Quasi-Rayleigh Waves in Transversely Isotropic Half-Space with Inclined Axis of Symmetry

A method for determination of characteristics of quasi-Rayleigh (qR) wave in a transversely isotropic homogeneous half-space with inclined axis of symmetry is outlined. The solution is obtained as a superposition of qP, qSV and qSH waves, and surface wave velocity is determined from the boundary conditions at the free surface and at infinity, as in case of Rayleigh wave in an isotropic half-space. Though the theory is simple enough, a numerical procedure for calculation of surface wave velocity presents some difficulties. The difficulty is caused by necessity to calculate complex roots of a non-linear equation, which in turn contains functions determined as roots of non-linear equations with complex coefficients. Numerical analysis shows that roots of the equation corresponding to the boundary conditions do not exist in the whole domain of azimuths and inclinations of the symmetry axis. The domain of existence of qR wave depends on the ratio of the elastic parameters: for some strongly anisotropic models the wave cannot exist at all. For some angles of inclination qR-wave velocities deviate from those calculated on the basis of the perturbation method valid for weak anisotropy, though they have the same tendency of variation with azimuth. The phase of qR wave varies with depth unlike Rayleigh wave in an isotropic half-space. Unlike Rayleigh wave in an isotropic half-space, qR wave has three components - vertical, radial and transverse. Particle motion in horizontal plane is elliptic. Direction of the major axis of the ellipsis coincides with the direction of propagation only in azimuths 0° (180°) and 90° (270°).     

平面P波在饱和半空间中洞室周围的散射(II):数值结果

本文通过数值计算研究了入射平面P波在饱和半空间中洞室周围散射问题,分析了入射波频率和角度、边界渗透条件、孔隙率、泊松比等参数对散射的影响。研究表明,平面P波入射情况下,饱和半空间和单相(干土)半空间中洞室附近地表位移幅值的差别很大,干土情况的水平位移幅值相对较大,饱和情况的竖向地表位移幅值相对较大;由于波在洞室附近的干涉,饱和情况与干土情况的地表位移出现相位漂移。随着孔隙率的增大,洞室附近水平地表位移幅值逐渐减小,竖向地表位移幅值则逐渐增大;当孔隙率较低时,边界渗透条件对地表位移幅值的影响很小,而当孔隙率较大时,边界渗透条件的影响则不可忽视,不透水情况下,水平和竖向地表位移幅值的峰值均相对较大;随着入射频率的增加,孔隙率的影响逐渐增大,而且不透水情况下孔隙率的影响相对较大。随着泊松比的增大,洞室附近水平地表位移幅值逐渐降低,竖向地表位移幅值则逐渐增大;泊松比较小时,边界渗透条件对位移幅值的影响较大,泊松比较大时,边界渗透条件对位移幅值的影响则较小;随着入射频率的增加,泊松比的影响逐渐增大。当孔隙率较小时,半空间地表和洞室表面孔隙水压幅值较小,但空间变化比较剧烈,随着孔隙率的增大,孔隙水压逐渐增大但空间变化逐渐平缓;随着入射频率的增加,孔隙水压幅值逐渐增大,且孔隙水压的空间变化逐渐变得复杂。     

Spatial dispersion of surface waves at the free surface of a general anisotropic elastic medium

A new technique relates the wave velocity of the surface waves in anisotropic elastic medium to its elastic constants. Anisotropic propagation of surface waves is studied in a half-space occupied by a general anisotropic elastic solid. The phase velocity expressions of quasi-waves, in three-dimensional space, are used to derive the secular equation of surface waves. The complex secular equation is resolved, analytically, into real and imaginary parts and is then solved, numerically, for phase velocity along a given phase direction on the surface. The complete procedure is thus analogous to the one used for conventional Rayleigh waves in isotropic medium. A non-linear equation relates the ray direction of the surface waves to its phase direction on the (plane) surface of the medium. The analytical differentiation of secular equation yields the directional derivative of phase velocity. This derivative is used to calculate the wave velocity of surface waves. Spatial variations of phase velocity, wave velocity and ray direction over the free plane surface are plotted for the numerical models of crustal rocks with orthorhombic, monoclinic and triclinic anisotropies.     

Propagation characteristics of blast-induced shock waves in a jointed rock mass

The propagation characteristics of blast-induced shock waves in a jointed rock mass have been monitored and studied. Accelerometers were set up on a rock surface along three lines, at 0°, 45° and 90° with respect to the orientation of the predominant joint strikes. Cylindrical charges were detonated in a charge hole, and ground accelerations in both vertical and radial directions at various points on the rock surface were recorded. Results show that rock joints have significant effects on the propagation characteristics of blast-induced shock waves. The amplitude and principal frequency of shock waves attenuate with the increase of distance from the charge centre, and the increase of incident angle between the joint strike and the wave propagation path. The measured data were compared with the empirical equations of shock wave attenuation proposed by other authors. The mechanism of rock joint effect, the attenuation of shock waves in relation to the propagation distance, the charge weight and the incident angle, are discussed in this paper.     

