On the wave propagation in inhomogeneous non-isotropic media

Summary FollowingEason, we have discussed here the propagation of elastic waves in non-homogeneous spheres and cylinders when the curved surface is given a uniform normal loading. The material is assumed to be transversely isotropic with respect to a direction of symmetry, the stress and displacement components within the body may be assumed to depend on one space co-ordinate and time alone. The particular case in which the elastic parameters are proportional to (radius) ⁿ has been considered.     

Rayleigh wave propagation in a two-layer anisotropic media

Summary The possibility of propagation of Rayleigh waves in an incompressible crust of constant density and rigidity varying exponentially with depth lying on a semi-infinite transversely isotropic base has been discussed in this paper. Frequency equation has been derived and numerical calculations are made. The result obtained in this case is compared with that ofNewlands [3]²) andDutta [4].     

Surface waves in a micropolar elastic solid containing voids

The present paper investigates the effect of voids on the propagation of surface waves in a homogeneous micropolar elastic solid medium which contains a distribution of vacuous pores (voids). The general theory for surface wave propagation in micropolar elastic media containing voids has been presented. Particular cases of surface waves (Rayleigh’s, Love’s and Stoneley’s) in micropolar media which contain vacuous pores have been deduced from the above general theory. Discussions have been made in each case to highlight the effect of voids and micropolar character of the material medium separately. Their joint effect has also been studied in details. Modulation of Rayleigh wave velocity has been studied numerically. It is observed that Love waves are not affected by the presence of voids.     

Elastische Multipolwellen in einem Medium,das aus homogenen,isotropen, planparallelen Schichten besteht

Zusammenfassung Gegeben ist ein festes, elastisches Medium, das aus homogenen, isotropen, planparallelen Schichten aufgebaut ist. Im Innern einer Schicht wird ein punktförmiger Störungsherd von Multipolcharakter angenommen. Die Störung setzt zur Zeitt=0 ein. Unter Anwendung der Laplace-Transformation werden nach dem Verfahren vonCagniard «Exponentialkoeffizienten» und «Übertragungsfaktoren» berechnet, die gestatten, die elastischen Verschiebungen der an den verschiedenen Unstetigkeitsflächen reflektierten und gebrochenen Wellen zu bestimmen.

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Summary (Elastic waves of multiplet type in a solid consisting of homogeneous isotropic layers with plane parallel boundaries) — An unlimited elastic solid consists of homogeneous isotropic layers, separated by plane parallel boundaries. A point in the interior of one layer is a multiple source of elastic waves. The disturbance begins att=0. By aid of Laplace-transforms, «exponential coefficients» and «transmission factors» have been calculated according to the method shown byCagniard. By aid of the «transmission factors» we can determine the elastic displacements of the waves reflected and refracted at the boundaries.

     

Transient disturbance produced in an elastic half-space by impulsive shearing traction distributed over a circular region

The displacement produced in a semi-infinite, homogeneous, isotropic elastic medium by the application of shearing traction over a circular portion of the half-space has been evaluated in exact form by Cagniard's Technique (Cagniard, 1962;Gakenheimer andMiklowitz, 1969).     

On disturbances produced in an elastic medium by twists applied on the inner surface of a spherical cavity

Summary In this note, disturbances generated in an isotropic elastic medium by some simple types of twisting forces applied on the inner surface of a spherical cavity have been investigated. Prof.S. K. Chakraborty, Surendranath College, 24 Harrison Road,Calcutta (India).     

Disturbances of cylindrical origin in an isotropic elastic medium

Summary The paper deals with the investigations of disturbances in an infinite homogeneous isotropic and perfectly elastic medium produced by a source operating on a small portion of the inner surface of a cylindrical hole. Prof.S. K. Chakraborty, M. Sc., Surendranath College, 24 Harrison Road,Calcutta-9, India.     

