Reflection and transmission of obliquely incident Rayleigh waves at a vertical discontinuity between two welded quarter-spaces

1979 ◽  
Vol 69 (5) ◽  
pp. 1409-1423
Author(s):  
Thomas C. Chen ◽  
Leonard E. Alsop
Keyword(s):  
Free Surface ◽  
Phase Velocity ◽  
Rayleigh Waves ◽  
Velocity Ratio ◽  
Normal Incidence ◽  
Angle Of Incidence ◽  
Reflection And Transmission ◽  
Obliquely Incident ◽  
Vertical Boundary ◽  
Diffraction Effects

abstract We use an approximate method to study the reflection and transmission of obliquely incident Rayleigh waves on a vertical boundary between two welded quarter-spaces. For two media with a phase velocity ratio of 1.16 our calculation shows that the transmitted energy follows a reciprocity relation and decreases from near 100 per cent at normal incidence to 50 per cent at about 40°. The reflected energy is less than 1 per cent for angles of reflection less than 40°. When the Rayleigh wave impinges upon the less rigid medium, the reflected energy decreases as the angle of incidence increases; whereas for incidence at the more rigid medium, the reflected energy decreases at first, and then it increases as the angle of incidence increases. Since boundary conditions on the free surface are not taken into account by our method, diffraction effects are ignored. The effect of neglecting the free surface requirement is difficult to quantify, but we believe that it is small since the calcualted and experimental results agree well at normal incidence.

Collimated Sputtering of Titanium and Titanium Nitride Films

MRS Bulletin ◽  
1995 ◽  
Vol 20 (11) ◽  
pp. 42-45 ◽  
Author(s):  
James G. Ryan ◽  
Stephen B. Brodsky ◽  
Tomio Katata ◽  
Makoto Honda ◽  
Naohiro Shoda ◽  
...  
Keyword(s):  
Physical Vapor Deposition ◽  
Normal Incidence ◽  
Angle Of Incidence ◽  
Tin Films ◽  
Sputtered Films ◽  
Obliquely Incident ◽  
Grain Structures ◽  
Pass Through ◽  
Full Face ◽  
Contact Adhesion

Collimated sputtering is a physical vapor deposition (PVD) method where a collimator is inserted between a conventional “full-face-erosion” sputtering target and a substrate (Figure 1). The collimator is a plate of hexagonal cells that acts as a filter to remove obliquely incident atoms before they arrive at the substrate. Only material with a nearly normal incidence trajectory may pass through the collimator and deposit on the substrate. Collimated sputtering was initially evaluated for conductor-level depositions in order to improve the filling of recessed features. Although the method has been successfully used to fill damascene structures, depositing thick conductor films is inefficient because most of the sputtered material is captured by the collimator, causing the collimator to clog quickly, necessitating frequent replacement.A more common use of collimated sputtering is associated with the deposition of thin “liner” films. For example, thin, collimated aluminum alloy films have been used as underlayers for aluminum reflow processes. Also, collimated Ti/TiN films are used as contact/adhesion layers for chemically vapor-deposited (CVD) W metallization. Collimation provides better bottom and sidewall coverage for small, high aspect-ratio features than conventionally sputtered films do.Coliimated sputtered films often exhibit unique properties because the angle of incidence of depositing atoms is controlled. Collimated AlMg alloys have superior electromigration resistance compared to noncollimated AlMg films. Collimated TiN films appear to exhibit denser grain structures when compared to films deposited with higher amounts of obliquely incident flux (Figure 2).

Scattering of surface waves obliquely incident on a fixed half immersed circular cylinder

1984 ◽  
Vol 96 (2) ◽  
pp. 359-369 ◽  
Author(s):  
B. N. Mandal ◽  
S. K. Goswami
Keyword(s):  
Integral Equation ◽  
Surface Water ◽  
Circular Cylinder ◽  
Water Waves ◽  
Wave Number ◽  
Angle Of Incidence ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Obliquely Incident ◽  
Surface Water Waves

AbstractThe problem of scattering of surface water waves obliquely incident on a fixed half immersed circular cylinder is solved approximately by reducing it to the solution of an integral equation and also by the method of multipoles. For different values of the angle of incidence and the wave number the reflection and transmission coefficients obtained by both methods are evaluated numerically and represented graphically to compare the results obtained by the respective methods.