A note on the approximate determination of the dispersion of Love waves in a laterally nonhomogeneous layer lying over a homogeneous half-space

Rayleigh's principle and the concept of the local wave number have been utilised for the approximate determination of the dispersion of Love waves propagating in a laterally heterogeneous layer lying over a homogeneous half-space. The shear wave velocity and the rigidity in the surface layer have been assumed to decrease with the increase of the lateral distance from the origin. The range of validity of the dispersion equation obtained by this method has been examined critically. It was found that: (a) for existence of Love waves the minimum value of shear wave velocity in the layer must be less than that in the matter below, and (b) the phase velocity of Love waves decreases with the increase of the lateral distance from the origin.     

A new series solution method for two-dimensional elastic wave scattering along a canyon in half-space

This paper presents a semi-analytical method for studying the two-dimensional problem of elastic wave scattering by surface irregularities in a half-space. The new method makes use of the member of a c-completeness family of wave functions to construct the scattering fields, and then applies equal but opposite tractions to those of the foregoing constructed scattering fields on the horizontal surface of the half-space to produce additional scattering fields. These additional scattering fields are a series of Lamb's solutions. Thus the whole scattering field constructed in the series automatically satisfies the Navier equations, the condition of zero traction on the half-space surface, and the radiation boundary conditions at infinity. Using the traction-free conditions along the canyon surface, the coefficients of the series solutions are determined via a least-squares method. For incident P, SV, and Rayleigh waves, the numerical results are presented for the scattering displacements in the vicinity of a semi-circular canyon in the half-space.     

Two-dimensional scattering of SH waves in a soil layer underlain with a sloping bedrock

A two-dimensional analysis is applied to examine the effect that a sloping bedrock half-space has on the amplification of an anti-plane shear wave. The direct boundary integral equation method is used for the two-dimensional analysis. The particular soil–rock configuration investigated includes a homogeneous soil layer underlain by a sloping rock half-space. The rock half-space dips for a horizontal distance L and then becomes horizontal so that the overlying soil layer has a thickness H that remains constant from this point to infinity. The materials in the soil–rock configuration are considered viscoelastic except in the rock half-space below soil layer thickness H, which is considered elastic. This limitation in damping is due to the correction used for the truncation of the half-space boundary. Four cases are used to study the relationship between rock slope and surface displacement, vertical, 1:2, 1:4, 1:8. Surface displacements are determined for each of these cases for half-space incidence angles of 90, 75, and 60°. To allow for applicability to a wide range of problems, results are determined as a function of dimensionless parameters. In addition, solutions from a one-dimensional analysis are compared with the results of the two-dimensional analysis to establish limits outside of which a one-dimensional analysis suffices.     

Effects of random variations of soil properties on site amplification of seismic ground motions

This paper investigates the effects of random variations of soil properties on site amplification of seismic waves. First, based on attenuation laws and the filtered Tajimi–Kanai spectrum, seismic motion at the base rock of a soil site is stochastically generated according to an assumed earthquake with a given magnitude and epicentral distance. Motions on the surface of this layered random soil site are calculated by nonlinear wave propagation methods, and by assuming the incoming seismic wave consisting of SH wave or combined P and SV waves. Soil properties, including shear modulus, damping ratio and mass density, as well as ground water level are considered as random in the numerical calculation. The Rosenblueth method is used to solve the random dynamic responses of the soil site. Parametric calculations are performed to investigate the effects of various parameters on site amplification of seismic waves. The mean and maximum ground motions on surface of the site are estimated. Numerical results indicate that the estimated surface motions differ substantially if the random variations of soil properties and soil saturation level are taken into consideration in the analysis.     

地基半空间等效集总体系的比拟法与实测分析

先略述莱斯默比拟法的形成;再由半空间理论等效为质弹体系,得出辐射阻尼比、刚度及参振土质量,并论述两体系的结合;最后经实测、分析和使用,考虑土体非匀质性折减阻尼比以作修正,使其更为实用。这有助于消除在我国长期认为阻尼比大而不安全、不便使用的疑虑,以便推动半空间理论在我国的实用化。     

Two-dimensional scattering of in-plane body waves due to a discontinuity in bedrock

The direct boundary integral equation technique is used to study in-plane surface amplification of in-plane seismic body waves for the case of an inhomogeneity in a bedrock half-space. In the studied soil configuration, a soil layer rests on a rock half-space which includes a rock inclusion. The rock inclusion considered is a semi-infinite horizontal rock layer in which its upper boundary borders the soil layer. Materials in the soil–rock configuration are considered viscoelastic except for the section of the rock half-space below the level of the rock inclusion which is considered elastic. A parametric study is performed to determine controlling factors for surface displacement due to in-plane body waves. The study investigates varying the stiffness and the thickness of the rock inclusion for a range of frequencies and wave incidence angles. Anti-plane waves for this type of soil-rock configuration have been addressed in a previous article by Heymsfield (Earthquake Engng. Struct. Dyn. 28 : 841–855 (1999)). Copyright © 1999 John Wiley & Sons, Ltd.     

