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L. Knopoff

Bio: L. Knopoff is an academic researcher from University of California, Los Angeles. The author has contributed to research in topics: Love wave & Lamb waves. The author has an hindex of 7, co-authored 8 publications receiving 310 citations.

Papers
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Journal ArticleDOI
TL;DR: In this paper, the transmission and reflection coefficients for Rayleigh waves normally incident upon the corner of a homogeneous elastic wedge formed by two stress-free planes are computed using a Green9s function method of approximation.
Abstract: Using a Green9s function method of approximation, transmission and reflection coefficients are computed for the problem of Rayleigh waves normally incident upon the corner of a homogeneous elastic wedge formed by two stress-free planes. The Rayleigh waves are incident from infinity and travel along one surface of the wedge. The transmitted waves on the second surface and the reflected waves on the first surface are calculated by the application of Huygens9 principle. A pair of coupled integral equations for the displacements are obtained by means of a representation theorem. Neglecting the diffracted body waves near the corner, the coupled integral equations are reduced to a pair of algebraic equations. A new feature of the calculation involves consideration of diffracted surface waves travelling toward the vertex. Numberical values of the phase shifts and attenuation factors in the transmitted and reflected waves are computed as functions of the wedge angle. Comparison with experimental results show considerably better agreement than has been obtained previously.

54 citations

Journal ArticleDOI
TL;DR: In this paper, self-consistent results on the properties of inhomogeneous materials are compared with results based on multiple scattering calculations for a medium containing either spherical inclusions or cylindrical inclusions.
Abstract: Self-consistent results on the properties of inhomogeneous materials are compared with results based on multiple scattering calculations for a medium containing either spherical inclusions or cylindrical inclusions. Inconsistencies are found in terms of order c2, where c is the concentration of inclusions. Thus the claims of O'Connell and Budiansky and others regarding the validity of self-consistent calculations for properties of inhomogeneous materials at nondilute concentrations are not supported.

45 citations

Journal ArticleDOI
01 Nov 1980
TL;DR: In this paper, the overall velocities as well as the specific attenuation coefficients of plane harmonic compressional and shear waves are calculated for low frequencies and dilute concentration of the cracks.
Abstract: The problem of the determination of the overall dynamic elastic moduli of an elastic solid permeated by uniformly distributed penny-shaped cracks is considered. The cracks are assumed to be filled with a viscoelastic material. The orientations of the cracks may be either parallel or perfectly random. The overall velocities as well as the specific attenuation coefficients of plane harmonic compressional and shear waves are calculated for low frequencies and dilute concentration of the cracks.

19 citations


Cited by
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TL;DR: A review of wave propagation in anisotropic media can be found in this paper, where the authors present consistent theoretical formulations for the numerical solution of a number of propagation problems, including body and surface wave polarizations.

771 citations

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TL;DR: In this paper, the problem of determining the elastic properties of composite materials (polycrystals, polycrystals and porous or cracked solids) is approached in several ways, via scattering theory, through variational principles, or by the assumption of specific geometries for the material under consideration.
Abstract: The determination of the elastic properties of composite materials (multiphase aggregates, polycrystals, and porous or cracked solids) from the elastic properties of the components may be approached in several ways. The problem may be treated statistically, via scattering theory, through variational principles, or by the assumption of specific geometries for the material under consideration. Each of these methods is reviewed in turn. The widely used Voigt-Reuss-Hill average can be a poor approximation for both two-phase composites and polycrystals, and its replacement by the two Hashin-Shtrikman bounds is recommended. For pore-containing or crack-containing media, specific geometry models must be considered if useful results are to be obtained. If aggregate theory is used to estimate the moduli of individual components of a composite whose bulk properties are known, the shear moduli of the component phases must be matched (within a factor of 2 or 3) for the method to be useful. Results for nonlinear composites (which allow calculation of the pressure variation of aggregate moduli) have been obtained for only a few special cases.

714 citations

Journal ArticleDOI
01 Sep 1980
TL;DR: In this paper, the authors derived the differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions, which consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term.
Abstract: The differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions is derived. It consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term. This implies the second-order interactions between scatterers are included but those of the third order are not.The formulae are specialized to the case of thin cracks, either aligned in a single direction or randomly oriented. Expressions for the overall elastic constants are derived for the case of long wavelengths. These expressions are accurate to the second order in the number density of scatterers.

699 citations

Journal ArticleDOI
TL;DR: In this article, the dispersion of Love and Rayleigh waves in the period range 17-167 s is used to detect the change in the structure of the upper mantle as the age of the sea-floor increases away from the mid-ocean ridge.
Abstract: Summary The dispersion of Love and Rayleigh waves in the period range 17–167 s is used to detect the change in the structure of the upper mantle as the age of the sea-floor increases away from the mid-ocean ridge Using the single station method, the group and phase velocities of Rayleigh waves were measured for 78 paths in the east Pacific In order to describe the observed Rayleigh wave dispersion, both a systematic increase in velocities with the age of the sea-floor and anisotropy of propagation are required The maximum change in velocity with age is about 5 per cent, with the contrast between age zones decreasing with increasing period The greatest change occurs in the first few million years, due to the rapid cooling and solidification of the upper part of the lithosphere In the 0–5 My age zone, the average thickness of the lithosphere can be no greater than 30 km, including the water and crustal layers Within 10 My after formation, the lithosphere reaches a thickness of about 60 km As the mantle continues to cool, the shear velocity within the lithosphere increases Within the area of this study, no change occurs in the upper mantle deeper than about 80 km Rayleigh waves travel fastest in the direction of spreading The degree of anisotropy in Rayleigh wave propagation is frequency-dependent, reaching a maximum of 2–0±0–2 per cent at a period of about 70 s Several models are constructed which can reproduce this frequency-dependent anisotropy The regional phase velocities of the fundamental and first higher Love modes have been simultaneously measured using a new technique The Love wave data is inconsistent with the Rayleigh wave data unless SH velocity is higher than SV velocity within the uppermost 125 km of the mantle Anisotropy deeper than 250 km is suggested, but not required, by the data

451 citations

Journal ArticleDOI
TL;DR: In this article, the authors proposed a linear combination of a double couple and compensated linear-vector dipole (RL-D dipole) sources for high-frequency seismic motions.
Abstract: Models of earthquake sources that have no volume change, no net force, and no net torque as criteria for the radiation of first motions, have five degrees of freedom in their spatial orientation. The usual double-couple model has only three degrees of freedom. The most general source of high-frequency seismic motions must be a linear combination of a double couple and another source called the compensated linear-vector dipole. A radiation pattern of amplitudes of first motions on the focal sphere cannot be uniquely decomposed into the radiation patterns due to the two sources.

422 citations