Plane Wave Reflection Coefficient Research Articles

Reciprocal volume and surface scattering integrals for anisotropic elastic media

Linear scattering theory in anisotropic media is useful for describing modelling, inversion and migration algorithms. The single-scattering or Born approximation leads to a volume scattering integral which is further simplified by using the geometrical ray approximation (GRA) for Green's functions from the source and receiver to the scattering point. Discontinuities of the medium parameters which are confined to smooth surfaces will reflect and refract the propagating waves. This is often described by the Kirchhoff-Helmholtz integral, which uses Green's representation of the reflected field and the Kirchhoff approximation for the field and its normal derivative at the surface. The reflected field and its derivative are often approximated by multiplying the corresponding parts of the incoming field with the plane-wave reflection coefficient computed for the angle between the incoming ray and the surface normal (Kirchhoff approximation). Besides the inconsistency of imposing both the field and its normal derivative on the surface to represent the field away from it, the Kirchhoff-Helmholtz integral gives rise to a reflected response which is non-reciprocal. The Born and Kirchhoff-Helmholtz integrals have traditionally been treated as completely separate formulations in the studies of reflection and transmission of waves due to smooth interfaces. A simple use of the divergence theorem applied to the Born volume integral gives a reciprocal surface scattering integral, which can be seen as a natural link between the two formulations. This unifying integral has been recently derived in the context of inversion. We call it the Born-Kirchhoff-Helmholtz (BKH) integral. The properties of the BKH integral are investigated by a stationary-phase evaluation, and the result is interpreted in ray theoretical terms. For isotropic media, explicit expressions are given.

Green's functions for Maxwell's equations: application to spontaneous emission

We present a new formalism for calculating the Green's function for Maxwell's equations. As our aim is to apply our formalism to light scattering at surfaces of arbitrary materials, we derive the Green's function in a surface representation. The only requirement on the material is that it should have periodicity parallel to the surface. We calculate this Green's function for light of a specific frequency and a specific incident direction and distance with respect to the surface. The material properties entering the Green's function are the reflection coefficients for plane waves at the surface. Using the close relationship between the Green's function and the density of states (DOS), we apply our method to calculate the spontaneous emission rate as a function of the distance to a material surface. The spontaneous emission rate can be calculated using Fermi's Golden Rule, which can be expressed in terms of the DOS of the optical modes available to the emitted photon. We present calculations for a finite slab of cylindrical rods, embedded in air on a square lattice. It is shown that the enhancement or suppression of spontaneous emission strongly depends on the frequency of the light.

Scattering of EM waves from a slightly rough surface of a generally anisotropic plane-layered halfspace

The problem of plane wave scattering by a slightly rough boundary of an arbitrarily layered medium with general anisotropy of electric and magnetic properties is considered using boundary conditions transference method and a small perturbation approximation combined with the Green's functions technique. It is shown that in the Born approximation, the second statistical moments of the fluctuation field scattered from an inhomogeneous anisotropic halfspace can be expressed exclusively in terms of the external parameters referring to a flat boundary and the spectral density of roughness. In the present circumstance, the set of the external parameters includes the four plane-wave reflection coefficients from a flat boundary and the (limiting) values of constitutive parameters of the adjacent media at said boundary, Within the framework of the outlined approach, the covariance dyads, Poynting's vector, scattering coefficients and Stokes parameters of the fluctuation field, and the Mueller matrix for a rough boundary are calculated.

