Response of an Elastic Half Space to Expanding Surface Loads

1971 ◽  
Vol 38 (1) ◽  
pp. 99-110 ◽  
Author(s):  
D. C. Gakenheimer
Keyword(s):  
Rayleigh Wave ◽  
Exact Solutions ◽  
Half Space ◽  
Poisson’S Ratio ◽  
Elastic Half Space ◽  
Poisson's Ratio ◽  
Constant Rate ◽  
Different Depths ◽  
Surface Loads

A class of elastic half-space problems involving axisymmetric, normally applied, surface loads is investigated. Each load is assumed to suddenly emanate from a point on the surface and expand radially at a constant rate. The cases of loads shaped like a ring and a disk are considered in detail. Exact solutions are derived for the displacements at every point in the half space in terms of single integrals. Each integral is identified as a specific wave. The integrals are evaluated analytically and numerically for different depths in the half space, for loads expanding at superseismic, transeismic, and subseismic rates, and for different values of Poisson’s ratio. Moreover, the interaction of the loads and the Rayleigh wave is described. Then solutions are obtained for loads of other shapes by convoluting the ring and disk load solutions.

THE REFLECTION OF RAYLEIGH WAVES BY A HIGH IMPEDANCE OBSTACLE ON A HALF‐SPACE

Geophysics ◽  
1960 ◽  
Vol 25 (6) ◽  
pp. 1195-1202 ◽  
Author(s):  
R. W. Fredricks ◽  
L. Knopoff
Keyword(s):  
Reflection Coefficient ◽  
Rayleigh Wave ◽  
Half Space ◽  
Poisson’S Ratio ◽  
Vertical Displacement ◽  
Elastic Half Space ◽  
Poisson's Ratio ◽  
Dual Integral Equations ◽  
High Impedance ◽  
Theoretic Solution

The reflection of a time‐harmonic Rayleigh wave by a high impedance obstacle in shearless contact with an elastic half‐space of lower impedance is examined theoretically. The potentials are found by a function—theoretic solution to dual integral equations. From these potentials, a “reflection coefficient” is defined for the surface vertical displacement in the Rayleigh wave. Results show that the reflected wave is π/2 radians out of phase with the incident wave for arbitrary Poisson’s ratio. The modulus of the “reflection coefficient” depends upon Poisson’s ratio, and is evaluated as [Formula: see text] for σ=0.25.

The azimuthal and polar radiation patterns obtained from a horizontal stress applied at the surface of an elastic half space

1962 ◽  
Vol 52 (1) ◽  
pp. 27-36
Author(s):  
J. T. Cherry
Keyword(s):  
Surface Wave ◽  
Half Space ◽  
Rayleigh Waves ◽  
Poisson’S Ratio ◽  
Horizontal Stress ◽  
Elastic Half Space ◽  
The Body ◽  
Body Waves ◽  
Poisson's Ratio ◽  
A Value

Abstract The body waves and surface waves radiating from a horizontal stress applied at the free surface of an elastic half space are obtained. The SV wave suffers a phase shift of π at 45 degrees from the vertical. Also, a surface wave that is SH in character but travels with the Rayleigh velocity is shown to exist. This surface wave attenuates as r−3/2. For a value of Poisson's ratio of 0.25 or 0.33, the amplitude of the Rayleigh waves from a horizontal source should be smaller than the amplitude of the Rayleigh waves from a vertical source. The ratio of vertical to horizontal amplitude for the Rayleigh waves from the horizontal source is the same as the corresponding ratio for the vertical source for all values of Poisson's ratio.

scholarly journals A single Rayleigh mode may exist with multiple values of phase-velocity at one frequency

2020 ◽  
Vol 222 (1) ◽  
pp. 582-594
Author(s):  
Thomas Forbriger ◽  
Lingli Gao ◽  
Peter Malischewsky ◽  
Matthias Ohrnberger ◽  
Yudi Pan
Keyword(s):  
Phase Velocity ◽  
Rayleigh Wave ◽  
Half Space ◽  
Wave Velocity ◽  
Poisson’S Ratio ◽  
P Wave ◽  
Poisson's Ratio ◽  
Wave Number ◽  
Homogeneous Layer ◽  
Rayleigh Mode

