Propagation and attenuation of Rayleigh waves in a semi-infinite unsaturated poroelastic medium

An analytical model for describing the propagation and attenuation of Rayleigh waves along the free surface of an elastic porous medium containing two immiscible, viscous, compressible fluids is developed in the present study based on the poroelastic equations formulated by Lo et al. [Lo WC, Sposito G, Majer E. Wave propagation through elastic porous media containing two immiscible fluids. Water Resour Res 2005;41:W02025]. The dispersion equation obtained is complex-valued due to viscous dissipation resulting from the relative motion of the solid to the pore fluids. As an excitation frequency is stipulated, the dispersion equation that is a cubic polynomial is numerically solved to determine the phase speed and attenuation coefficient of Rayleigh waves in Columbia fine sandy loam permeated by an air–water mixture. Our numerical results show that, corresponding to three dilatational waves, there is also the existence of three different modes of Rayleigh wave in an unsaturated porous medium, which are designated as the R1, R2, and R3 waves in descending order of phase speed, respectively. The phase speed of the R1 wave is non-dispersive (frequency-independent) in the frequency range we examined (10 Hz–10 kHz) and decreases as water saturation increases, whose magnitude ranges from 20% to 49% of that of the first dilatational wave with respect to water content. However, it is revealed numerically that the R2 and R3 waves are functions of excitation frequency. Given the same water saturation and excitation frequency, the phase speeds of the R2 and R3 waves are found to be approximately 90% of those of the second and third dilatational waves, respectively. The R1 wave has the lowest attenuation coefficient whereas the R3 wave attenuates highest.     

Propagation of Rayleigh waves on visco-elastic cylindrical surfaces placed in a magnetic field

Summary This paper studies axial Rayleigh waves in visco-elastic cylinder surrounded by vacuum and for body of same material with a cylindrical cavity. Magneto-elastic equations of motion for wave propagation in the radial and axial direction have been solved to obtain frequency equations.     

Propagation of Rayleigh waves in soils

Summary Propagation of Rayleigh type waves in soils is considered in this paper. It is a well known fact that soils do not behave like an ordinary isotropic elastic medium where the ratio of Young's modulus to the modulus of rigidity is much less than that in sandy soils. Considering the velocity of Rayleigh type wave as less than that of distortional wave (which is an observed fact) a probable value of this ratio is determined, and also assuming the value of this ratio based on some experimental data, the velocity of wave propagation in the medium is deduced.     

利用瑞利激光雷达观测北京地区上平流层地形重力波活动

本文利用中国科学院空间科学与应用研究中心的瑞利激光雷达首次观测到了平流层地形重力波活动的现象,并结合美国国家环境预报中心(NCEP)的全球预报系统(GFS)的风场数据分析了该地形重力波的基本参数.与惯性重力波相比较,地形重力波的密度扰动没有下传的相位,在同一高度上,其扰动相位保持不变.北京空间科学与应用研究中心瑞利激光雷达自2012年开始观测实验以来,已经观测到多起地形重力波活动事件.本文以2013年11月11日的观测数据为例,研究北京上空的地形重力波活动,并结合GFS风场数据分析了北京上平流层地形重力波的波长、传播方向、传播速度等参量.通过分析得到在2013年11月11日北京上空存在一列传播方向为北偏西52.4°,水平波长为5.5km,平均垂直波长约为6.0km的地形重力波.     

黏弹性介质中瑞利波频散曲线和衰减系数曲线的反演

瑞利波具有能量大、信噪比高等特点,可以用来反演介质内部的力学信息,近年来在浅层地球物理勘探、深层地震学研究以及超声波无损检测等多个领域都有较广泛的应用。目前大多数瑞利波的应用中都假设介质是弹性的,然而实际中岩石、土壤和金属等介质都在一定程度上体现出了黏弹性。当介质的黏弹性较强时仍然采用弹性假设研究其中瑞利波的反演将增大误差,因此有必要考虑黏弹性介质中的瑞利波反演,但是目前这方面的研究仍不够深入。本文研究黏弹性介质中瑞利波频散曲线和衰减系数曲线的反演问题,给出其在半空间中联合反演的方法,并对该方法的误差进行分析。     

