RAYLEIGH‐WAVE MOTION IN AN ELASTIC HALF‐SPACE

Geophysics ◽  
1965 ◽  
Vol 30 (1) ◽  
pp. 97-101 ◽  
Author(s):  
W. A. Sorge
Keyword(s):  
Rayleigh Wave ◽  
Half Space ◽  
Rayleigh Waves ◽  
Wave Motion ◽  
Elastic Half Space ◽  
Two Dimensional ◽  
Seismic Model

Measurements made on Rayleigh waves below the surface of a simulated elastic half‐space confirm in detail the behavior predicted by theory. These measurements, made by means of a two‐dimensional seismic model, show that the amplitude of the Rayleigh wave falls off rapidly with increasing depth.

Exact Rayleigh-Wave Solution from a Point Source in Homogeneous Elastic Half-Space

2020 ◽  
Vol 110 (2) ◽  
pp. 783-792
Author(s):  
Jie Zhou ◽  
Haiming Zhang
Keyword(s):  
Point Source ◽  
Rayleigh Wave ◽  
Half Space ◽  
Rayleigh Waves ◽  
Wave Motion ◽  
Elastic Half Space ◽  
Far Field ◽  
Residue Theorem ◽  
Geometrical Spreading ◽  
Analytical Expressions

ABSTRACT A Rayleigh wave, often the most visible component in the far-field seismograms, is an important type of seismic-wave motion associated with the Earth’s surface. In this study, we explore some of the general properties of the Rayleigh wave in a homogeneous elastic half-space. Starting from the displacement expressed in the form of a wavenumber integral in the frequency domain, we extract the contribution from the pole in the complex wavenumber plane to obtain the excitation formulae of the Rayleigh wave by the residue theorem for complex integrals. Numerical results are compared with the full wavefield solutions to validate our solutions. By examining the analytical expressions obtained, we explore some basic properties of Rayleigh waves such as the particle motion and geometrical spreading. We also demonstrate that these properties of the Rayleigh wave excited by a point source are slightly different from but mostly consistent with the well-known classical properties of plane Rayleigh waves.

Model study of explosion-generated Rayleigh waves in a half space

1964 ◽  
Vol 54 (2) ◽  
pp. 475-484
Author(s):  
I. N. Gupta ◽  
C. Kisslinger
Keyword(s):  
Rayleigh Wave ◽  
Half Space ◽  
Rayleigh Waves ◽  
Radial Component ◽  
Two Dimensional ◽  
Source Depth ◽  
Dimensional Model ◽  
Retrograde Motion ◽  
Model Study ◽  
Two Dimensional Model

ABSTRACT The Rayleigh waves generated by an explosion on or in the interior of a two-dimensional model show that the source acts as a downward impulse when the shot is on or just below the surface, and as a buried source of compression for deeper shots. The seismograms are in agreement with established theory for the line source on or in a half-space. The source depth corresponding to the reversal of polarity of the Rayleigh wave is small, and appears to be equal to the radius of the zone of inelastic failure around the shot. The polarity reversal is a true indication of a change in the mechanism of Rayleigh wave generation, and is not related to the change from retrograde motion at the free surface to prograde motion in the interior associated with the change in sign of the radial component at depth.

scholarly journals Approximate formula for the H/V ratio of Rayleigh waves in incompressible orthotropic half-spaces coated by a thin elastic layer

2017 ◽  
Vol 39 (4) ◽  
pp. 365-374
Author(s):  
Pham Chi Vinh ◽  
Tran Thanh Tuan ◽  
Le Thi Hue
Keyword(s):  
Free Surface ◽  
Rayleigh Wave ◽  
Half Space ◽  
Rayleigh Waves ◽  
Elastic Layer ◽  
Approximate Formula ◽  
Elastic Half Space ◽  
Displacement Amplitude ◽  
Vertical Displacements ◽  
Thin Elastic Layer

This paper is concerned with the propagation of Rayleigh waves in an incompressible orthotropic elastic half-space coated with a thin incompressible orthotropic elastic layer. The main purpose of the paper is to establish an approximate formula for the Rayleigh wave H/V ratio (the ratio between the amplitudes of the horizontal and vertical displacements of Rayleigh waves at the traction-free surface of the layer). First, the relations between the traction amplitude vector and the displacement amplitude vector of Rayleigh waves at two sides of the interface between the layer and the half-space are created using the Stroh formalism and the effective boundary condition method. Then, an approximate formula for the Rayleigh wave H/V ratio of third-order in terms of dimensionless thickness of the layer has been derived by using these relations along with the Taylor expansion of the displacement amplitude vector of the thin layer at its traction-free surface. It is shown numerically that the obtained formula is a good approximate one. It can be used for extracting mechanical properties of thin films from measured values of the  Rayleigh wave H/V ratio.