Reflection and transmission coefficients for three-dimensional plane waves in elastic media

Problems for transient line and point load sources in a multilayered elastic medium may be treated by the method of generalized ray. In this method an integral representation of the Laplace-transformed multiply reflected and/or transmitted cylindrical/spherical wave, known as a ray integral, is constructed by linear superposition of the Laplace-transformed plane waves. The inverse Laplace transform of the ray integral can be found in closed form by applying the Cagniard method. For problems in the Cartesian coordinates for line load sources emitting cylindrical waves consistent with either the plane strain conditions or the antiplane strain conditions and for problems in the cylindrical coordinates for axisymmetric and asymmetric point load sources emanating spherical waves, it is well known that: (1) the system of incident, reflected, and transmitted cylindrical/spherical waves at an interface separating two dissimilar media can be divided into two independent of each other, if both present, parts: the coupled P and SV waves, and the SH waves, (2) the reflected and transmitted ray integrals representing the Laplace-transformed reflected and transmitted cylindrical/spherical waves can be constructed by linear superposition of the Laplace-transformed plane P and SV waves, or the plane SH waves, and (3) the potential reflection and transmission coefficients for the plane P, SV, and S H waves are basic to such a superposition. In the present paper we treat the asymmetric three-dimensional problem in the Cartesian coordinates for an arbitrary oriented point force radiating the spherical P and S waves. For this problem all four functions representing the displacement potentials are coupled in the boundary conditions at the interface, the total wave motion at the interface is composed of the coupled spherical P and S waves, and the Laplace-transformed reflected and transmitted spherical waves are therefore constructed by linear superposition of the three-dimensional coupled plane P and S waves. Since such a superposition requires the knowledge of the potential reflection and transmission coefficients for the three-dimensional coupled plane P and S waves, the purpose of the present paper is to derive systematically these coefficient formulas.

A normal mode model for acousto-elastic ocean environments

A normal mode method for propagation modeling in acousto-elastic ocean waveguides is described. The compressional (p-) and shear (s-) wave propagation speeds in the multilayer environment may be constant or have a gradient (1/c2 linear) in each layer. Mode eigenvalues are found by analytically computing the downward- and upward-looking plane wave reflection coefficients R1 and R2 at a reference depth in the fluid and searching the complex k plane for points where the product R1R2=1. The complex k-plane search is greatly simplified by following the path along which |R1R2|=1. Modes are found as points on the path where the phase of R1R2 is a multiple of 2π. The direction of the path is found by computing the derivatives d(R1R2)/dk analytically. Leaky modes are found, allowing the mode solution to be accurate at short ranges. Seismic interface modes such as the Scholte and Stonely modes are also found. Multiple ducts in the sound speed profile are handled by employing multiple reference depths. Use of Airy function solutions to the wave equation in each layer when computing R1 and R2 results in computation times that increase only linearly with frequency.

Numerical absorbing boundary conditions for the scalar and vector wave equations

Electromagnetic field computation may be carried out conveniently by using the finite element method (FEM). When solving open region problems using this technique, it becomes necessary to enclose the scatterer with an outer boundary upon which an absorbing boundary condition (ABC) is applied; analytically-derived ABCs have been used extensively for this purpose. Numerical absorbing boundary conditions (NABCs) have been proposed as alternatives to analytical ABCs. For the two-dimensional (2-D) Helmholtz equation, it has been demonstrated analytically that these NABCs become equivalent to many of the existing analytical ABs in the limit as the cell size tends to zero. In addition, the numerical efficiency of these NABCs has been evaluated by using as an indicator the reflection coefficient for plane and cylindrical waves incident upon an arbitrary boundary. We have extended this procedure to the study of the NABCs derived, for the three-dimensional (3-D) scalar and vector wave equations from the point of view of their numerical implementation in node- and edge-based FEM formulations, respectively.

Precision ultrasonic reflection studies in fluid-coupled plates

An approximate, but highly accurate, analysis is presented for the reflection of bounded acoustic beams—and their detection by finite receivers—from fluid-loaded structures, demonstrating the relationship between the plane-wave reflection coefficient and the transducer voltage. The correspondence of the reflection coefficient zeros, in both angle and frequency, with the observed voltage minima is found to depend critically on the experimental geometry, especially the lateral transducer placement. In measurements with two identical transducers, the beam shape, its sidelobes, and far-field or near-field conditions are found to play a relatively minor role in the receiver voltage. Instead, the size of the transducers and their relative lateral positioning with respect to the sample are the major geometrical factors determining the appearance, width, and depth of reflection minima. A series of precision, swept-frequency experiments on a uniform, homogeneous plate to validate the calculation and expose any possible limitations is also reported here. In general, excellent detailed agreement between the efficient approximate calculation and the results of the experiments has been obtained. An expression is derived to predict the optimal transducer positioning for accurate estimation of reflection coefficient zeros from voltage measurements. These results have significance for material property extraction from reflection measurements on fluid-loaded structures.

