Equations For Reflection Coefficients Research Articles

“Zoeppritz” rationalized, and generalized to anisotropic media

This paper is meant to be a guide through a labyrinth of reflectivity and transmissivity at interfaces between anisotropic elastic media. The Zoeppritz equations for isotropic plane-wave reflection and transmission coefficients are cast in a form from which explicit elementary expressions for the reflectivity and transmissivity matrices are derived in terms of four submatrices of the Zoeppritz coefficient matrix, two submatrices associated with each of the two media. The two submatrices associated with a medium are also the building blocks for a simple construction of the propagator matrix for a layer of that medium. All expressions are applicable to anisotropic media, subject to the requirement that the media involved must be at least monoclinic with a mirror plane of symmetry parallel to the reflecting plane. The submatrices associated with a medium depend only on the elastic properties of that medium and are functions of the eigenvalues and eigenvectors of the Christoffel equations governing plane-wave propagation in the medium. The advantages of this approach to reflectivity are (a) heuristic and (b) it allows a programmer to further modularize computer codes needed for computation of reflection coefficients and propagator matrices in anisotropic elastic layered media programs. Some curious results concerning reflectivity between isotropic and orthorhombic media are presented.

Quantum-ohmic resistance fluctuation in disordered conductors—An invariant imbedding approach

It is now well known that in extreme quantum limit, dominated by the elastic impurity scattering and the concomitant quantum interference, the zero-temperature d.c. resistance of a strictly one-dimensional disordered system is non-additive and non-self-averaging. While these statistical fluctuations may persist in the case of a physically thin wire, they are implicitly and questionably ignored in higher dimensions. In this work, we have re-examined this question. Following an invariant imbedding formulation, we first derive a stochastic differential equation for the complex amplitude reflection coefficient and hence obtain a Fokker-Planck equation for the full probability distribution of resistance for a one-dimensional continuum with a Gaussian white-noise random potential. We then employ the Migdal-Kadanoff type bond moving procedure and derive the d-dimensional generalization of the above probability distribution, or rather the associated cumulant function –‘the free energy’. For d=3, our analysis shows that the dispersion dominates the mobilitly edge phenomena in that (i) a one-parameter B-function depending on the mean conductance only does not exist, (ii) an approximate treatment gives a diffusion-correction involving the second cumulant. It is, however, not clear whether the fluctuations can render the transition at the mobility edge ‘first-order’. We also report some analytical results for the case of the one dimensional system in the presence of a finite electric fiekl. We find a cross-over from the exponential to the power-low length dependence of resistance as the field increases from zero. Also, the distribution of resistance saturates asymptotically to a poissonian form. Most of our analytical results are supported by the recent numerical simulation work reported by some authors.

Optical bistability in nonlinear media: An exact method of calculation

The method of invariant imbedding is used to obtain exact equations for reflection and transmission coefficients for a plane wave incident on a slab of nonlinear material. These equations are solved for thin slabs of InSb and clear bistability is obtained.

REFLECTION OF ELASTIC WAVES FROM PERIODICALLY STRATIFIED MEDIA WITH INTERFACIAL SLIP*

AbstractA periodically stratified elastic medium can be replaced by an equivalent homogeneous transverse isotropic medium in the long wavelength limit. The case of a homogeneous medium with equally spaced parallel interfaces along which there is imperfect bonding is a special instance of such a medium. Slowness surfaces are derived for all plane wave modes through the equivalent medium and reflection coefficients for a half‐space of such a medium are found. The slowness surface for the SH mode is an ellipsoid. The exact solution for the reflection of SH‐waves from a half‐space with parallel slip interfaces is found following the matrix method of K. Gilbert applied to elastic waves. Explicit results are derived and in the long wavelength limit, shown to approach the results for waves in the equivalent homogeneous medium. Under certain conditions, a half‐space of a medium with parallel slip interfaces has a reflection coefficient independent of the angle of incidence and thus acts like an acoustic reducing mirror. The method for the reflection of P‐ and SV‐waves is fully outlined, and reflection coefficients are shown for a particular example. The solution requires finding the eigenvalues of a 4 × 4 transfer matrix, each eigenvalue being associated with a particular wave. At higher frequencies, unexpected eigenvalues are found corresponding to refracted waves for which shear and compressional parameters are completely coupled. The two eigenvalues corresponding to the transmitted wavefield give amplitude decay perpendicular to the stratification along with up‐ and downgoing phase propagation in some other direction.Much of this work was performed while the author was at the Department of Geophysics and Planetary Sciences, Tel‐Aviv University, Ramat‐Aviv, Israel. The author is grateful for illuminating discussions with K. Helbig and K. Gilbert.