Pseudospectral modeling and dispersion analysis of Rayleigh waves in viscoelastic media

Multichannel Analysis of Surface Waves (MASW) is one of the most widely used techniques in environmental and engineering geophysics to determine shear-wave velocities and dynamic properties, which is based on the elastic layered system theory. Wave propagation in the Earth, however, has been recognized as viscoelastic and the propagation of Rayleigh waves presents substantial differences in viscoelastic media as compared with elastic media. Therefore, it is necessary to carry out numerical simulation and dispersion analysis of Rayleigh waves in viscoelastic media to better understand Rayleigh-wave behaviors in the real world. We apply a pseudospectral method to the calculation of the spatial derivatives using a Chebyshev difference operator in the vertical direction and a Fourier difference operator in the horizontal direction based on the velocity–stress elastodynamic equations and relations of linear viscoelastic solids. This approach stretches the spatial discrete grid to have a minimum grid size near the free surface so that high accuracy and resolution are achieved at the free surface, which allows an effective incorporation of the free surface boundary conditions since the Chebyshev method is nonperiodic. We first use an elastic homogeneous half-space model to demonstrate the accuracy of the pseudospectral method comparing with the analytical solution, and verify the correctness of the numerical modeling results for a viscoelastic half-space comparing the phase velocities of Rayleigh wave between the theoretical values and the dispersive image generated by high-resolution linear Radon transform. We then simulate three types of two-layer models to analyze dispersive-energy characteristics for near-surface applications. Results demonstrate that the phase velocity of Rayleigh waves in viscoelastic media is relatively higher than in elastic media and the fundamental mode increases by 10–16% when the frequency is above 10 Hz due to the velocity dispersion of P and S waves.     

Wave-induced steady streaming,mass transport and net sediment transport in rough turbulent ocean bottom boundary layers

The interaction between two important mechanisms which causes streaming has been investigated by numerical simulations of the seabed boundary layer beneath both sinusoidal waves and Stokes second order waves, as well as horizontally uniform bottom boundary layers with asymmetric forcing. These two mechanisms are streaming caused by turbulence asymmetry in successive wave half-cycles (beneath asymmetric forcing), and streaming caused by the presence of a vertical wave velocity within the seabed boundary layer as earlier explained by Longuet-Higgins. The effect of wave asymmetry, wave length to water depth ratio, and bottom roughness have been investigated for realistic physical situations. The streaming induced sediment dynamics near the ocean bottom has been investigated; both the resulting suspended load and bedload are presented. Finally, the mass transport (wave-averaged Lagrangian velocity) has been studied for a range of wave conditions. The streaming velocities beneath sinusoidal waves (Longuet-Higgins streaming) is always in the direction of wave propagation, while the streaming velocities in horizontally uniform boundary layers with asymmetric forcing are always negative. Thus the effect of asymmetry in second order Stokes waves is either to reduce the streaming velocity in the direction of wave propagation, or, for long waves relative to the water depth, to induce a streaming velocity against the direction of wave propagation. It appears that the Longuet-Higgins streaming decreases as the wave length increases for a given water depth, and the effect of wave asymmetry can dominate, leading to a steady streaming against the wave propagation. Furthermore, the asymmetry of second order Stokes waves reduces the mass transport (wave-averaged Lagrangian velocity) as compared with sinusoidal waves. The boundary layer streaming leads to a wave-averaged transport of suspended sediments and bedload in the direction of wave propagation.     

Thermoelastic waves in non-simple media

Summary The present paper is an attempt to investigate the propagation of thermoelastic waves propagated in non-simple media and study the distinction between propagation in simple and nonsimple media. Plane progressive waves and Rayleigh waves have been discussed.     

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.     

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.     

Wave propagation in two layered medium under initial stresses

Summary In this paper, the frequency equation for phase velocity of waves propagated in a laminated medium consisting of two eleastic layers of finite thickness under initial stresses, has been obtained. It has been shown that when wave length becomes very small compared to the thickness of each layer, the wave approaches two Rayleigh waves at the two outer surfaces with the possibility of Stoneley waves at the interface. The propagation ofSH-waves in the composite medium under initial stresses has also been discussed. A particular case has been taken to find the velocity of Love wave in the homogeneous half space under initial compressive stresses.Biot's incremental deformation theory has been used.     

