Rotational cone models for a soil layer on flexible rock half-space

The unfolded cone model used for calculating the dynamic response of a disk on the surface of a soil layer resting on flexible rock for translational motion is extended to rotational motion. The method is analogous to that for a layer on rigid rock, the only modification being that the reflection coefficient – α replaces the coefficient of total reflection – 1. The modified value of – α, which, in general, is frequency-dependent, is determined by considering one-dimensional wave propagation along the cone for the first impingement at the layer–rock interface. The low- and high-frequency limits of – α for the rotational motion are the same as for translational motion. As these limits do not depend on frequency, the dynamic analysis using cones can be performed in the familiar time domain. The transfer function constructed by addressing the reflections–refractions at the soil–rock interface and the reflections at the free surface in the unfolded cone model is highly accurate, resulting in the same accuracy of the dynamic response of a disk on a layer resting on flexible rock as that on a homogeneous half-space modelled with a cone.     

Cone models for nearly incompressible soil

Cones can be used to model soil in a unified strength-of-materials approach. For the vertical and rocking motions involving predominantly compressional-extensional deformation, the corresponding dilatational wave velocity tends to infinity for Poisson's ratio approaching 1/2. Based on the rigorous solution for the dynamic stiffness of a rigid disk for all frequencies, whereby the partition of the power among P-, S- and Rayleigh waves is also discussed, two special features are necessary for the vertical and rocking motions for nearly incompressible soil with Poisson's ratio between 1/3 and 1/2: (1) The appropriate wave velocity is selected as twice the shear wave velocity and not as the dilatational wave velocity; (2) A trapped mass which increases linearly with Poisson's ratio is introduced. The trapped mass can be assigned to the base mat, allowing the cone model to be constructed in the same way for all Poisson's ratios. The realization of cone models for surface foundations on a homogeneous half-space and on a layer on a flexible half-space and for embedded and pile foundations is addressed.     

Dynamic stiffness of foundation embedded in layered halfspace based on wave propagation in cones

An Erratum has been published for this article in Earthquake Engineering & Structural Dynamics 33(6) 2004, 793. The dynamic stiffness of a foundation embedded in a multiple‐layered halfspace is calculated postulating one‐dimensional wave propagation in cone segments. In this strength‐of‐materials approach the sectional property of the cone segment increases in the direction of wave propagation. Reflections and refractions with waves propagating in corresponding cone segments occur at layer interfaces. Compared to rigorous procedures the novel method based on cone segments is easy to apply, provides conceptual clarity and physical insight in the wave propagation mechanisms. This method postulating one‐dimensional wave propagation in cone segments with reflections and refractions at layer interfaces is evaluated, calculating the dynamic stiffness of a foundation embedded in a multiple‐layered halfspace. For sites resting on a flexible halfspace and fixed at the base, engineering accuracy (deviation of ±20%) is achieved for all degrees of freedom with a vast parameter variation. The behaviour below the cut‐off frequency in an undamped site fixed at its base is also reliably predicted. The accuracy is, in general, better than for the method based on cone frustums, which can lead to negative damping. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

来自海底高速层径向波的理论地震图研究

本文利用各向异性反射率技术计算理论地震图,提出海底高速薄层会产生沿高速层水平传播的波（简称径向波）,这种波在水层中作为P波,在固液界面激发下行横波,该均匀横波以临界角入射高速薄层,在层内作为超临界角的非均匀横波水平传播,再以临界角转换为上行传播的均匀横波,最终在固液界面上行透射转换为水层中P波.高速薄层传播的径向波不同于界面折射波,也不同于具有频散的面波和通道波.理论地震图的研究表明,径向波具有线性时距,能与海底强反射具有同等振幅水平;径向波有其振幅、时距位置和斜率这些观测记录参数,分别对应高速层的厚度、深度和近似的横波速度;径向波可以克服折射波解释中遇到的振幅强弱和高速层速度等困难.径向波可作为探测海底高速薄层的有力工具,对于研究高速层屏蔽、海底反射类型的多样性和相应的资料处理解释有重要意义.     

Scattering of internal waves in a two-layer fluid flowing through a channel with small undulations

Using two-dimensional linear water wave theory, we consider the problem of normal water wave (internal wave) propagation over small undulations in a channel flow consisting of a two-layer fluid in which the upper layer is bounded by a fixed wall, an approximation to the free surface, and the lower one is bounded by a bottom surface that has small undulations. The effects of surface tension at the surface of separation is neglected. Assuming irrotational motion, a perturbation analysis is employed to calculate the first-order corrections to the velocity potentials in the two-layer fluid by using Green’s integral theorem in a suitable manner and the reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom undulation. Two special forms of the shape function are considered for which explicit expressions for reflection and transmission coefficients are evaluated. For the specific case of a patch of sinusoidal ripples having the same wave number throughout, the reflection coefficient up to the first order is an oscillatory function in the quotient of twice the interface wave number and the ripple wave number. When this quotient approaches one, the theory predicts a resonant interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of the incident wave energy occurs if this number is large. Again, when a patch of sinusoidal ripples having two different wave numbers for two consecutive stretches is considered, the interaction between the bed and the interface near resonance attains in the neighborhood of two (singular) points along the x-axis (when the ripple wave number of the bottom undulation become approximately twice as large as the interface wave number). The theoretical observations are presented in graphical form.     