Electromagnetic dyadic Green's function in cylindrically multilayered media

A spectral-domain dyadic Green's function for electromagnetic fields in cylindrically multilayered media with circular cross section is derived in terms of matrices of the cylindrical vector wave functions. Some useful concepts, such as the effective plane wave reflection and transmission coefficients, are extended in the present spectral domain eigenfunction expansion. The coupling coefficient matrices of the scattering dyadic Green's functions are given by applying the principle of scattering superposition. The general solution has been applied to the case of axial symmetry (n=0, n is eigenvalue parameter in /spl phi/ direction) where the scattering coefficients are decoupled between TM and TE waves. Two specific geometries, i.e., two- and three-layered media that are frequently employed to model the practical problems are considered in detail, and the coupling coefficient matrices of their dyadic Green's functions are given, respectively.

Antenna design with the use of photonic band-gap materials as all-dielectric planar reflectors

A finite thickness slab of two-dimensional photonic band-gap (PBG) material is analyzed to determine the plane-wave reflection and transmission coefficients as functions of the angle of incidence. It is shown that within the band-gap, where there is total reflection, the reflection is equivalent to that from a plane surface located within the PBG material (the reflection plane concept). The two-dimensional PBG material is used as an all-dielectric reflector for an electric dipole antenna. Field patterns computed with the use of the reflection plane concept are compared with measurements made on an experimental model and are found to be in good agreement. A three-dimensional PBG material with a BCC lattice is also used as a reflector for an electric dipole antenna. Field patterns calculated for this structure using the finite-difference-time-domain (FDTD) method are also in good agreement with measurements. © 1996 John Wiley & Sons, Inc.

Reflection and transmission coefficients for a layered fluid sediment overlying a uniform solid substrate

Simple expressions are found for plane-wave reflection and transmission coefficients for a layered (continuously varying density and sound speed) fluid sediment layer sandwiched between a uniform fluid above (representing the ocean) and a uniform solid substrate below; exact solutions are derived first in terms of the wave functions in the sediment layer, and these are then approximated using Langer’s generalization of the WKB method. The approximate solutions, valid for monotonic but otherwise arbitrary sediment profiles, are evaluated for a class of quasilinear sound-speed profiles and shown to be very accurate for the cases considered; for some special cases they turn out to be exact. Bottom loss predictions are presented for a clay sediment overlying a basalt substrate, examining in particular the influence of different density and sound-speed profiles on the excitation of Stoneley waves at the sediment–substrate boundary.

Weak-contrast approximation of the elastic scattering matrix in anisotropic media

The plane-wave reflection and transmission coefficients at a plane interface between two anisotropic media constitute the elements of the elastic scattering matrix. For a 1-D anisotropic medium the eigenvector decomposition of the system matrix of the transformed elasto-dynamic equations is used to derive a general expression for the scattering matrix. Depending on the normalization of the eigenvectors, the expressions give scattering coefficients for amplitudes or for vertical energy flux.

Born-type schemes for the acoustic probing of 1-D fluid media from time-harmonic planar reflection coefficients at two incidences

The probing of a 1-D (depth-) inhomogeneous fluid half-space is carried out from a limited knowledge of plane-wave reflection coefficients with respect to frequency at two precritical incidences. Born-type inversion algorithms are investigated. First, index profiles as a function of a density-dependent pseudodepth are retrieved at two incidences; then density and speed-of-sound profiles versus depth are extracted. A Born-iterative scheme builds up the solution as the limit of a sequence of solutions of approximate, constrained linear problems. A Born-extended scheme directly yields a linearized solution. Two variants of the Born-iterative scheme are developed, the first one when data are known in both amplitude and phase, the second one with amplitude-only data. Possible—but not limiting—applications are in underwater acoustics, once the planar reflection coefficient of a seabed has been found, e.g., from the depth-dependent Green’s function in the spectral domain. Numerical simulations illustrate the schemes. Good index profiles are obtained with reasonable behavior versus noise or other causes of limitations. As is expected, separate probing of speed and density profiles remains difficult. The computationally costly exact scheme is not better than the cheaper approximate one when good data are available, but handles more complicated situations.

