Generalized Reflection Coefficients Research Articles

The effect of a second-order velocity discontinuity on elastic waves near their turning point

Summary. The rather abrupt changes in velocity gradient which have some- times been proposed, notably in the upper mantle and near the base of the mantle, have an effect equivalent to that of one or more second-order discon- tinuities, where partial reflection occurs due to a change in curvature of the wavefront across these discontinuities. The effect is ignored in the classical WKBJ approximation to the wave functions, but it can be explicitly demon- strated by applying the extended WKBJ method (Langer's approximation) to a piecewise smooth layered model. For the purpose of ths study it is convenient to represent the response of such a model by a generalized reflection coefficient. For a model of one or a system of several second-order discontinui- ties (approximating a change in velocity gradient over a finite depth interval), the reflection coefficient can be perhaps surprisingly large for long-period waves near their turning point. It is shown that this effect can significantly alter the amplitude decay of sfj waves diffracted around the core, in models where a change in velocity gradient near the core-mantle boundary consti- tutes a low-velocity zone at the base of the mantle; such models have recently been proposed. With the same velocity gradients, the effect on P diffraction is less important. The results for SH diffraction in these models support the conclusion that a small amplitude decay must be explained by a velocity decrease with depth, i.e. a low-velocity zone at the base of the mantle.

Exact decomposition of bottom interacting paths via a generalized reflection coefficient expansion

The exact range independent acoustic field for an arbitrarily structured multilayered medium may be expressed as a Fourier‐Bessel integral involving generalized reflection coefficients. One obtains upon evaluation the total result due to all possible bottom interactions. It is desirable for some applications to decompose this total field result into path type contributions without incurring the approximations associated with ray theory. This is accomplished by first recognizing that the generalized reflection coefficient may be written in terms of partial local reflection coefficients. An infinite series of integrals results when the term in the kernel containing the local coefficients is expanded. Each term in this expansion then contains combinations of various local reflection coefficients which can be readily interpreted as the desired path contributions. The spatial amplitude and phase dependence is obtained by evaluating each term separately using the fast field program technique. Examples will be p...

High-frequency acoustic scattering from submerged cylindrical shells coated with viscoelastic absorbing layers

We analyze the high-frequency scattering of a plane-wave incident on a submerged air-filled cylindrical steel shell coated with a viscoelastic layer. The scattered pressure field consists of a geometrical acoustic contribution and another part due to creeping waves. We analyze the resulting generalized reflection coefficient as a function of azimuthal angle, for two shell sizes and various viscosity levels of the coating. We also obtain the locations of roots of the 10×10 characteristic determinant in the complex ν-plane for the two shell sizes of interest. These zeroes are compared to the zeroes of rigid and soft cylinders and to the roots of a simpler 3×3 determinant corresponding to the case of a solid sound-absorbing cylinder of the same size. The results show that the region in the ν-plane where many of the roots are usually concentrated for bare structures, contain no roots after the absorbing layer is bonded to the steel shell. All the above information is then used to compare the normal-mode solution for the sonar cross section of the coated cylindrical shell found in an earlier study of ours (and extended here to higher k1c values), and the corresponding expression obtained to lowest order from an application of the Watson–Sommerfeld transformation. This comparison, as exhibited in various graphs, is found to be very favorable. It verifies that the lowest-order term in the Watson–Sommerfeld transformed solution is capable of accurately reproducing the exact scattering solution at high frequencies for the cylindrical coated target considered here.

Generalized reflection coefficient in the diffraction of a plane electromagnetic wave at an impedance wedge

Generalized reflection coefficient in the diffraction of a plane electromagnetic wave at an impedance wedge

2-D MULTIPLE REFLECTIONS

Starting with a 1-D subsurface model, a method is developed for modeling and inverting the class of multiple reflections involving the near‐perfect reflector at the free surface. A solution to the practical problem of estimating the source waveform is discussed, and application of the 1-D algorithm to field data illustrates the successful elimination of seafloor and peg‐leg multiples. Extending the analysis to waves in two dimensions, we make the approximation that the subsurface behaves as an acoustic medium. Based on several numerical and theoretical considerations, the scalar wave equation is split into two separate partial differential equations: one governing propagation of upcoming waves and a second describing downgoing waves. The result is a pair of propagation equations which are coupled where reflectors exist. Finite difference approximations to the initial boundary value problem are developed to integrate numerically the surface reflection seismogram. Use of the 2‐D algorithm for modeling free‐surface multiple reflections is illustrated by several reflector models. The 2-D inverse problem of simultaneously migrating primary reflections and inverting diffracted multiples consists of reversing the forward calculation with the data as boundary conditions. Causal directions of propagation are related to downward continuation of surface data. Reflector mapping principles are used to develop a general reflection coefficient estimator. The inverse algorithm is illustrated using the results of the 2-D forward calculation as the boundary conditions.

Standing waves on beaches

Simultaneous measurements of run-up spectra and nearshore wave spectra associated with standing waves have been made on a natural beach. Digital wave staffs and bottom-mounted pressure sensors were placed in a line at various distances offshore from a digital run-up meter. The run-up meter, which measured the position of the edge of the water, consisted of a series of electrical contacts suspended above the beach face and spaced 30 cm apart. Because of the possible influence of nonlinear interactions on the spectra, bicoherence was estimated, and the frequency range appropriate to a linearized model was defined. In this low-frequency region of the spectrum the cross-spectra phase angles between data records had values near 0 or π at frequencies of high coherence. The energy spectrum levels and the observed phase differences between records were in good agreement with the linear shallow water wave theory (Lamb, 1932) for standing waves on a plane beach. The general reflection coefficient for the data, estimated from the disagreement between data and standing wave theory, was found to be approximately 0.7. This result indicates that the theory of Miche (1944) for reflection from a plane beach predicts a reflection coefficient that is too high by a factor of approximately 1.5.

Approximate backscatter of electromagnetic waves from an inhomogeneous lunar surface

Intensity calculations were made for the field scattered from a nonhomogeneous lunar crater model for wavelengths ranging from ultraviolet to microwave. General Fresnel reflection coefficients are utilitized in the Kirchhoff integral to calculate the monochromatic field scattered from a heterogeneous surface whose shape may be approximated by straight lines. Each straight line then constitutes a Fresnel elemental scattering surface of different material properties. The straight-line approximation of the arbitrary surface allows a first-order analytical solution for the scattered field.