Weakly coupled potentials for high-frequency elastic waves in continuously stratified media

1974 ◽  
Vol 64 (5) ◽  
pp. 1575-1588
Author(s):  
Paul G. Richards
Keyword(s):  
Coupling Coefficient ◽  
Elastic Waves ◽  
High Frequency ◽  
Wave Equations ◽  
Asymptotic Methods ◽  
Body Waves ◽  
Radial Dependence ◽  
Spherically Symmetric ◽  
Weakly Coupled ◽  
Seismic Body Waves

Abstract A simple construction is given, in terms of potentials, for assigning P and S components of elastic-wave displacement in media with depth-dependent density and Lame parameters. Attention is focused on spherically symmetric media, with radial dependence of material properties, and the asymptotic methods used are expected to be accurate for seismic body waves and surface waves with period less than about 1 min. Novel features of the construction include: (a) its completeness (all possible displacements are represented by the potentials), (b) the development of second-order wave equations, for the P and S potentials, which explicitly display a coupling coefficient, and (c) demonstration of the way in which P and SV components of displacement decouple as frequency increases. Previous work has given constraints on the medium such that P and SV decouple completely: these constraints are here simplified and are seen to arise naturally in the context of the present potential representation. The coupling coefficient (between P and SV) is examined in a variety of earth models.

Asymptotic theory of body waves in a radially heterogeneous Earth

1969 ◽  
Vol 59 (5) ◽  
pp. 2039-2059
Author(s):  
Sarva Jit Singh ◽  
Ari Ben-Menahem
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
High Frequency ◽  
Asymptotic Theory ◽  
Heterogeneous Medium ◽  
Body Waves ◽  
Spherically Symmetric ◽  
Vector Wave ◽  
Vector Wave Equation ◽  
Earth Models

abstract Various aspects of elastic wave propagation in a spherically symmetric, non-gravitating, isotropic, inhomogeneous medium are considered. It is shown through a simple example that the high frequency decoupling conditions of the vector wave equation may be approximately satisfied by real Earth models. An asymptotic theory is developed for the decoupled potential equations. This theory is applied to the case of a shear dislocation and to that of a center of compression in a radially heterogeneous medium. Explicit expressions are obtained for the ray-theoretical displacements.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

A dynamic model for far-field acceleration

1982 ◽  
Vol 72 (4) ◽  
pp. 1049-1068
Author(s):  
John Boatwright
Keyword(s):  
Stress Drop ◽  
High Frequency ◽  
Dynamic Stress ◽  
The Body ◽  
Body Waves ◽  
Far Field ◽  
Rupture Velocity ◽  
Frequency Level ◽  
Average Change ◽  
Dynamic Stress Drop

abstract A model for the far-field acceleration radiated by an incoherent rupture is constructed by combining Madariaga's (1977) theory for the high-frequency radiation from crack models of faulting with a simple statistical source model. By extending Madariaga's results to acceleration pulses with finite durations, the peak acceleration of a pulse radiated by a single stop or start of a crack tip is shown to depend on the dynamic stress drop of the subevent, the total change in rupture velocity, and the ratio of the subevent radius to the acceleration pulse width. An incoherent rupture is approximated by a sample from a self-similar distribution of coherent subevents. Assuming the subevents fit together without overlapping, the high-frequency level of the acceleration spectra depends linearly on the rms dynamic stress drop, the average change in rupture velocity, and the square root of the overall rupture area. The high-frequency level is independent, to first order, of the rupture complexity. Following Hanks (1979), simple approximations are derived for the relation between the rms dynamic stress drop and the rms acceleration, averaged over the pulse duration. This relation necessarily depends on the shape of the body-wave spectra. The body waves radiated by 10 small earthquakes near Monticello Dam, South Carolina, are analyzed to test these results. The average change of rupture velocity of Δv = 0.8β associated with the radiation of the acceleration pulses is estimated by comparing the rms acceleration contained in the P waves to that in the S waves. The rms dynamic stress drops of the 10 events, estimated from the rms accelerations, range from 0.4 to 1.9 bars and are strongly correlated with estimates of the apparent stress.

Polarization analysis of high-frequency, three-component seismic data

1991 ◽  
Vol 81 (2) ◽  
pp. 622-642
Author(s):  
K. Bataille ◽  
J. M. Chiu
Keyword(s):  
High Frequency ◽  
Time Windows ◽  
Random Noise ◽  
Polarization State ◽  
Principal Component ◽  
Small Time ◽  
Body Waves ◽  
Local Data ◽  
Principal Component Approach ◽  
Component Approach

Abstract We present a method to determine the polarization of body waves from three-component, high-frequency data and examples of its application. The method is based on the principal component approach. One advantage of this approach is that the polarization state can be determined for small time windows compared with the predominant period of the wave. This is particularly useful for identifying converted waves within the crust. The stability of the result is analyzed with synthetic cases by adding simultaneous arrivals from waves and random noise. The method works well with both synthetic and local data in the detection of the polarization of the wave by separating arrivals from different directions. From the local data, some seismic phases related to crustal conversions are observed that require strong lateral variations.

Reflection and transmission of elastic waves in a system of corrugated layers

1967 ◽  
Vol 57 (3) ◽  
pp. 393-419
Author(s):  
A. Levy ◽  
H. Deresiewicz
Keyword(s):  
Incident Wave ◽  
Elastic Waves ◽  
Scattered Field ◽  
Single Layer ◽  
Body Waves ◽  
Numerical Computations ◽  
Plane Body ◽  
Reflection And Transmission ◽  
Plane Parallel

abstract The scattered field generated by normally incident body waves in a system of layers having small, but otherwise arbitrary, periodic deviations from plane parallel boundaries is shown to consist of superposed plane body and surfacetype waves. Results of numerical computations for two like half-spaces separated by a sinusoidally corrugated single layer, and by two layers, reveal the variation of the amplitude of the field with ratios of velocities, densities, impedances, and with those of depth of layers and wavelength of the boundary corrugations to the wavelength of the incident wave.

scholarly journals Asymptotic analysis for dispersion relations and travel times in noise cross-correlations: spherically symmetric case

2018 ◽  
Vol 474 (2218) ◽  
pp. 20180382
Author(s):  
Tianshi Liu ◽  
Haiming Zhang
Keyword(s):  
Asymptotic Analysis ◽  
Surface Waves ◽  
Green's Functions ◽  
Green’S Functions ◽  
Dispersion Relations ◽  
Body Waves ◽  
Spherically Symmetric ◽  
Travel Times ◽  
Cross Correlations ◽  
The Cross

The cross-correlations of ambient noise or earthquake codas are massively used in seismic tomography to measure the dispersion curves of surface waves and the travel times of body waves. Such measurements are based on the assumption that these kinematic parameters in the cross-correlations of noise coincide with those in Green's functions. However, the relation between the cross-correlations of noise and Green's functions deserves to be studied more precisely. In this paper, we use the asymptotic analysis to study the dispersion relations of surface waves and the travel times of body waves, and come to the conclusion that for the spherically symmetric Earth model, when the distribution of noise sources is laterally uniform, the dispersion relations of surface waves and the travel times of SH body-wave phases in noise correlations should be exactly the same as those in Green's functions.

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