Reflection and Transmission Coefficients for an Incident Plane Shear Wave at an Interface Separating Two Dissimilar Poroelastic Solids

Using Biot’s poroelasticity theory, we derive expressions for the reflection and transmission coefficients for a plane shear wave incident on an interface separating two different poroelastic solids. The coefficients are formulated as a function of the wave incidence angle, frequency and rock properties. Specific cases calculated include the boundary between water-saturated sand and water-saturated sandstone and the gas–water interface in sand. The results show a very different interface response to that of an incident P wave. Plane SV wave incidence does not significantly excite the Biot slow P wave if the frequency of the wave is below the transition frequency. Above this frequency, an incident plane SV wave can generate a mode-converted slow Biot P wave which is actually a normal propagating wave and not highly attenuating as in the usual (diffusive) case. For an incident SV wave onto a gas–water interface, even at very high frequency, there is no significant Biot second P wave produced. For small incident angles, the gas–water interface is essentially transparent. With increasing angles, there can arise an unusual "definitive angle" in the reflection/transmission coefficient curves which is related to the change of fluid viscosity on both sides of the interface and provides a possible new means for underground fluid assessment.     

Seismic modelling study of a subglacial lake

We characterize the seismic response of Lake Vostok, an Antarctic subglacial lake located at nearly 4 km depth below the ice sheet. This study is relevant for the determination of the location and morphology of subglacial lakes. The characterization requires the design of a methodology based on rock physics and numerical modelling of wave propagation. The methodology involves rock-physics models of the shallow layer (firn), the ice sheet and the lake sediments, numerical simulation of synthetic seismograms, ray tracing, τ–p transforms, and AVA analysis, based on the theoretical reflection coefficients. The modelled reflection seismograms show a set of straight events (refractions through the firn and top-ice layer) and the two reflection events associated with the top and bottom of the lake. Theoretical AVA analysis of these reflections indicates that, at near offsets, the PP-wave anomaly is negative for the ice/water interface and constant for the water/sediment interface. This behaviour is shown by AVA analysis of the synthetic data set. This study shows that subglacial lakes can be identified by using seismic methods. Moreover, the methodology provides a tool for designing suitable seismic surveys.     

含黏性流体孔隙介质分界面上反透射系数特征研究

地下岩石由岩石骨架和孔隙流体组成,通常流体含黏性.地震波在地下介质中传播时受岩石骨架和黏性流体的影响会呈现出复杂的变化.本文将流、固体位移和应力连续作为边界条件,推导出含黏性流体孔隙介质分界面上反透射系数方程;通过建立上层为饱油、下层为饱盐水的砂岩孔隙介质模型,开展反透射系数特征研究,分别分析不同频率、不同黏滞系数条件下,含黏性流体孔隙介质分界面上反透射系数随入射角的变化.研究表明,孔隙介质分界面上和等效介质分界面上的反透射系数分别随入射角的变化趋势基本一致,说明方程推导和数值计算的正确性;快纵波反透射系数受频率、流体黏性的影响较小,而快横波反透射系数在一定入射角范围内受频率、流体黏性的影响比较大;由于黏性孔隙流体的作用,慢纵波和慢横波的反透射系数受入射角、频率及流体黏性的影响都很大.     

Seismic modelling for geological fractures

Based on knowledge of a commutative group calculation of the rock stiffness and on some geophysical assumptions, the simplest fractured medium may be regarded as a fracture embedded in an isotropic background medium, and the fracture interface can be simulated as a linear slip interface that satisfies non‐welded contact boundary conditions: the kinematic displacements are discontinuous across the interface, whereas the dynamic stresses are continuous across the interface. The finite‐difference method with boundary conditions explicitly imposed is advantageous for modelling wave propagation in fractured discontinuous media that are described by the elastic equation of motion and non‐welded contact boundary conditions. In this paper, finite‐difference schemes for horizontally, vertically, and orthogonally fractured media are derived when the fracture interfaces are aligned with the boundaries of the finite‐difference grid. The new finite‐difference schemes explicitly have an additional part that is different from the conventional second‐order finite‐difference scheme and that directly describes the contributions of the fracture to the wave equation of motion in the fractured medium. The numerical seismograms presented, to first order, show that the new finite‐difference scheme is accurate and stable and agrees well with the results of previously published finite‐difference schemes (the Coates and Schoenberg method). The results of the new finite‐difference schemes show how the amplitude of the reflection produced by the fracture varies with the fracture compliances. Later, comparisons with the reflection coefficients indicate that the reflection coefficients of the fracture are frequency dependent, whereas the reflection coefficients of the impedance contrast interface are frequency independent. In addition, the numerical seismograms show that the reflections of the fractured medium are equal to the reflections of the background medium plus the reflections of the fracture in the elastic fractured medium.     