On the propagation of Rayleigh waves in three dimensions in alluvial soils

Summary The propagation of Rayleigh waves in three dimensions in alluvial soils which do not behave like ordinary isotropic elastic solids have been discussed in this paper. The frequency equation has been solved for different soil constants.     

Effect of surface wave propagation in a four-layered oceanic crust model

Dispersion of Rayleigh type surface wave propagation has been discussed in four-layered oceanic crust. It includes a sandy layer over a crystalline elastic half-space and over it there are two more layers—on the top inhomogeneous liquid layer and under it a liquid-saturated porous layer. Frequency equation is obtained in the form of determinant. The effects of the width of different layers as well as the inhomogeneity of liquid layer, sandiness of sandy layer on surface waves are depicted and shown graphically by considering all possible case of the particular model. Some special cases have been deduced, few special cases give the dispersion equation of Scholte wave and Stoneley wave, some of which have already been discussed elsewhere.     

Propagation and attenuation of Rayleigh waves in generalized thermoelastic media

This study considers the propagation of Rayleigh waves in a generalized thermoelastic half-space with stress-free plane boundary. The boundary has the option of being either isothermal or thermally insulated. In either case, the dispersion equation is obtained in the form of a complex irrational expression due to the presence of radicals. This dispersion equation is rationalized into a polynomial equation, which is solvable, numerically, for exact complex roots. The roots of the dispersion equation are obtained after removing the extraneous zeros of this polynomial equation. Then, these roots are filtered out for the inhomogeneous propagation of waves decaying with depth. Numerical examples are solved to analyze the effects of thermal properties of elastic materials on the dispersion of existing surface waves. For these thermoelastic Rayleigh waves, the behavior of elliptical particle motion is studied inside and at the surface of the medium. Insulation of boundary does play a significant role in changing the speed, amplitude, and polarization of Rayleigh waves in thermoelastic media.     

含孔隙介质的分层半空间表面瑞利波的衰减特性

以分层半空间内部含有一层孔隙介质为物理模型进行数值计算,研究半空间表面瑞利波的传播和衰减特性.为更加接近实际,结合瑞利波的激发特性,确定了瑞利波的主衰减曲线,并主要以此进行规律分析.针对速度递增和含低速层这两种典型的地质模型,讨论了瑞利波的传播衰减在不同地质模型下的特性,并分析了各自的规律.结果表明,在这两种模型下瑞利波的主衰减曲线都受孔隙介质所处空间位置影响产生比较明显的变化,但衰减系数极大值对应的波长与模型的表层厚度存在较明显的线性对应关系,利用这一关系,可以在实际勘探中快速得到表层介质厚度.另外,通过对比分析还可以看到,瑞利波主衰减曲线随孔隙介质的孔隙度和渗透率的变化都强于主频散曲线的变化,表明衰减曲线对孔隙度和渗透率的变化更加敏感,理论上更加适合进行介质参数反演工作.综合对比结果,我们认为瑞利波主衰减曲线中包含了更丰富的介质参数信息,如果能够有效利用,将可以提高瑞利波勘探的准确性和应用范围.     

两种不同势函数下半空间饱和多孔介质中Rayleigh波求解比较

分别对"考虑两种压缩波和幅值比例系数"和"考虑一种压缩波(P1或P2波)但不考虑幅值比例系数"两种不同势函数下的半空间饱和多孔介质中Rayleigh波求解进行详细推导,理论分析表明"考虑两种压缩波和幅值比例系数"下Rayleigh波求解推导更为严密,与饱和多孔介质中存在两种压缩波的事实相一致。在研究半空间饱和多孔介质中Rayleigh波时应采用"考虑两种压缩波和幅值比例系数"的势函数。     

Spatial behaviour of Rayleigh waves in layered half‐spaces under active surface sources

