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.

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.

Note on the use of reciprocity theorem for obtaining radiation patterns

1965 ◽  
Vol 55 (2) ◽  
pp. 277-281 ◽  
Author(s):  
Indra N. Gupta
Keyword(s):  
Free Surface ◽  
Half Space ◽  
Elastic Half Space ◽  
Reciprocity Theorem ◽  
Far Field ◽  
Radiation Patterns ◽  
Vertical Displacements ◽  
Field Radiation ◽  
P And Sv Waves

Abstract Expressions for the horizontal and vertical displacements at the surface of an elastic half space when plane harmonic P or SV waves are incident at any given angle are already known. On the basis of the reciprocity theorem, these expressions are used to obtain “far-field” radiation patterns of P and SV waves due to horizontal and vertical forces applied at the free surface.

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.

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 Rayleigh waves in compressible orthotropic half-space overlaid by a thin un-coaxial orthotropic layer

2021 ◽  
Vol 15 (4) ◽  
pp. 54-64
Author(s):  
Trinh Thi Thanh Hue ◽  
Phan Thi Thu Phuong ◽  
Pham Hong Anh
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Rayleigh Waves ◽  
Secular Equation ◽  
Traditional Approach ◽  
Elastic Half Space ◽  
Third Order ◽  
Effective Boundary ◽  
Effective Boundary Conditions ◽  
Thin Elastic Layer

The problem of Rayleigh waves in compressible orthotropic elastic half-space overlaid by a thin elastic layer of which principal material axes are coincident have been researched by many scientists. However, the problem with the conditions that the half-space and the layer have only one common principal material axis that perpendicular to the layer while the remains are not identical has not gotten enough attention. This paper presents a traditional approach to obtain an approximate secular equation by approximately replacing the thin layer by effective boundary conditions of third-order. The wave then is considered as a Rayleigh wave propagating in an orthotropic half-space, without coating, subjected to the effective boundary conditions. This explicit approximate secular equation is potentially useful in non-damage assessment studies.

scholarly journals An approximate secular equation of Rayleigh waves in an elastic half-space coated by a thin weakly inhomogeneous elastic layer

2015 ◽  
Vol 37 (1) ◽  
pp. 71-80
Author(s):  
Pham Chi Vinh ◽  
Vu Thi Ngoc Anh
Keyword(s):  
Half Space ◽  
Rayleigh Waves ◽  
Elastic Layer ◽  
Secular Equation ◽  
Elastic Half Space ◽  
Effective Boundary ◽  
Inhomogeneous Elastic Layer ◽  
Boundary Condition Method ◽  
Effective Boundary Condition ◽  
Thickness Variable

In this paper, the propagation of Rayleigh waves in a homogeneous isotropic elastic half-space coated with a thin weakly inhomogeneous isotropic elastic layer is investigated. The material parameters of the layer is assumed to  depend arbitrarily continuously on  the thickness variable. The contact between the layer and the half space is  perfectly bonded. The main purpose of the paper is to establish an approximate secular equation of the wave. By applying the effective boundary condition method an approximate secular equation of second order in terms of the dimensionless thickness of the layer is derived. It is shown that the obtained approximate secular equation has good accuracy.

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