三维层状黏弹性半空间中球面SH、P和SV波源自由场

采用刚度矩阵方法结合Hankel积分变换,求解了层状黏弹性半空间中球面SH、P和SV波的自由波场.首先,在柱坐标系下建立层状黏弹性半空间的反轴对称(柱面SH波)和轴对称(柱面P-SV波)情况精确动力刚度矩阵.进而由Hankel变换将空间域内的球面波展开为波数域内柱面波的叠加,然后将球面波源所在层的上下端面固定,求得固定层内的动力响应和固定端面反力,将固端反力反向施加到层状黏弹性半空间上,采用直接刚度法求得固端反力的动力响应,叠加固定层内和固端反力动力响应,求得波数域内球面波源动力响应.最后由Hankel积分逆变换求得频率-空间域内球面波源自由场,时域结果由傅里叶逆变换求得.文中验证了方法的正确性,并以均匀半空间和基岩上单一土层中球面SH、P和SV波为例分别在频域和时域内进行了数值计算分析.研究表明基岩上单一土层中球面波自由场与均匀半空间情况有着本质差异;基岩上单一土层中球面波位移频谱峰值频率与场地固有频率相对应,基岩面的存在使得基岩上单一土层地表点的位移时程非常复杂,振动持续时间明显增长;阻尼的增大显著降低了动力响应的峰值,同时也显著减少了波在土层的往复次数.     

Propagation of oblique waves over small bottom undulation in an ice-covered two-layer fluid

Scattering of oblique incident waves by small bottom undulation in a two-layer fluid, where the upper layer has a thin ice-cover while the lower one has the undulation, is investigated within the framework of linearized water wave theory. The ice-cover is being modeled as an elastic plate of very small thickness. There exist two modes of time-harmonic waves–one with lower wave number propagating along the ice-cover (ice-cover mode) and the other with higher wave number along the interface (interfacial mode). A perturbation analysis is employed to solve the corresponding boundary value problem governed by modified Helmholtz equation and thereby evaluating the reflection and transmission coefficients approximately up to first order for both modes. A patch of sinusoidal ripples, having two different wave numbers over two consecutive stretches, is considered as an example and the related coefficients are determined. It is observed that when the wave is incident on the ice-cover surface we always find energy transfer to the interface, but for interfacial incident waves there are parameter ranges for which no energy transfer to the ice-cover surface is possible. Also it is observed that for small angles of incidence, the reflected energy is more as compared to the other angles of incidence. These results are demonstrated in graphical form. From the derived results, the solutions for problems with free surface can be obtained as particular cases.     

Seismic reflection and transmission coefficients at an air-water interface of saturated porous soil

Based on the modified Biot＇s theory of two-phase porous media, a study was presented on seismic reflection and transmission coefficients at an air-water interface of saturated porous soil media. The major differences between air-saturated soils and water-saturated soils were theoretically discussed, and the theoretical formulas of reflection and transmission coefficients at an air-water interface were derived. The characteristics of propagation and attenuation of elastic waves in air-saturated soils were given and the relations among the frequency, the angle of incidence and the reflection, transmission coefficients were analyzed by using numerical methods. Numerical results show that the propagation characteristic of the wave in air-saturated soils is great different from that in water-saturated soils. The frequency and the angle of incidence can have great influences on the reflection and transmission coefficients at interface. Some new cognition about the wave propagation is obtained and the study suggests that we may carefully pay attention to the influence of air on the dynamic analysis of seismic wave.     

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

The direct boundary integral equation technique is applied to determine the impact on surface amplification caused by an inhomogeneity in a bedrock half-space. The particular soil-rock configuration studied is one in which a soil layer rests on a rock half-space which includes a rock inclusion. The particular rock inclusion considered for this study is a semi-infinite rock layer with its upper boundary bordering the soil layer. Materials 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 a vertically incident shear wave. The study includes varying the stiffness and the thickness of the inclusion for a range of frequencies. Solutions from a one-dimensional analysis are compared with the results of a two-dimensional analysis in order to establish limits inside of which a two-dimensional analysis is required. Copyright © 1999 John Wiley & Sons, Ltd.     