Modeling semi-anechoic electromagnetic measurement chambers

Previous studies developed a model to predict theoretically the low-frequency plane-wave reflection coefficient of an array of pyramid cone absorbers such as those used to line anechoic electromagnetic measurement chambers. The present authors apply this model in a geometrical optics approach to predict the electromagnetic field in a chamber lined with cone absorbers in the frequency range of 30-300 MHz. The results are compared with site attenuation measurements for two actual semi-anechoic chambers.

Sound scattering at a rough water–sediment interface in shallow water

The method of smoothing is used to study sound propagation and scattering in a realistic shallow-water environment with a randomly rough water–sediment interface. A mean field wave equation for the coherent acoustic field is obtained that has a complex-valued attenuation coefficient that is localized at the mean interface. The solution of the mean field equation, under highly idealized conditions, is derived and discussed. The branch point and pole singularities are found to be related to the head wave mode and the Biot–Tolstoy rough boundary wave mode, respectively. Also obtained is the coherent plane-wave reflection coefficient which is found to vanish when the acoustic frequency and the incident angle of the plane wave satisfy a certain resonant condition. For testing the mean field equation, a stochastic full-wave forward propagation PE model is developed for Monte Carlo simulations. The numerical results show that the coherent field obtained by an ensemble average over many realizations is in agreement with that obtained by solving the mean field equation just once. The Monte Carlo simulations also yield additional useful information about higher-order moments of the acoustic field. [Work supported by ONR.]

Wave scattering by a two-dimensional pressure-release surface based on a perturbation of the Green’s function

An approximate method is described, to calculate the two-dimensional acoustic scattering, from a slightly rough, time-independent, pressure-release surface. The formulation is based on a perturbation of the Green’s function allowing an approximation of the propagator in the kernel of the Helmholtz integral equation which reduces the integral equation to a convolution equation. The complexity of the problem is reduced from 𝒪(MN2) (for an iterative method where M is the number of iterations) to 𝒪(N ln N), where N is the number of discretizations used to describe the surface numerically. The specific surface examined has a sinusoidal profile, although the method may be applied to any slightly rough surface (hK≪1). We use this solution to generate plane-wave reflection coefficients from the Fourier transform of the pressure gradient on the surface. These may be used to calculate the scattered pressure. The existence of the exact solution for periodic surfaces, due to Holford [J. Acoust. Soc. Am. 70, 1116–1128 (1981)], allows comparison of the convolution approximation with the exact solution. The reflection coefficients and scattered energy are then compared with the approximate solutions of Rayleigh and Kirchhoff.

A theoretical study of numerical absorbing boundary conditions

Electromagnetic field computation involving inhomogeneous, arbitrarily-shaped objects may be carried out conveniently by using partial differential equation techniques, e.g., the finite element method (FEM). When solving open region problems using these techniques, it becomes necessary to enclose the scatterer with an outer boundary on which an absorbing boundary condition (ABC) is applied, and analytically-derived ABCs, e.g., the Bayliss-Gunzburger-Turkel and Engquist-Majda boundary conditions have been used extensively for this purpose. Numerical absorbing boundary conditions (NABCs) have been proposed as alternatives to analytical ABCs, and they are based upon a numerically-derived relationship that links the values of the field at the boundary nodes to those at the neighboring nodes. In the paper the authors demonstrate, analytically, that these NABCs become equivalent to many of the existing analytical ABCs in the limit as the cell size tends to zero. In addition, one can evaluate the numerical efficiency of these NABCs by using as an indicator the reflection coefficient for plane and cylindrical waves incident upon an arbitrary boundary.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Spatial Fourier-transform method for measuring reflection coefficients at oblique incidence. II. Experimental results

