Standing waves in a layered half space

1966 ◽  
Vol 56 (5) ◽  
pp. 1153-1161 ◽  
Author(s):  
I. N. Gupta
Keyword(s):  
Half Space ◽  
Body Wave ◽  
Standing Waves ◽  
Elastic Half Space ◽  
Layered System ◽  
Wave Type ◽  
Sh Waves ◽  
Earthquake Motion ◽  
Vertically Propagating ◽  
Layered Half Space

Abstract In a homogeneous elastic half space, plane harmonic waves give rise to standing waves only when on reflection there is no conversion from one wave-type to another. The existence of standing waves in a horizontally-layered half space is established for vertically propagating plane harmonic P, SV or SH waves. Expressions are derived for the particle displacements at the free surface and at any given depth. The layered system acts as a complicated filter suppressing certain periods while amplifying others. The results obtained may be helpful in an understanding of the ‘ground factor’, ambient seismic body-wave noise, and the vibration problem of a structure due to earthquake motion.

Body wave radiation patterns from force applied within a half space

1966 ◽  
Vol 56 (1) ◽  
pp. 173-183 ◽  
Author(s):  
Indra N. Gupta
Keyword(s):  
Half Space ◽  
Body Wave ◽  
Elastic Half Space ◽  
Reciprocity Theorem ◽  
Numerical Results ◽  
Wave Radiation ◽  
Far Field ◽  
Radiation Patterns ◽  
Sh Waves ◽  
Vertical Displacements

abstract Expressions are derived for the horizontal and vertical displacements at an arbitrary depth within a homogeneous, isotropic, elastic half space when plane harmonic P, SV or SH waves are incident at any given angle. On the basis of the reciprocity theorem, these expressions represent also the far-field polar radiation patterns of P, SV and SH waves due to horizontal and vertical forces applied at a point within the half space. Numerical results for a few selected values of depth are shown for a solid half space.

Guided Surface Waves on an Elastic Half Space

1971 ◽  
Vol 38 (4) ◽  
pp. 899-905 ◽  
Author(s):  
L. B. Freund
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Surface Waves ◽  
Half Space ◽  
Body Wave ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Normal Displacement ◽  
Rigid Barrier ◽  
Wave Disturbances

Three-dimensional wave propagation in an elastic half space is considered. The half space is traction free on half its boundary, while the remaining part of the boundary is free of shear traction and is constrained against normal displacement by a smooth, rigid barrier. A time-harmonic surface wave, traveling on the traction free part of the surface, is obliquely incident on the edge of the barrier. The amplitude and the phase of the resulting reflected surface wave are determined by means of Laplace transform methods and the Wiener-Hopf technique. Wave propagation in an elastic half space in contact with two rigid, smooth barriers is then considered. The barriers are arranged so that a strip on the surface of uniform width is traction free, which forms a wave guide for surface waves. Results of the surface wave reflection problem are then used to geometrically construct dispersion relations for the propagation of unattenuated guided surface waves in the guiding structure. The rate of decay of body wave disturbances, localized near the edges of the guide, is discussed.

Multiple Region Contact Solutions for a Flat Indenter on a Layered Elastic Half Space: Plane-Strain Case

1989 ◽  
Vol 56 (2) ◽  
pp. 251-262 ◽  
Author(s):  
T. W. Shield ◽  
D. B. Bogy
Keyword(s):  
Plane Strain ◽  
Half Space ◽  
Contact Region ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Geometrical Parameters ◽  
Restricted Range ◽  
Complete Contact ◽  
The One ◽  
Layered Half Space

The plane-strain problem of a smooth, flat rigid indenter contacting a layered elastic half space is examined. It is mathematically formulated using integral transforms to derive a singular integral equation for the contact pressure, which is solved by expansion in orthogonal polynomials. The solution predicts complete contact between the indenter and the surface of the layered half space only for a restricted range of the material and geometrical parameters. Outside of this range, solutions exist with two or three contact regions. The parameter space divisions between the one, two, or three contact region solutions depend on the material and geometrical parameters and they are found for both the one and two layer cases. As the modulus of the substrate decreases to zero, the two contact region solution predicts the expected result that contact occurs only at the corners of the indenter. The three contact region solution provides an explanation for the nonuniform approach to the half space solution as the layer thickness vanishes.

Dynamic Response of a Hemispherical Foundation Embedded in an Elastic Half-Space

1998 ◽  
Vol 120 (4) ◽  
pp. 343-348 ◽  
Author(s):  
C.-S. Yeh ◽  
T.-J. Teng ◽  
W.-I. Liao
Keyword(s):  
Dynamic Response ◽  
Half Space ◽  
Ground Surface ◽  
Integral Equation Method ◽  
Elastic Half Space ◽  
Least Square ◽  
Sh Waves ◽  
Soil Foundation ◽  
External Forces ◽  
Impedance Functions

The dynamic response of a massless rigid hemispherical foundation embedded in a uniform homogeneous elastic half-space is considered in this study. The foundation is subjected to external forces, moments, plane harmonic P and SH waves, respectively. The series solutions are constructed by three sequences of Lamb’s singular solutions which satisfy the traction-free conditions on ground surface and radiation conditions at infinity, automatically, and their coefficients are determined by the boundary conditions along the soil-foundation interface in the least square sense. The fictitious eigen-frequencies, which arise in integral equation method, will not appear in the numerical calculation by the proposed method. The impedance functions which characterize the response of the foundation to external harmonic forces and moments at low and intermediate frequencies are calculated and the translational and rocking responses of the foundation when subjected to plane P and SH waves are also presented and discussed in detail.

On raytracing in an elastic-anelastic medium

1991 ◽  
Vol 81 (2) ◽  
pp. 667-686 ◽  
Author(s):  
E. S. Krebes ◽  
M. A. Slawinski
Keyword(s):  
Travel Time ◽  
Half Space ◽  
Body Wave ◽  
Seismic Wave Propagation ◽  
Elastic Half Space ◽  
Elastic Plane ◽  
Ray Method ◽  
The One ◽  
Ray Parameter ◽  
Method Of Steepest Descent

Abstract In this article, we investigate seismic wave propagation in a medium consisting of a stack of anelastic layers sandwiched between two half-spaces. The upper half-space is perfectly elastic, and the lower half-space is anelastic. The source is in the upper elastic half-space. To compute a ray going from the source to the receiver (which can be anywhere in the medium), we examine two approaches. The first involves an evaluation of the Sommerfeld wavefield integral by the method of steepest descent, and we refer to the resulting ray as the stationary ray. The second involves assuming that the attenuation vector A1 of the initial ray segment emerging from the source in the elastic half-space is zero (an assumption often made in the literature), and we refer to the resulting ray as the conventional ray. We find that the stationary and conventional rays are, in general, not identical, in that the stationary ray has (a) a complex, rather than real, ray parameter; (b) a smaller travel time; (c) an initial ray segment that corresponds to an inhomogeneous elastic plane body wave (A1 ≠ 0); and (d) a substantially different value for the ray amplitude. The stationary ray actually has the smallest travel time of all possible rays, and hence it is the one that satisfies Fermat's principle of least time. Our results suggest that the stationary ray method is the correct method and that the conventional ray method is generally incorrect. The results might also find application in marine seismology, since water is practically a lossless medium.

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