scholarly journals Seismic Rayleigh waves on an exponentially graded, orthotropic half-space

2006 ◽  
Vol 463 (2078) ◽  
pp. 495-502 ◽  
Author(s):  
Michel Destrade
Keyword(s):  
Half Space ◽  
Rayleigh Waves ◽  
Seismic Waves ◽  
Symmetry Axis ◽  
Mass Density ◽  
Homogeneous Case ◽  
Marked Contrast ◽  
Homogeneous Half Space ◽  
Shear Horizontal ◽  
Exponentially Graded

Efforts at modelling the propagation of seismic waves in half-spaces with continuously varying properties have mostly been focused on shear-horizontal waves. Here, a sagittally polarized (Rayleigh type) wave travels along a symmetry axis (and is attenuated along another) of an orthotropic material with stiffnesses and mass density varying in the same exponential manner with depth. In contrast to what could be expected at first sight, the analysis is very similar to that of the homogeneous half-space, with the main and capital difference that the Rayleigh wave is now dispersive. The results are illustrated numerically for (i) an orthotropic half-space typical of horizontally layered and vertically fractured shales and (ii) for an isotropic half-space made of silica. In both the examples, the wave travels at a slower speed and penetrates deeper than in the homogeneous case. In the second example, the inhomogeneity can force the wave amplitude to oscillate as well as decay with depth, in marked contrast with the homogeneous isotropic general case.

Surface-wave dispersion computations

1970 ◽  
Vol 60 (2) ◽  
pp. 321-344 ◽  
Author(s):  
Fred Schwab ◽  
Leon Knopoff
Keyword(s):  
Half Space ◽  
Rayleigh Waves ◽  
Wave Dispersion ◽  
Reduction Technique ◽  
Surface Wave Dispersion ◽  
Single Precision ◽  
Homogeneous Half Space ◽  
Full Control ◽  
Significant Figures ◽  
Rayleigh Wave Dispersion

abstract Fundamental-mode Love- and Rayleigh-wave dispersion computations for multilayered, perfectly-elastic media were studied. The speed of these computations was improved, and the accuracy brought under full control. With sixteen decimal digits employed in these computations, fifteen significant-figure accuracy was found possible with Love waves and twelve to thirteen figure accuracy with Rayleigh waves. In order to ensure that the computed dispersion is correct to a specified accuracy, say σ significant figures, (σ + 1)/4 wavelengths of layered structure must be retained above a homogeneous half-space. To this accuracy, the homogeneous half-space is a sufficient model of the true layering it replaces. Using this result, it was possible to refine the usual layer-reduction technique so as to ensure retention of the specified accuracy while employing reduction. With this reduction technique in effect, and with σ specified below single-precision accuracy, the program can be run entirely in single precision; the specified accuracy is maintained without overflow or loss-of-precision problems being encountered during calculations.

Analysis of gravity and non-homogeneity on Rayleigh wave in anisotropic elastic-viscoelastic half space of higher order

2015 ◽  
Vol 11 (1) ◽  
pp. 120-130 ◽  
Author(s):  
Rajneesh Kakar
Keyword(s):  
Phase Velocity ◽  
Half Space ◽  
Rayleigh Waves ◽  
Mass Density ◽  
Wave Number ◽  
Inhomogeneous Layer ◽  
Third Order ◽  
Content Type ◽  
Anisotropic Elastic ◽  
Mathematical Techniques

Purpose – The purpose of this paper is to illustrate the propagation of Rayleigh waves in an anisotropic inhomogeneous layer placed over an isotropic gravitational viscoelastic half space of third order. Design/methodology/approach – It is considered that the mass density and the elastic coefficients of the layer are space dependent. Dispersion properties of waves are derived with the simple mathematical techniques. Graphs are plotted between phase velocity ‘k’ and wave number ‘c’ for different values of inhomogeneity parameters for a particular model and the effects of inhomogeneity and gravity are studied. Findings – The wave analysis indicates that the phase velocity of Rayleigh waves is affected quite remarkably by the presence of inhomogeneity, gravity and strain rates of strain parameters in the half space. The effects of inhomogeneity and depth on the phase velocity are also shown in corresponding figures. Originality/value – The results presented in this study may be attractive and useful for mathematicians, seismologists and geologists.