弹性层状半空间中凸起地形对入射平面SH波的放大作用

对Wolf理论进行拓展,使之可解决凸起地形对波的散射问题,进而利用间接边界元法,求解了弹性层状半空间中凸起地形对入射平面SH波的放大作用问题。本文模型的显著特点之一是考虑了层状半空间的动力特性以及层状半空间和凸起地形的阻尼;特点之二是计算精度高。文中以基岩上单一土层中半圆凸起地形对入射平面SH波的放大作用为例进行了数值计算分析。研究表明,基岩上单一土层中凸起地形对入射平面SH波放大作用和均匀半空间中凸起地形有着本质的差别;土层动力特性不仅影响凸起地形地表位移的幅值,还会影响地表位移的频谱;阻尼会显著降低凸起地形对高频波的放大作用。     

Torsional response of a rigid circular foundation on a saturated half-space to SH waves

An analytical approach is used to study the torsional vibrations of a rigid circular foundation resting on saturated soil to obliquely incident SH waves. Biot’s poroelastic dynamic theory is considered to characterize the saturated soil below the foundation, which is solved by Hankel transform later. In order to consider the scattering phenomena caused by the existence of the foundation, the total wave field in soil is classified into free-field, rigid-body scattering field and radiation scattering field. According to the classification of wave field and the mixed boundary-value conditions between the soil and the foundation, torsional vibrations of the foundation are formulated in two sets of dual integral equations. Then, the dual integral equations are reduced to Fredholm integral equation of the second kind to be solved. Combining with the dynamic equilibrium equations of the foundation, the expressions for the torsional vibrations of the foundation are obtained. Numerical results are presented to demonstrate the influence of excitation frequency, incident angle, the torsional inertia moment of the foundation and permeability of the saturated half-space on the torsional vibrations of the foundation.     

Long waves due to a symmetrical wind stress on the surface of a rotating ocean

Abstract

The problem of unsteady long waves generated by any horizontal and symmetrically distributed, time-periodic surface wind on a rotating ocean is analysed for large times and distances. Uniform asymptotic estimates of the surface displacement in the unsteady state are obtained. The steady-state wave and velocity fields at any distance are also determined. Some characteristics of the unsteady and steady motions are described. Also noted are the features that distinguish the motion from its one-dimensional analogue for which a non-uniform analysis in the unsteady state along with a large-distance form of the surface elevation are already known.     

PKPdif波-内核低速层存在的地震学证据

通过对中国数字地震仪台网(CDSN)11个台1993—2012年间30个极远震记录的分析,识别出了PKP波入射到内外核界面(ICB)上的衍射波PKPdif波。PKPdif波在震中距120°左右时出现在PKIKP波之前,在震中距150°以后出现在PKIKP波之后,是一个长周期波。为了解释PKPdif波超前于PKIKP波的原因,设想在距地面约5 156.1～5 372.2 km的深度,即内外核界面下有可能存在一个PKIKP波的低速层(厚度约216.1 km)。该设想符合Jeffreys速度模型,给出的PKPdif波在震中距119.4°～176.1°间的走时表填补了目前使用的《IASPEI1991地震波走时表》的空白。讨论PKPdif波的运动学特征有助于对内外核界面物理性质的认识,有助于提高分析震相的水平和积累震相分析经验。     

Dynamic response of a group of flexible foundations to incident seismic waves

The dynamic response of a finite number of flexible surface foundations subjected to harmonic incident Rayleigh or SH waves is presented. The foundations are assumed to be resting on an elastic half-space. The results show that the foundation stiffness has a marked effect on the vertical response, while there is only a minor effect on the horizontal displacements. In general, the dynamic response decreases with increasing foundation stiffness. In cases of Rayleigh wave incidence, the existence of an adjacent foundation generates a certain amount of horizontal response in the direction perpendicular to the incident wave and subsequently causes the system to undergo a torsional motion; while in cases of horizontally incident SH waves, a vertical response has been observed and its magnitude is comparable to the response in the direction of the incident wave.     

