spacing or its square, the design relationship between desired accuracy and the required array manifold grid spacing is given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the exact Kirchhoff and Rayleigh-Sommerfeld theories for the focusing of an infinite scalar spherical wave-field

This communication discusses the exact analytic solutions to three well-known scalar diffraction theories - the Kirchhoff and first and second Rayleigh-Sommerfeld theories - for the focusing of a monochromatic spherical incident wave in an infiniteaperture system.

Multicomponent Seismic Reflection Profiling–Some Scale Model Experiments

Reflection experiments have been carried out on simple two-dimensional laboratory models to demonstrate the advantages of multicomponent profiling over conventional scalar wavefield recording. The 2 ´ 2 matrix of observations were taken with a biaxial piezoelectric receiver (vertical and horizontal receivers) and two piezoelectric source types ? vertical force (P wave) and horizontal force (S wave) ? to record the complete vector wavefield. The four component data were processed using a controlled direction reception (CDR) filter, which transforms the data into four sections for interpretation: PP, PS, SP and SS, including all mode conversions. In these sections not only is the specified linearly polarised wave type enhanced but coherent noise such as Rayleigh waves (elliptic polarisation) and incoherent background noise (isotropically polarised) is suppressed. With a single component source and a single component receiver, recording and interpretation of the various wavefields is difficult. Using vector wavefield recording and processing, it is easy to separate and identify compressional-wave reflections, shear-wave reflections, and all mode converted waves (PS reflections and SP reflections). The availability of S wave sections, in addition to conventional P wave sections, lends new insight to seismic interpretation. It improves the accuracy in prediction/seismic mapping and yields better vertical and horizontal resolution of the structure. In addition, the multicomponent sections contain diagnostic information on lithology, porosity, fluid content and anisotropy of the medium.

The Contributions of Homogeneous and Evanescent Plane Waves to the Scalar Optical Field: Exact Diffraction Formulae

Abstract The angular spectrum representation of a scalar wave field separates naturally into two parts–the contribution of homogeneous plane waves and the contribution of inhomogeneous (or evanescent) plane waves. Exact diffraction formulae are derived for the separate contributions for an arbitrary incident field, when the field is monochromatic, and also when the field has arbitrary time-dependence.

Diffraction by conical surfaces at high frequencies

We consider the scalar wave field described by the Helmholtz equation generated by a point source or by a plane wave in the presence of a conical obstacle of a rather arbitrary cross-section. The solution is constructed in form of Watson's integral. The latter is investigated in a high-frequency approximation, and as a result we obtain formulae for the scattering amplitude of the spherical wave diffracted by the vertex of a cone. We illustrate the approach proposed by considering some examples dealing with diffraction by cones.

A spectral domain method for multiple scattering in discrete random media

In an earlier paper (see ibid., vol.36, p.1114-28, 1988) a spectral-domain method was developed for analyzing multiply scattered scalar wavefields propagating in continuous random media. This method is extended to accommodate vector wavefields propagating in discrete random media. The two-dimensional Fourier spectra of vector wavefields propagating in the forward and backward directions are characterized by a pair of coupled first-order differential equations. Dyadic scattering functions characterize the local interaction of the wavefields with the random medium. The results are restricted to sparse distributions whereby the dyadic scattering functions are easily computed. The first- and second-order moments of the vector wavefields can be computed by invoking an assumption essentially equivalent to the Markov approximation as it is applied to scalar wavefields propagating in continuous random media. A complete solution for the coherent wavefield is derived and compared to known results. The results are essentially equivalent to those obtained by using the effective field approximation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The edge wave superposition method (2-D scalar problem)

SUMMARY A method of computation of high-frequency asymptotic scalar wavefields in 2-D inhomogeneous media is suggested. It is based on the fact that the wave equation has two linearly independent high-frequency asymptotic fundamental solutions. The first one coincides with zero-order ray theory solution, the second is the edge wave. Unlike the Gaussian beam method, based on superposition of the first fundamental solutions, the presented method exploits the existence of the second fundamental solution. The wavefield is expressed by the superposition of edge waves, scattering from each point of an arbitrary contour where the initial data are known. A multicontour superposition technique allows consideration of diffractions, caused by unsmooth interfaces. The computing algorithm is expressed in matrix product form which is very convenient for wave modelling.