Seismic waves from finite faults in layered media

1979 ◽  
Vol 69 (6) ◽  
pp. 1693-1714
Author(s):  
F. Abramovici ◽  
J. Gal-Ezer
Keyword(s):  
Point Source ◽  
Half Space ◽  
Seismic Waves ◽  
Time Dependent ◽  
Homogeneous Layer ◽  
Two Dimensional ◽  
Homogeneous Half Space ◽  
Finite Line ◽  
Finite Line Source ◽  
Time Dependent Solution

abstract The time-dependent solution for a multipolar source in a structure consisting of a homogeneous layer over a homogeneous half-space is obtained as a sum of generalized rays. Numerical seismograms are calculated for a horizontal strikeslip and a horizontal dip-slip for a point-source, a finite line-source, and a finite two-dimensional source in the form of a rectangle. For comparison, the displacements in a homogeneous space and half-space are also calculated. The seismograms for finite sources are similar to those for a point-source but show less conspicuous phases, the arriving pulses being wider and less sharp.

Rayleigh wave equations with couple stress: modeling and dispersion characteristic

Geophysics ◽  
2021 ◽  
pp. 1-64
Author(s):  
Yanqi Wu ◽  
Jianwei Ma
Keyword(s):  
Wave Propagation ◽  
Half Space ◽  
Scale Effect ◽  
Linear Elasticity ◽  
Rayleigh Waves ◽  
Couple Stress ◽  
Characteristic Length ◽  
Couple Stress Theory ◽  
Stress Theory ◽  
Homogeneous Half Space

In elastostatics, the scale effect is a phenomenon in which the elastic parameters of a medium vary with specimen size when the specimen is sufficiently small. Linear elasticity cannot explain the scale effect because it assumes that the medium is a continuum and does not consider microscopic rotational interactions within the medium. In elastodynamics, wave propagation equations are usually based on linear elasticity. Thus, nonlinear elasticity must be introduced to study the scale effect on wave propagation. In this work, we introduce one of the generalized continuum theories—couple stress theory—into solid earth geophysics to build a more practical model of underground medium. The first-order velocity-stress wave equation is derived to simulate the propagation of Rayleigh waves. Body and Rayleigh waves are compared using elastic theory and couple stress theory in homogeneous half- space and layered space. The results show that couple stress causes the dispersion of surface waves and shear waves even in homogeneous half-space. The effect is enhanced by increasing the source frequency and characteristic length, despite its insufficiently clear physical meaning. Rayleigh waves are more sensitive to couple stress effect than body waves. Based on the phase-shifting method, it was determined that Rayleigh waves exhibit different dispersion characteristics in couple stress theory than in conventional elastic theory. For the fundamental mode, the dispersion curves tend to move to a lower frequency with an increase in characteristic length l. For the higher modes, the dispersion curves energy is stronger with a greater characteristic length l.

Arbitrary Load Distribution on a Layered Half Space

2000 ◽  
Vol 122 (4) ◽  
pp. 672-681 ◽  
Author(s):  
N. Schwarzer
Keyword(s):  
Half Space ◽  
Load Distribution ◽  
Elastic Field ◽  
Stress Fields ◽  
Method Of Images ◽  
Elementary Functions ◽  
Homogeneous Case ◽  
Arbitrary Load ◽  
Homogeneous Half Space ◽  
Layered Half Space

This paper develops a method which allows one to calculate the complete elastic field (stress field and displacements) of layered materials of transverse and complete isotropy under given load conditions. It is assumed that the layered body consists of an infinite half-space and various infinite planes which are all ideally bonded to each other. Thus, the interfaces are parallel to the surface of the resulting “coated half space.” The approach is based on the method of images in classical electrostatics. The final solution for an arbitrary load problem can be presented as a series of potential functions, where corresponding functions may be interpreted as “image loads” the analogous to “image charges.” The solution for the elastic field for any arbitrary stress distribution on the surface of the coated half space can be obtained in a relatively straightforward manner by using the method described here as long as the corresponding solution for the homogeneous half space is known. Further, if this solution of the homogeneous case may be expressed in terms of elementary functions, then the solution for the coated half space is elementary, too. Explicit formulas for the stress fields for some particular examples are given. [S0742-4787(00)01204-2]

scholarly journals Horizontal Acoustic Barriers for Protection from Seismic Waves