In the free state, Rayleigh waves are assumed to travel in the form of planar wavefronts. Under such an assumption, the propagation behaviour of the modes of Rayleigh waves in layered half‐spaces is only frequency dependent. The frequency behaviour, which is often termed as dispersion, is determined by the shear wave velocity profile of layered soils within the depth related to wavelength (or frequency). According to this characteristic, the shear wave velocity profile can be back‐analysed from the dispersion. The technique is widely used in the surface wave testing. However, the wavefronts of Rayleigh waves activated by the surface sources are non‐planar. The geometric discrepancy could result in Rayleigh waves manifesting distance‐dependent behaviour, which is referred to as spatial behaviour in this paper. Conventional analysis ignoring this spatial behaviour could introduce unexpected errors. In order to take the effects of sources on the propagation behaviour into account, a new mathematical model is established for Rayleigh waves in layered elastic media under vertical disc‐like surface sources using the thin‐layer method. The spatial behaviour of the activated modes and the apparent phase velocity, which is the propagation velocity of Rayleigh waves superposed by the multiple modes, are then analysed. Aspects of the spatial behaviour investigated in this paper include the equilibrium path, the particle orbit, and the geometric attenuation of the activated Rayleigh waves. The results presented in this paper can provide some guidelines for developing new inverse mathematical models and algorithms.     

Pseudo Rayleigh wave in a partially saturated non-dissipative porous solid

Propagation of surface waves is studied at the pervious boundary of a porous solid saturated with a mixture of two immiscible fluids. An approach, based on continuum mixture theory, is used to derive a secular equation for the propagation of harmonic waves at the stress-free plane surface of this non-dissipative medium. Numerical analysis shows that this secular equation may not represent the propagation of true surface wave in the porous aggregate. Then, this equation is solved numerically for the propagation of pseudo Rayleigh wave or the leaky surface waves. To ensure the existence of pseudo Rayleigh wave, capillary effect between two (wetting and non-wetting) pore-fluids is related to the partial saturation. Effects of porosity and partial saturation coupled with capillary effect are observed on the phase velocity of pseudo Rayleigh waves in sandstone saturated with water-CO₂ mixture.     

Vector attenuation: elliptical polarization, raypaths and the Rayleigh-window effect

Waves in dissipative media exhibit elliptical polarization. The direction of the major axis of the ellipse deviates from the propagation direction. In addition, Snell's law does not give the raypath, since the propagation (wavevector) direction does not coincide with the energy‐flux direction. Each of these physical characteristics depends on the properties of the medium and on the inhomogeneity angle of the wave. The calculations are relevant for multicomponent surveys, where the receivers are placed on the ocean‐floor. An example of the role played by inhomogeneous waves is given by the Rayleigh‐window effect, which implies a significant amplitude reduction of the reflection coefficient of the ocean‐bottom.     

Anomalous Rayleigh-Wave Propagation Along Oceanic Trench

A clear later phase of amplitude larger than the direct surface wave packet was observed at stations in Hokkaido, Japan, for several events of the December 1991 off-Urup earthquake swarm in the Kuril Islands region. From its particle motion, this phase is likely to be a fundamental Rayleigh wave packet that arrived with an azimuth largely deviated from each great-circle direction. As its origin, Nakanishi (1992) proposed that the sea-trench topography in this area as deep as 10 km may produce a narrow zone of low velocity for Rayleigh waves of periods around 15 sec. Following this idea, we compute ray paths and estimate how Rayleigh waves would propagate if we assume that lateral velocity variations are caused only by seafloor topography. We confirm that thick sea water in the trench indeed produces the phase velocity of Rayleigh waves to be smaller than in a surrounding area by the degree over 100%. Such a low-velocity zone appears only in a period range from 12 to 20 sec. Although this strong low-velocity zone disturbs the direction of Rayleigh wave propagation from its great circle, the overall ray paths are not so affected as far as an epicentre is outside this low-velocity zone, that is, off the trench axis. In contrast, the majority of rays are severely distorted for an event within the low-velocity zone or, in other words, in the neighborhood of the trench axis. For such an event, a part of wave energy appears to be trapped in this zone and eventually propagates outwards due to the curvature or bend of trench geometry, resulting in very late arriving waves of large amplitude with an incident direction clearly different from great circles. This phenomenon is observed only at a very limited period range around 16 sec. These theoretical results are consistent with the above mentioned observation of Nakanishi (1992).