On the stability of a non-convolutional perfectly matched layer for isotropic elastic media

The multi-axial perfectly matched layer (M-PML) is a material boundary condition for wave propagation problems in unbounded domains. It is obtained by extending the formulation of the split-field perfectly matched layer to a more general absorptive medium, for which damping profiles are specified along all dimensions of the problem. Under the hypothesis of small damping, it has been demonstrated that the stability of the system of partial differential equations of the M-PML can be related to the ratio of the damping profiles, and stable M-PML terminations for isotropic and orthotropic elastic media have been constructed. In the present work, we use the Routh–Horwitz determinants to demonstrate that the conclusions regarding the stability of M-PML for isotropic media for small damping are in fact valid for the more general case of damping coefficients of any (positive) value. The effectiveness of the M-PML is demonstrated by constructing stable terminations for the abovementioned media. The stability analysis is presented for 2-D in-plane (P-SV) wave propagation in elastic isotropic continua.     

Effect of gravity on steady-state response to moving loads

Summary The object of the paper is to investigate the effect of gravity on steady-state response to moving loads on the free-plane boundary of a semi-infinite isotropic elastic solid medium. The classical equations of motion for the propagation of elastic waves in a solid medium have been modified by Biot's theory of initial stress by supposing the force of gravity to create a type of initial stress of hydrostatic nature.     

Motion in a non-homogeneous elastic medium by a twisting impulsive force on the surface of a spherical cavity

Summary In this paper, the propagation of disturbance due to a twisting impulsive force on the surface of a spherical cavity, in a non-homogeneous isotropic medium, in which the rigidity and density vary inversely as the distance from the centre is investigated. Numerical solution is done by the method suggested byKromm.     

Propagation of Love type waves in a non-homogeneous layer lying over a vertically semi-infinite homogeneous isotropic medium

Summary The aim of this paper is to study the propagation of love-type waves in a homogeneous half-space overlain by a heterogeneous crust with various types of heterogenety. Frequency equations are obtained in each of the cases and numerical calculations are done in some cases and the results thus obtained are compared with those done byJeffreys andMitra [3]²).     

Propagation of a fracture in an infinite medium

Summary The displacement produced in an infinite homogeneous, isotropic elastic medium due to a steadily progressing crack has been obtained in a closed form by using Laplace transforms andCagniard's method. Numerical values of the displacement have been obtained and presented graphically.     

On the propagation of spherical waves due to large underground explosion

Summary When a high explosive pressure acts on the boundary of a spherical cavity in an elastic medium, the material in a region around the charge behaves plastically and thereafter an elastic wave is propagated. Not only that but the elastic plastic boundary moves also with a certain velocity which is less than the elastic wave velocitySabodash [3]²). In this paper, firstly the case when the elastic plastic boundary moves with a velocityV ₀. (V ₀<c) has been dealth with. The pressure in the elastic medium on this boundary has been assumed to be constant and equal toP ₀. Secondly the case when the elastic plastic boundary moves with an exponentially decaying velocity has been considered. The pressure in the elastic medium on this boundary has also been assumed to be of exponential character.     

Piezomagnetic monitoring in the South Pacific region

Data from an array of piezomagnetic stations in the South Pacific Islands indicate that noise limitations on piezomagnetic stress monitoring reported for California (Mueller andJohnston, 1981;Johnston et al., 1984) are probably pessimistic since they seem to include significant diurnal residual. Raw station differences could be significant at the 1.0 nT level out to distances of several hundred kilometres. Two large earthquakes have occurred within the network, and no coseismic anomalies were observed. However, changes in observed magnetic field for several stations within 200 km of the earthquakes indicate that stress propagation effects are observable prior to the events, and that these effects dominate the simple elastic effects previously investigated which accompany the stress drops for moderate to large earthquakes.     

On the propagation of finite Love Waves

Summary The problem of the propagation of finite Love Waves in a heterogeneous elastic half space lying over a homogeneous elastic half space, using the quasilinear stress-strain relation due toS. Ferhst [4] is considered in detail. The variations of the parameter in the layer assumed to be of the form 1= 0e ^z, 0e ^z where is a constant andz is distance measured from the surface into the layer.     