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Sergey V. Kuznetsov ◽  
Aybek E. Nafasov
Keyword(s):  
Surface Layer ◽  
Numerical Simulations ◽  
Half Space ◽  
Rayleigh Waves ◽  
Seismic Waves ◽  
Specific Region ◽  
Love Waves ◽  
Thin Surface Layer

The basic idea of a seismic barrier is to protect an area occupied by a building or a group of buildings from seismic waves. Depending on nature of seismic waves that are most probable in a specific region, different kinds of seismic barriers can be suggested. Herein, we consider a kind of a seismic barrier that represents a relatively thin surface layer that prevents surface seismic waves from propagating. The ideas for these barriers are based on one Chadwick's result concerning nonpropagation condition for Rayleigh waves in a clamped half-space, and Love's theorem that describes condition of nonexistence for Love waves. The numerical simulations reveal that to be effective the length of the horizontal barriers should be comparable to the typical wavelength.

scholarly journals Scattering by a Cavity in an Exponentially Graded Half-Space

2009 ◽  
Vol 76 (3) ◽  
Author(s):  
P. A. Martin
Keyword(s):  
Differential Equation ◽  
Half Space ◽  
Circular Cavity ◽  
Homogeneous Half Space ◽  
Governing Differential Equation ◽  
Proper Account ◽  
Exponentially Graded

An inhomogeneous half-space containing a cavity is bonded to a homogeneous half-space. Waves are incident on the interface and the problem is to calculate the scattered waves. For a circular cavity in an exponentially graded half-space, it is shown how to solve the problem by constructing an appropriate set of multipole functions. These functions are singular on the axis of the cavity, they satisfy the governing differential equation in each half-space, and they satisfy the continuity conditions across the interface between the two half-spaces. Seven recent publications are criticized: They do not take proper account of the interface between the two half-spaces.

Transmission of Rayleigh waves past a step change in elevation

1965 ◽  
Vol 55 (2) ◽  
pp. 319-334 ◽  
Author(s):  
A. K. Mal ◽  
L. Knopoff
Keyword(s):  
Surface Waves ◽  
Half Space ◽  
Green's Function ◽  
Rayleigh Waves ◽  
Function Method ◽  
Step Change ◽  
Reflection Coefficients ◽  
Homogeneous Half Space ◽  
Transmission And Reflection Coefficients ◽  
Transmission And Reflection

Abstract Using a Green's function method of approximation, transmission and reflection coefficients are computed for the problem of Rayleigh waves incident upon a step change in the elevation of the surface of a homogeneous half-space. Several versions of these approximations are given, differing principally by the method in which the surface waves on the vertical face of the step are taken into account.

On a technique for deriving explicit transfer matrices of orthotropic layers

2015 ◽  
Vol 37 (4) ◽  
pp. 303-315 ◽  
Author(s):  
Pham Chi Vinh ◽  
Nguyen Thi Khanh Linh ◽  
Vu Thi Ngoc Anh
Keyword(s):  
Wave Propagation ◽  
Explicit Form ◽  
Half Space ◽  
Transfer Matrix ◽  
Rayleigh Waves ◽  
Secular Equation ◽  
Transfer Matrices ◽  
Orthotropic Layer ◽  
Wave Propagation Problem ◽  
Arbitrary Thickness

This paper presents  a technique by which the transfer matrix in explicit form of an orthotropic layer can be easily obtained. This transfer matrix is applicable for both the wave propagation problem and the reflection/transmission problem. The obtained transfer matrix is then employed to derive the explicit secular equation of Rayleigh waves propagating in an orthotropic half-space coated by an orthotropic layer of arbitrary thickness.

Sign in / Sign up

Export Citation Format

Share Document