Reflection and Transmission of Obliquely Incident Surface Waves by an Edge of a Quarter Space: Theory and Experiment

1992 ◽  
Vol 59 (2) ◽  
pp. 349-355 ◽  
Author(s):  
Z. L. Li ◽  
J. D. Achenbach ◽  
I. Komsky ◽  
Y. C. Lee
Keyword(s):  
Surface Wave ◽  
Singular Integral Equations ◽  
Angle Of Incidence ◽  
Theoretical Formulation ◽  
Reflection And Transmission ◽  
Translational Invariance ◽  
Transmission Coefficients ◽  
Obliquely Incident ◽  
Quarter Space ◽  
Definition Of

The reflection and transmission of a plane time-harmonic surface wave which is obliquely incident on the edge of a quarter space is investigated theoretically, numerically, and experimentally. The theoretical formulation of the problem, which takes advantage of the translational invariance along the edge of the quarter space, is reduced to a system of singular integral equations along axes normal to the edge, for the defracted displacement components on the faces of the quarter space axes normal to the edge. The truncation of these equations leads to the definition of reflection and transmission coefficients, R and T. The equations are solved for R, T, and the diffracted displacements by the use of the boundary element method. A self-calibrated experimental technique is proposed which deploys four surface wave transducers, and which removes the effects of variable coupling between the transducers and the faces of the quarter space as the positions of the transducers are varied. The technique is particularly suited for the measurement of |R/T| as a function of the angle of incidence. Excellent agreement is observed between numerically and experimentally obtained values.

On: “Reflection and transmission of plane compressional waves by R. D. Tooley, T. W. Spencer, and H. F. Sagori, (GEOPHYSICS, 30, 552, April, 1965).

Geophysics ◽  
1984 ◽  
Vol 49 (12) ◽  
pp. 2195-2195 ◽  
Author(s):  
L. R. Denham ◽  
R. A. R. Palmeira
Keyword(s):  
Wave Velocity ◽  
Seismic Wave ◽  
Velocity Ratio ◽  
The Other ◽  
P Wave ◽  
Angle Of Incidence ◽  
Reflection And Transmission ◽  
Compressional Waves ◽  
P Wave Velocity ◽  
Simplified Formula

One of the most widely reproduced figures for the partition of energy of a seismic wave at an interface between two media is from Tooley et al.’s 1965 paper. Recently we referred to this paper and noticed some anomalies in the curves shown there in Figure 11. We looked at the values of the curves at zero angle of incidence and noted that while the values of a P-wave velocity ratio of 2, 3, and 4 agree with the values given by the simplified formula {(V2−V1)/(V2+V1)}**2, the values shown for the P-wave velocity ratios less than 1 do not agree. For a ratio of 0.25, the coefficient at zero angle should be 0.36; it is shown as about 0.72. A ratio of 0.5 should give a coefficient of 0.11; 0.52 is shown. There are similar discrepancies for all the other ratios less than 1.

scholarly journals Approximate reflection coefficients for a thin VTI layer

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. C1-C11 ◽  
Author(s):  
Qi Hao ◽  
Alexey Stovas
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Normal Incidence ◽  
Reflection And Transmission Coefficients ◽  
Reflection Coefficients ◽  
Angle Of Incidence ◽  
Isotropic Layer ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
P And Sv Waves

We have developed an approximate method to derive simple expressions for the reflection coefficients of P- and SV-waves for a thin transversely isotropic layer with a vertical symmetry axis (VTI) embedded in a homogeneous VTI background. The layer thickness is assumed to be much smaller than the wavelengths of P- and SV-waves inside. The exact reflection and transmission coefficients are derived by the propagator matrix method. In the case of normal incidence, the exact reflection and transmission coefficients are expressed in terms of the impedances of vertically propagating P- and S-waves. For subcritical incidence, the approximate reflection coefficients are expressed in terms of the contrast in the VTI parameters between the layer and the background. Numerical examples are designed to analyze the reflection coefficients at normal and oblique incidence and investigate the influence of transverse isotropy on the reflection coefficients. Despite giving numerical errors, the approximate formulas are sufficiently simple to qualitatively analyze the variation of the reflection coefficients with the angle of incidence.