Simultaneous measurement and inversion of surface wave dispersion and attenuation curves

Surface wave tests are non-invasive seismic techniques that have traditionally been used to determine the shear wave velocity (i.e. shear modulus) profile of soil deposits and pavement systems. Recently, Rix et al. [J. Geotech. Geoenviron. Engng 126 (2000) 472] developed a procedure to obtain near-surface values of material damping ratio from measurements of the spatial attenuation of Rayleigh waves. To date, however, the shear wave velocity and shear damping ratio profiles have been determined separately. This practice neglects the coupling between surface wave phase velocity and attenuation that arises from material dispersion in dissipative media. This paper presents a procedure to measure and invert surface wave dispersion and attenuation data simultaneously and, thus, account for the close coupling between the two quantities. The methodology also introduces consistency between phase velocity and attenuation measurements by using the same experimental configuration for both. The new approach has been applied at a site in Memphis, TN and the results obtained are compared with independent measurements.     

sPL,一个近距离确定震源深度的震相

实际地震波形观测表明,对于大陆结构相对简单的地壳中的地震而言,有一震相出现在P 波和S波之间.一般在30~50 km附近发育得较好,其能量主要集中在径向分量,而垂向分量的振幅相对径向要小,切向分量上的振幅很弱,且波形以低频为主,通常没有P波尖锐.在利用FK方法计算合成地震图的基础上,发现该震相是由S波入射到自由地表形成水平传播的P波(文献称为surface P wave,自由地表P波)或者包括S波入射到地表后形成的多次P波或其散射震相.由于该震相是由S波和P波之间耦合而形成,本文将其定义为sPL(s coupled into P) 震相.理论波形研究表明,sPL相对直达P波的到时差对震中距离不敏感,而随着震源深度的增加几乎呈线性增加,因此可以很好的约束震源深度.本文以2005年江西九江地震为例,证实了sPL确定震源深度的可行性和可靠性.在观测到sPL震相的情况下,离震源50 km以内的一个三分量地震台站的波形就可以帮助获得可靠的震源深度,而不需要精确的震中距离.由于sPL震相出现距离较近,对于较小(三级以上)的地震也可以应用,因此在稀疏台网布局情形下sPL对于确定中小地震深度应该具有很好的应用意义.     

Scattered Surface Wave Energy in the Seismic Coda

One of the many important contributions that Aki has made to seismology pertains to the origin of coda waves (Aki, 1969; Aki and Chouet, 1975). In this paper, I revisit Aki's original idea of the role of scattered surface waves in the seismic coda. Based on the radiative transfer theory, I developed a new set of scattered wave energy equations by including scattered surface waves and body wave to surface wave scattering conversions. The work is an extended study of Zeng et al. (1991), Zeng (1993) and Sato (1994a) on multiple isotropic-scattering, and may shed new insight into the seismic coda wave interpretation. The scattering equations are solved numerically by first discretizing the model at regular grids and then solving the linear integral equations iteratively. The results show that scattered wave energy can be well approximated by body-wave to body wave scattering at earlier arrival times and short distances. At long distances from the source, scattered surface waves dominate scattered body waves at surface stations. Since surface waves are 2-D propagating waves, their scattered energies should in theory follow a common decay curve. The observed common decay trends on seismic coda of local earthquake recordings particular at long lapse times suggest that perhaps later seismic codas are dominated by scattered surface waves. When efficient body wave to surface wave conversion mechanisms are present in the shallow crustal layers, such as soft sediment layers, the scattered surface waves dominate the seismic coda at even early arrival times for shallow sources and at later arrival times for deeper events.     

Statistics of gravity waves in the lower stratosphere over Beijing based on high vertical resolution radiosonde

Atmospheric gravity waves, with small to medium scales, prevail in the atmosphere and have global ef- fects. Many researches show that gravity waves are the main source that causes the variation of wind and temperature field in the stratosphere, and that the break-up of upward propagating gravity waves is the dominant sources of small scale turbulent and mixing processes in the middle atmosphere. Theories and ob- servations indicate that the redistribution of momen- tum, caused by the generati…     

A NUMERICAL STUDY OF LAMB'S PROBLEM*

This paper considers propagation of elastodynamic waves in an imperfectly elastic half-space. Two different excitation modes are investigated: a buried source of compressional waves and a vertically directed areal load applied to the surface. Numerical integration of the analytical solution of the wave equation allows study of the vertical and horizontal components of displacement and/or particle velocity anywhere in the half-space. One case of particular interest concerns the examination of particle displacement and velocity at the surface in a circular area above the source. In another application seismograms generated by an explosive buried source are contrasted with seismograms generated by the transient application of a vertically directed load to the free surface. Still another application of considerable practical interest concerns the study of the nongeometrical pS—wave, in particular its characteristics as functions of range and depth. Finally, in the last application the behavior of a rarely observed wave (denoted here by the letter U) is studied in both elastic and visco-elastic half-spaces.