Elastic waves in periodically stratified media with interfacial slip

A periodically stratified elastic media can be replaced by an equivalent homogeneous transverse isotropic medium in the long wavelength limit. The case of a homogeneous medium with equally spaced parallel interfaces along which there is imperfect bonding [M. Schoenberg, J. Acoust. Soc. Am. 68, 1516–1521 (1980)] is a special case of such a medium. Slowness surfaces are derived for all plane-wave modes through the equivalent medium and reflection coefficients for a half-space of such a medium are found. The slowness surface for the SH mode is an ellipsoid. The exact solution for waves in a medium with parallel slip interfaces is found using the matrix method for periodic media [K. Gilbert, J. Acoust. Soc. Am. Suppl. 1 66, S73 (1979)] applied to elastic waves. Explicit results are derived and in the long wavelength limit, shown to approach the results for waves in the equivalent homogeneous medium. Under certain conditions, a half-space of a medium with parallel slip interfaces has a reflection coefficient independent of the angle of incidence and thus acts like an acoustic reducing mirror.

Discontinuity radiation in surface waveguides*

Diffraction occurs when a surface waveguide (such as a dielectric slab guide or an optical fiber) passes from one medium to another. Integral equations for transmission, reflection, and diffraction coefficients are derived. These equations are then solved asymptotically for the case of a small discontinuity.

Analysis of 3-cm radio height-gain curves taken over rough terrain

This report describes the effect of terrain and meteorological conditions on the height-gain pattern of 3.2-cm radio waves over various short transmission paths. Equivalent reflection coefficients are obtained and potential reflection areas are investigated. A study of the time variations in the height of nulls in the signal strength pattern is made and the relationship between movement of the nulls and the corresponding refractive index distribution is considered.

Experimental Analysis of Equivalent Reflection Coefficient from Various Reflecting Rough Surfaces in Microwave Propagation

Experimental Analysis of Equivalent Reflection Coefficient from Various Reflecting Rough Surfaces in Microwave Propagation

Vertical Distribution of Radio Meteorological M-curve for near the Snow-Covered Terrain

The variation of the equivalent reflection coefficient (ρe) is one of the most important factors which determine the properties of K-Type Fading at microwave propagation over snow-covered terrain. And ρe's physical meaning is somewhat complicated by the reason that it contains many factors (surface deposited snow's dielectric properties, radio meteorology, etc.). So we observed ρe's daily variation at the model propagation test (at very short distance 100m, 4000) and annexed the radio meteorological condition's measurements.This paper shows the method of this experiments and the analytic results on the radio meteorological M-curve (M-profile lies on the very low layer near upon the model test's reflection point). The vertical profiles of measured atmospheric temperature and M-curve are represented in Fig. 4 and ΔM (M-inversion) in Fig. 6 respectively.In general, the variation of ΔM is smaller than over the land and sea, especially at night time. So it is found that the dielectric properties of deposited snow at reflection surface had probably a pretty large influence on the variation of ρe.

The Scattering of Radio Waves in the Lower and Middle Atmosphere

The evidence relating to the reflection of radio waves from levels below 80 kilometers is considered and apparatus used to investigate the reflection coefficients of these regions is described. The new experimental results here presented are not in agreement with those of earlier workers, but indicate that reflections from region B (below 10 kilometers) and region C (35 to 60 kilometers) are very weak and are due to scattering patches rather than reflecting strata. It is shown that reflections from region B are probably due to water molecules and that echoes with time delays corresponding to semipaths of 10-25 kilometers probably originate at scattering centers within the troposphere. The equivalent reflection coefficient of region C is discussed and the mechanism of formation of this region of ionization is briefly considered in connection with atmospheric temperature gradients.

Ionospheric Disturbances, Fadeouts and Bright Hydrogen Solar Eruptions

F0R some months we have been recording daily the equivalent heights and reflection coefficients of the F2 region of the ionosphere, for radio frequencies near to the critical penetration frequency of the region. Under these conditions, the recorded quantities are exceptionally sensitive to small changes in the electron density of the F2 region.