Reflection and transmission of inhomogeneous waves with particular application to Rayleigh waves

1974 ◽  
Vol 64 (6) ◽  
pp. 1635-1652
Author(s):  
L. E. Alsop ◽  
A. S. Goodman ◽  
S. Gregersen
Keyword(s):  
Rayleigh Wave ◽  
Half Space ◽  
Rayleigh Waves ◽  
Normal Incidence ◽  
Elastic Half Space ◽  
Small Computer ◽  
Good Representation ◽  
Inhomogeneous Wave ◽  
Reflection And Transmission ◽  
Inhomogeneous Waves

abstract The concept of inhomogeneous waves, which is quite common in the electromagnetic literature but not in the seismic, is reviewed. It is shown that the Rayleigh wave results from the constructive interference of a P inhomogeneous wave with an SV inhomogeneous wave. Of some interest is the fact that while the two inhomogeneous waves have prograde elliptical particle motions, they combine to yield the familiar retrograde motion near the surface. These two inhomogeneous waves are the only “allowed” inhomogeneous waves in a homogeneous elastic half-space. More can occur in a layered half-space but they are always discrete in number. However, “non-allowed” inhomogeneous waves often give a very good representation of the wave motion over a restricted area—a fact which is demonstrated for leaky modes and propagation of a Rayleigh wave on a three-quarter space. With this in mind, a formalism is developed for the transmission and reflection of the two allowed inhomogeneous waves making up the Rayleigh wave into nonallowed inhomogeneous waves. These are nonallowed in the sense that they do not fit the boundary conditions at a discrete number of points. Here the ray picture breaks down and diffractive effects occur. Our assumption is that for many interesting cases, these diffractive effects are small and can be ignored. The components of the resultant displacements and stresses from the nonallowed inhomogeneous waves on Rayleigh waves can be obtained by use of a relation due to Herrera, thus yielding the reflection and transmission coefficients. The agreement with previously published values is good. While only normal incidence is considered in this paper, the extension to non-normal incidence is straightforward. The required calculations are well within the capabilities of a small computer such as an IBM 1130.

Propagation of Surface Waves Across an Anisotropic Layer

1994 ◽  
Vol 61 (3) ◽  
pp. 596-604 ◽  
Author(s):  
E. N. Its ◽  
J. S. Lee
Keyword(s):  
Surface Waves ◽  
Rayleigh Waves ◽  
Differential Operators ◽  
Interface Layer ◽  
Reflection And Transmission Coefficients ◽  
Function Technique ◽  
Angle Of Incidence ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Matrix Differential

Propagation of elastic surface waves across a thin anisotropic interface layer between two vertically inhomogeneous isotropic quarter-spaces is considered. The relationship between the surface wave fields at the opposite sides of the layer is obtained in the form of matrix differential operators. Based on the Green’s function technique, an analytical method is developed to calculate reflection and transmission coefficients of Rayleigh waves at the layer. The reflection and transmission coefficients of Rayleigh waves at the interface layer are calculated as a function of the angle of incidence for various models of layers with hexagonal symmetry and results are discussed in some detail. Several isotropic layers of low or high velocity materials are also considered to examine the trade-off between anisotropy and inhomogeneity of the interface layer.

On resolving fine periodic microstructures in the optical microscope

1989 ◽  
Vol 47 ◽  
pp. 364-365
Author(s):  
W.S. Putnam ◽  
C. Viney
Keyword(s):  
Liquid Crystalline ◽  
Numerical Aperture ◽  
Polarized Light ◽  
Normal Incidence ◽  
Optical Microscope ◽  
Angle Of Incidence ◽  
Resolution Limit ◽  
Crystalline Materials ◽  
Periodic Microstructures ◽  
Liquid Crystalline Materials

Many sheared liquid crystalline materials (fibers, films and moldings) exhibit a fine banded microstructure when observed in the polarized light microscope. In some cases, for example Kevlar® fiber, the periodicity is close to the resolution limit of even the highest numerical aperture objectives. The periodic microstructure reflects a non-uniform alignment of the constituent molecules, and consequently is an indication that the mechanical properties will be less than optimal. Thus it is necessary to obtain quality micrographs for characterization, which in turn requires that fine detail should contribute significantly to image formation.It is textbook knowledge that the resolution achievable with a given microscope objective (numerical aperture NA) and a given wavelength of light (λ) increases as the angle of incidence of light at the specimen surface is increased. Stated in terms of the Abbe resolution criterion, resolution improves from λ/NA to λ/2NA with increasing departure from normal incidence.

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