A new approach to modeling the electromagnetic response of conductive media

We introduce a new and potentially useful method for computing electromagnetic (EM) responses of arbitrary conductivity distributions in the earth. The diffusive EM field is known to have a unique integral representation in terms of a fictitious wave field that satisfies a wave equation. We show that this integral transform can be extended to include vector fields. Our algorithm takes advantage of this relationship between the wave field and the actual EM field. Specifically, numerical computation is carried out for the wave field, and the result is transformed back to the EM field in the time domain. The proposed approach has been successfully demonstrated using two‐dimensional (2‐D) models. The appropriate TE‐mode diffusion equation in the time domain for the electric field is initially transformed into a scalar wave equation in an imaginary q domain, where q is a time‐like variable. The corresponding scalar wave field is computed numerically using an explicit q‐stepping technique. Standard finite‐difference methods are used to approximate the fields, and absorbing boundary conditions are implemented. The computed wave field is then transformed back to the time domain. The result agrees fairly well with the solution computed directly in the time domain. We also present an approach for general three‐dimensional (3‐D) EM problems for future studies. In this approach, Maxwell’s equations in the time domain are first transformed into a system of coupled first‐order wave equations in the q domain. These coupled equations are slightly modified and then cast into a “symmetric” and “divergence‐free” form. We show that it is to this particular form of equations that numerical schemes developed for solving wave equations can be applied efficiently.

Boundary element formulation for sound field problems

AbstractSince the sound field is a scalar field, the steady‐state field is expressed by a scalar Helmholtz equation. In application of the boundary element method to this problem, many techniques which have been developed in the two‐dimensional boundary element method can be used, and a technique of spherical trigonometry also is used in evaluating a solid angle. The radiation impedance of a breathing sphere which was evaluated by the present method agreed very well with the analytical value. The present method is applicable to other scalar wave fields as well as to vector fields if the boundary conditions are treated carefully.

Phase saddles and dislocations in two-dimensional waves such as the tides

A two-dimensional scalar wavefield of fixed frequency contains, in general, points where the amplitude is zero and the phase is indeter­minate. On a map of contours of equal phase these wave dislocations (interference nulls) are accompanied by saddles. When an external parameter is changed dislocations can be created in pairs or a pair can meet and destroy one another. For the simplest single-frequency wave equation it is a topological necessity that two saddles should participate in this event; moreover, they have to lie, in the final stage before annihilation, on the circle whose diameter is the line joining the dislocations. Examples are given to show how this basic pattern is always ultimately attained even when initially the configuration is quite different. In tidal theory, where the dislocations are amphidromic points, the external parameter that moves them can be the frequency. An example of an annihilation event occurs in the South Atlantic, and a close pair of amphidromic points may explain anomalous tidal observations from the Antarctic Peninsula. The tidal current, as distinct from the tidal rise and fall, provides an example of a two or three-dimensional vector field, and it is pointed out that the singularities in this field are precisely the same as those to be found in the polarization field of an electromagnetic wave.

A spectral-domain method for multiple scattering in continuous randomly irregular media

A spectral-domain method for computing the first and second-order moments of a scalar wavefield propagating in a continuous randomly irregular medium is described. The scattering is characterized by the moments of incremental forward and backward scattering functions. Approximate forms for these moments are given in terms of the spectral density function of the relative permittivity fluctuations. Solutions are developed for incident plane and spherical waves. Well-known results that are usually derived from the parabolic wave equation using the Markov approximation are easily recovered from the general solutions. The results show that backscatter enhancements depend on the correlation between the forward and backward scattered waves. Computations are performed to illustrate the effects of backscatter on VHF transionospheric radiowaves.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>