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.

scholarly journals Surface waves in multilayered elastic media I. Rayleigh and Love waves from buried sources in a multilayered elastic half-space

1964 ◽  
Vol 54 (2) ◽  
pp. 627-679
Author(s):  
David G. Harkrider
Keyword(s):  
Surface Waves ◽  
Frequency Domain ◽  
Half Space ◽  
Rayleigh Waves ◽  
Elastic Half Space ◽  
Love Waves ◽  
Displacement Fields ◽  
Matrix Formulation ◽  
Total Displacement ◽  
Rayleigh And Love Waves

ABSTRACT A matrix formulation is used to derive integral expressions for the time transformed displacement fields produced by simple sources at any depth in a multilayered elastic isotropic solid half-space. The integrals are evaluated for their residue contribution to obtain surface wave displacements in the frequency domain. The solutions are then generalized to include the effect of a surface liquid layer. The theory includes the effect of layering and source depth for the following: (1) Rayleigh waves from an explosive source, (2) Rayleigh waves from a vertical point force, (3) Rayleigh and Love waves from a vertical strike slip fault model. The latter source also includes the effect of fault dimensions and rupture velocity. From these results we are able to show certain reciprocity relations for surface waves which had been previously proved for the total displacement field. The theory presented here lays the ground work for later papers in which theoretical seismograms are compared with observations in both the time and frequency domain.

Rayleigh wave propagation at the boundary surface of dry sandy ($SiO_2$) thermoelastic solids

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Shishir Gupta ◽  
Rishi Dwivedi ◽  
Smita Smita ◽  
Rachaita Dutta
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Rayleigh Wave ◽  
Penetration Depth ◽  
Dispersion Equation ◽  
Half Space ◽  
Rayleigh Waves ◽  
Secular Equation ◽  
Rayleigh Surface Wave ◽  
Content Type

Purpose The purpose of study to this article is to analyze the Rayleigh wave propagation in an isotropic dry sandy thermoelastic half-space. Various wave characteristics, i.e wave velocity, penetration depth and temperature have been derived and represented graphically. The generalized secular equation and classical dispersion equation of Rayleigh wave is obtained in a compact form. Design/methodology/approach The present article deals with the propagation of Rayleigh surface wave in a homogeneous, dry sandy thermoelastic half-space. The dispersion equation for the proposed model is derived in closed form and computed analytically. The velocity of Rayleigh surface wave is discussed through graphs. Phase velocity and penetration depth of generated quasi P, quasi SH wave, and thermal mode wave is computed mathematically and analyzed graphically. To illustrate the analytical developments, some particular cases are deliberated, which agrees with the classical equation of Rayleigh waves. Findings The dispersion equation of Rayleigh waves in the presence of thermal conductivity for a dry sandy thermoelastic medium has been derived. The dry sandiness parameter plays an effective role in thermoelastic media, especially with respect to the reference temperature for η = 0.6,0.8,1. The significant difference in η changes a lot in thermal parameters that are obvious from graphs. The penetration depth and phase velocity for generated quasi-wave is deduced due to the propagation of Rayleigh wave. The generalized secular equation and classical dispersion equation of Rayleigh wave is obtained in a compact form. Originality/value Rayleigh surface wave propagation in dry sandy thermoelastic medium has not been attempted so far. In the present investigation, the propagation of Rayleigh waves in dry sandy thermoelastic half-space has been considered. This study will find its applications in the design of surface acoustic wave devices, earthquake engineering structural mechanics and damages in the characterization of materials.

Rayleigh-type surface wave on a rotating orthotropic elastic half-space with impedance boundary conditions

2020 ◽  
Vol 26 (21-22) ◽  
pp. 1980-1987
Author(s):  
Baljeet Singh ◽  
Baljinder Kaur
Keyword(s):  
Surface Wave ◽  
Boundary Conditions ◽  
Rayleigh Wave ◽  
Surface Waves ◽  
Half Space ◽  
Rotation Rate ◽  
Secular Equation ◽  
Elastic Half Space ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The propagation of Rayleigh type surface waves in a rotating elastic half-space of orthotropic type is studied under impedance boundary conditions. The secular equation is obtained explicitly using traditional methodology. A program in MATLAB software is developed to obtain the numerical values of the nondimensional speed of Rayleigh wave. The speed of Rayleigh wave is illustrated graphically against rotation rate, nondimensional material constants, and impedance boundary parameters.

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