SUMMARY Other than commonly assumed in seismology, the phase velocity of Rayleigh waves is not necessarily a single-valued function of frequency. In fact, a single Rayleigh mode can exist with three different values of phase velocity at one frequency. We demonstrate this for the first higher mode on a realistic shallow seismic structure of a homogeneous layer of unconsolidated sediments on top of a half-space of solid rock (LOH). In the case of LOH a significant contrast to the half-space is required to produce the phenomenon. In a simpler structure of a homogeneous layer with fixed (rigid) bottom (LFB) the phenomenon exists for values of Poisson’s ratio between 0.19 and 0.5 and is most pronounced for P-wave velocity being three times S-wave velocity (Poisson’s ratio of 0.4375). A pavement-like structure (PAV) of two layers on top of a half-space produces the multivaluedness for the fundamental mode. Programs for the computation of synthetic dispersion curves are prone to trouble in such cases. Many of them use mode-follower algorithms which loose track of the dispersion curve and miss the multivalued section. We show results for well established programs. Their inability to properly handle these cases might be one reason why the phenomenon of multivaluedness went unnoticed in seismological Rayleigh wave research for so long. For the very same reason methods of dispersion analysis must fail if they imply wave number kl(ω) for the lth Rayleigh mode to be a single-valued function of frequency ω. This applies in particular to deconvolution methods like phase-matched filters. We demonstrate that a slant-stack analysis fails in the multivalued section, while a Fourier–Bessel transformation captures the complete Rayleigh-wave signal. Waves of finite bandwidth in the multivalued section propagate with positive group-velocity and negative phase-velocity. Their eigenfunctions appear conventional and contain no conspicuous feature.

Deformation of a Flexible Disk Bonded to an Elastic Half Space—Application to the Lung

1980 ◽  
Vol 102 (3) ◽  
pp. 234-239 ◽  
Author(s):  
S. J. Lai-Fook ◽  
M. A. Hajji ◽  
T. A. Wilson
Keyword(s):  
Half Space ◽  
Poisson’S Ratio ◽  
Radial Strain ◽  
Lung Parenchyma ◽  
Elastic Half Space ◽  
Poisson's Ratio ◽  
Normal Displacement ◽  
Direct Estimate ◽  
Analytic Expressions ◽  
Flexible Disk

An analysis is presented of the deformation of a homogeneous, isotropic, elastic half space subjected to a constant radial strain in a circular area on the boundary. Explicit analytic expressions for the normal and radial displacements and the shear stress on the boundary are used to interpret experiments performed on inflated pig lungs. The boundary strain was induced by inflating or deflating the lung after bonding a flexible disk to the lung surface. The prediction that the surface bulges outward for positive boundary strain and inward for negative strain was observed in the experiments. Poisson’s ratio at two transpulmonary pressures was measured, by use of the normal displacement equation evaluated at the surface. A direct estimate of Poisson’s ratio was possible because the normal displacement of the surface depended uniquely on the compressibility of the material. Qualitative comparisons between theory and experiment support the use of continuum analyses in evaluating the behavior of the lung parenchyma when subjected to small local distortions.

Some numerical solutions for Lamb's problem

1974 ◽  
Vol 64 (2) ◽  
pp. 473-491
Author(s):  
Harold M. Mooney
Keyword(s):  
Rayleigh Wave ◽  
Poisson’S Ratio ◽  
Pulse Width ◽  
Numerical Solutions ◽  
Pulse Height ◽  
P Wave ◽  
Poisson's Ratio ◽  
Wave Form ◽  
Lamb’S Problem ◽  
Wave Forms

abstract We consider a version of Lamb's Problem in which a vertical time-dependent point force acts on the surface of a uniform half-space. The resulting surface disturbance is computed as vertical and horizontal components of displacement, particle velocity, acceleration, and strain. The goal is to provide numerical solutions appropriate to a comparison with observed wave forms produced by impacts onto granite and onto soil. Solutions for step- and delta-function sources are not physically realistic but represent limiting cases. They show a clear P arrival (larger on horizontal than vertical components) and an obscure S arrival. The Rayleigh pulse includes a singularity at the theoretical arrival time. All of the energy buildup appears on the vertical components and all of the energy decay, on the horizontal components. The effects of Poisson's ratio upon vertical displacements for a step-function source are shown. For fixed shear velocity, an increase of Poisson's ratio produces a P pulse which is larger, faster, and more gradually emergent, an S pulse with more clear-cut beginning, and a much narrower Rayleigh pulse. For a source-time function given by cos2(πt/T), −T/2 ≦ T/2, a × 10 reduction in pulse width at fixed pulse height yields an increase in P and Rayleigh-wave amplitudes by factors of 1, 10, and 100 for displacement, velocity and strain, and acceleration, respectively. The observed wave forms appear somewhat oscillatory, with widths proportional to the source pulse width. The Rayleigh pulse appears as emergent positive on vertical components and as sharp negative on horizontal components. We show a theoretical seismic profile for granite, with source pulse width of 10 µsec and detectors at 10, 20, 30, 40, and 50 cm. Pulse amplitude decays as r−1 for P wave and r−12 for Rayleigh wave. Pulse width broadens slightly with distance but the wave form character remains essentially unchanged.

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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