A method using spatial Fourier transforms for measuring the plane-wave reflection coefficient at oblique angles of incidence has been proposed in an earlier paper [M. Tamura, J. Acoust. Soc. Am. 88, 2259–2264 (1990)]. The present paper gives experimental verification of the method. Experimental results are shown for two types of samples: a perfectly absorbing plane (an imaginary plane in the air) and a layer of a polyurethane foam backed by a hard plywood board. The measured results are generally in good agreement with the theoretical value (for the first sample) or with the results calculated from the acoustical parameters of the material (for the second sample). The agreement shows the usefulness of the method for measuring acoustic properties of absorbent materials.

The influence of the modal properties of a stiff layer embedded in a solid medium on the field generated in the layer by a finite-sized transducer

This paper is the second of a pair which compare the modal properties of a stiff layer embedded in a solid medium with the minima of the reflection coefficient. In the first paper [J. Acoust. Soc. Am. 97, 1625–1637 (1995)] the modal properties and the plane-wave reflection coefficient are compared. In this paper a finite width incident beam is modeled and the reflected field is studied, the spatially realistic simulation enabling the comparisons to address the issues which will arise in practical measurements of reflection coefficient minima. The finite beam reflection minima can be caused both by minima of the plane-wave reflection coefficient and by interference of the leaking field produced by a propagating mode with the specular reflection. In systems in which there is a large acoustic impedance contrast between the layer and the surrounding medium, these phenomena occur under practically the same conditions. However, when the surrounding medium is similar to the layer, major changes in the reflection behavior are seen. Predictions are presented for single interfaces and embedded layers, assuming both fluid and solid embedding media. For a single interface it is shown that the reflection coefficient minima correlate directly to the modal properties since plane-wave reflection coefficient minima do not occur. For an embedded layer whose properties are similar to those of the embedding medium, the plane-wave reflection coefficient minima dominate the response and the (strongly leaking) modes of the layer are only excited very weakly by the incident beam.

Comparison of the modal properties of a stiff layer embedded in a solid medium with the minima of the plane-wave reflection coefficient

Waves which propagate along embedded layers offer potential for the nondestructive characterization of the layers and of the boundary conditions between the layers and the embedding medium. However, it is known that the minima of the reflection coefficient, popularly used to measure the modal properties of leaky waves in plates immersed in fluids, do not correlate precisely with the dispersion curves and may be particularly misleading when the embedding medium is solid and of the same order of impedance as the layer. This paper is the first of a pair which study the practical case of a thin solid layer which is embedded at the bondline of a diffusion bonded titanium joint and is slightly stiffer than the titanium. In this paper the modal properties of the layer are compared with the plane-wave reflectivity; the second paper [J. Acoust. Soc. Am. 97, 1638–1649 (1995)] addresses the near-field response of the system to a finite acoustic beam. It is demonstrated in this paper that the loci of the reflection coefficient minima are related to the dispersion curves in wave-type groupings, but that they do not match in a direct manner. The divergence of the loci from the dispersion curves appears to increase with the rate of leakage of the modes which in turn is related inversely to the acoustic impedance contrast between the layer and the embedding medium. Therefore the loci of the plane-wave reflection coefficient minima should not be used to plot the modal properties of the system. Nevertheless, it is recognized that the loci are potentially useful for the characterization of the embedded layer since they are sensitive to changes in the properties of the layer.

Plane-wave reflection and transmission coefficients for a three-layered elastic medium

Simple expressions are derived for the (p and s) plane-wave reflection and transmission coefficients for a three-layered elastic medium comprising a homogeneous transition layer between two half-spaces. The result, presented as a function of the well-known coefficients for a two-layered (solid–solid) medium, is in a simple explicit form, and is entirely equivalent to the implicit relationships obtained by conventional matrix methods. Reflection of sound from the seabed with a thin sediment layer is considered as an illustrative example.