A coupled fluid layer–rigid disk–poroelastic half-space vibration problem

This paper proposes a coupled fluid layer–foundation–poroelastic half-space vibration model to study how still water affects foundations operating underwater. As an example, we consider the problem of the vertical vibration of a rigid disk on a poroelastic half-space covered by a fluid layer having a finite depth. The solution of the disk vibration problem is obtained using the boundary conditions at the free surface of the fluid layer and the boundary conditions at the fluid layer–poroelastic medium interface. The solution is expressed in terms of dual integral equations that are converted into Fredholm integral equations of the second kind and solved numerically. Selected numerical results for the vertical dynamic impedance coefficient are examined based on different water depths, poroelastic materials, disk permeabilities and frequencies of excitation. Based on the numerical results, it is proposed that the hydrodynamic pressure caused by the foundation vibration is the intrinsic reason that the existence of a fluid layer has such a great effect on the dynamic characteristics of the foundation. In many cases, the hydrodynamic pressure caused by the foundation vibration cannot be ignored when designing dynamic underwater foundations. These results are helpful in understanding the dynamic response of foundations under still water without water waves, such as foundations in pools, lakes and reservoirs.     

Propagation properties of guided wave in the anchorage structure of rock bolts

We investigated the properties of guided wave propagating in grouted rock bolts and the formation of the interface wave through theoretical analysis along with experimental and numerical simulations. Experimental and numerical simulations reveal that the wave propagating in anchorage structure is related to boundary conditions within the range of excitation wave frequencies. Waves with different frequencies have different propagation velocities and attenuation characteristics. The optimal excitation wave occurs in grouted rock bolts with minimized attenuation and maximized propagation distance, and the end reflection of grouted rock bolts can be observed clearly. Longitudinal wave propagating in rock bolts is very sensitive to anchorage strength. With the increase of anchorage strength, longitudinal wave gradually attenuates and eventually disappears. Subsequently, interface wave appears and the velocity of wave propagating in the grouted part becomes that of the interface wave. Based on these studies, ultrasonic guided wave was used to study the end reflection of embedded rock bolts with different anchorage strengths and bonding lengths. The relationships among anchorage strength, bonding length and attenuation coefficient K, as well as the means to inspect the bonding quality of the embedded rock bolts were also evaluated.     

Response of unbounded soil in scaled boundary finite‐element method

The scaled boundary finite‐element method is a powerful semi‐analytical computational procedure to calculate the dynamic stiffness of the unbounded soil at the structure–soil interface. This permits the analysis of dynamic soil–structure interaction using the substructure method. The response in the neighbouring soil can also be determined analytically. The method is extended to calculate numerically the response throughout the unbounded soil including the far field. The three‐dimensional vector‐wave equation of elasto‐dynamics is addressed. The radiation condition at infinity is satisfied exactly. By solving an eigenvalue problem, the high‐frequency limit of the dynamic stiffness is constructed to be positive definite. However, a direct determination using impedances is also possible. Solving two first‐order ordinary differential equations numerically permits the radiation condition and the boundary condition of the structure–soil interface to be satisfied sequentially, leading to the displacements in the unbounded soil. A generalization to viscoelastic material using the correspondence principle is straightforward. Alternatively, the displacements can also be calculated analytically in the far field. Good agreement of displacements along the free surface and below a prism foundation embedded in a half‐space with the results of the boundary‐element method is observed. Copyright © 2001 John Wiley & Sons, Ltd.     

Effect of reservoir bottom on earthquake response of concrete dams

The absorption of hydrodynamic pressure waves at the reservoir bottom has dominant effects on the structural response of the dam when subjected to ground motion. In the present study, a model is proposed for the absorption effects of the reservoir bottom in the earthquake analysis of dams. The model utilizes the wave reflection coefficient approach and is based on the solution of the wave equation in a sediment layer of viscoelastic material with a constant thickness overlying an elastic, semi-infinite foundation. Numerical studies were conducted to evaluate the effect of the sediment layer thickness and material properties as well as the effect of reflection of waves from the underlying rock. It is shown that the current approach of assuming the wave reflection coefficient at the reservoir bottom based on the characteristics of the sediment material and excluding the effect of the reflected waves from the underlying rock, may significantly underestimate the seismic response of the dam.     

基于粘性-滑移界面接触模型半空间隧道衬砌对平面P波的散射

基于粘性-滑移模型模拟隧道衬砌和围岩之间的接触状态,采用间接边界积分方程法求解弹性半空间中隧道衬砌对平面P波的散射,着重考察了地表位移放大和衬砌动应力集中效应。结果表明:接触滑移边界的刚度系数和粘度系数对隧道衬砌的动应力分布影响较大,滑移刚度较小时,共振频率段动应力集中效果更加显著;随着粘度系数的增大,衬砌内部环向应力幅值逐渐减小。另外,界面刚度系数和粘度系数对隧道衬砌动力反应的影响程度也受控于入射频率和角度。