The Numerical Modeling of 3-D Elastic Wave Equation Using a High-Order,Staggered-Grid, Finite Difference Scheme

This article provides the application of the high-order, staggered-grid, finite-difference scheme to model elastic wave propagation in 3-D isotropic media. Here, we use second-order, temporal-and high-order spatial finite-difference formulations with a staggered grid for discretization of the 3-D elastic wave equations of motion. The set of absorbing boundary conditions based on paraxial approximations of 3-D elastic wave equations are applied to the numerical boundaries. The trial resuits for the salt model show that the numerical dispersion is decreased to a minimum extent, the accuracy high and diffracted waves abundant. It also shows that this method can be used for modeling wave propagation in complex media with the lateral variation of velocity.     

CodaQ as a combination of scattering and intrinsic attenuation: Numerical simulations with the boundary integral method

Numerical modelling ofSH wave seismograms in media whose material properties are prescribed by a random distribution of many perfectly elastic cavities and by intrinsic absorption of seismic energy (anelasticity) demonstrates that the main characteristics of the coda waves, namely amplitude decay and duration, are well described by singly scattered waves in anelastic media rather than by multiply scattered waves in either elastic or anelastic media. We use the Boundary Integral scheme developed byBenites et al. (1992) to compute the complete wave field and measure the values of the direct waveQ and coda wavesQ in a wide range of frequencies, determining the spatial decay of the direct wave log-amplitude relation and the temporal decay of the coda envelope, respectively. The effects of both intrinsic absorption and pure scattering on the overall attenuation can be quantified separately by computing theQ values for corresponding models with (anelastic) and without (elastic) absorption. For the models considered in this study, the values of codaQ ^–1 in anelastic media are in good agreement with the sum of the corresponding scatteringQ ^–1 and intrinsicQ ^–1 values, as established by the single-scattering model ofAki andChouet (1975). Also, for the same random model with intrinsic absorption it appears that the singly scattered waves propagate without significant loss of energy as compared with the multiply scattered waves, which are strongly affected by absorption, suggesting its dominant role in the attenuation of coda waves.     

Love waves in inhomogeneous and anisotropic earth — I

Summary The frequency equation is derived for the propagation of Love waves in the earth's crust, composed of transversely isotropic layers and overlying anisotropic and inhomogeneous mantle. The exact boundary value problem is solved for a single layer and extended to multilayered media by generalizing theHaskell's technique. In fact the problem of deriving the frequency equation has been reduced to finding out the solution of the equation of motion subject to the appropriate boundary conditions. To illustrate the method, the author has derived frequency equations of Love waves for linear, exponential and generalized power law variation of vertical shear wave velocity with depth in the half space overlain by transversely isotropic inhomogeneous stratum.     

流体饱和多孔隙介质弹性波方程边界元解法研究

基于流体饱和多孔隙各向同性介质模型,本文首先推导了流体饱和多孔隙介质中弹性波传播的频率域系统动力方程及边界积分方程,然后给出了流体饱和多孔隙介质弹性波方程的基本解,最后,利用本文给出的边界元方法对流体饱和多孔隙各向同性介质中的弹性波传播进行了数值模拟.结果表明：不论是从固相位移,还是液相位移的地震合成记录都能看到明显的慢速P波,本文提出的流体饱和多孔隙介质弹性波边界元法是有效可行的.     

双相各向异性介质中弹性波方程的有限元解法及波场模拟

基于双相各向异性介质模型,首先推导了双相各向异性介质中弹性波传播的动力学方程及其Galerkin变分方程和有限元运动方程,然后给出了孔隙弹性波方程的有限元数值解法以及二维双相PTL介质中波场模拟的人为吸收边界条件. 最后,利用本文给出的有限元方法对双相PTL介质和双相各向同性介质中的弹性波传播进行了数值模拟. 结果表明:有限元方法和吸收边界条件有效、可行,在理想相界条件下,不论是从固体位移,还是从流体位移的波场快照都能看到明显的慢速拟P波;在黏滞相界情况下,能否观察到慢速拟P波,与含流体地层介质的耗散性质有关.对实际含流体介质,从流体位移分量的波场快照比从固体位移波场快照更容易观察到慢速拟P波.