Propagation ofSH-waves in a heterogeneous visco-elastic layer overlying a homogeneous voigt medium

Summary The propagation ofSH-waves in a vosco-elastic layer having variable material properties has been studied to explain the damping phenomenon of the wave motion. The damping coefficient may be calculated and the effect of heterogeneity on such type of wave motion may be observed from the obtained frequency equation.     

On the propagation of love waves in a heterogeneous layer lying on a homogeneous medium

Summary The present note is devoted to find out the possibility of propagation of love waves in a heterogeneous layer of finite depth lying on a homogenous semi-infinite elastic medium, the rigidity and density in the upper layer varying as (1–sinz), where is a constant andz is the vertical distance from the interface. The numerical results for the phase velocity for some special Earth models are given.     

二维均匀介质中积分方程的快速精确算法

采用稳定型双共轭梯度快速Fourier变换(BCGS-FFT)算法精确计算二维均匀介质中的积分方程.采用一种新的插值函数作为基函数和试探函数对积分方程进行弱化离散,离散后的积分方程采用稳定型双共轭梯度迭代方法进行求解,从而得到异常体内电场的分布.计算时采用快速Fourier变换技术将积分方程内Green函数与电场的乘积表示成褶积形式以加快计算速度.数值计算举例说明了算法的精确性和有效性.     

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).     

Some characteristics of quasi-biennial geomagnetic oscillations of solar origin

Summary The monthly means of the geomagnetic index aa and of the relative sunspot numbers were used to analyse the quasi-biennial oscillations (QBO) in geomagnetic activity related to an analogous variation in solar activity. Statistical methods of time series analysis were applied: autocorrelation and power spectrum, cross-correlation, complex demodulation, and the digital filter technique. The aim of this investigation was not only the detection of QBO in geomagnetic activity, which has been done by various authors, but also the determination of the time variation of the period, phase and amplitude of these oscillations in the given time interval with reference to the results of Apostolov [2] and Apostolov and Letfus [4]. The following main results have been obtained: The characteristic period of QBO in geomagnetic activity is shorter and varies in a narrower interval than that in sunspot activity; the lifetime of the oscillations is about 22 years and roughly coincides with Hale's cycle; the amplitude of QBO in geomagnetic activity has its maximum on the decreasing branch of the sunspot cycle; the QBO in geomagnetic activity follows the analogous variation in sunspot activity with a delay of 2 to 6 months for various solar cycles, as compared with the 3 to 22 moths for the raw, unfiltered data. The causes of the differences in the characteristics of QBO in geomagnetic and sunspot activity are considered and it is shown that these differences do not contradict the solar control of this phenomenon.

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Analytical representation of Green’s functions of the Biot equations for elastic waves in a homogeneous two-phase medium

The system of Biot vector equations in the frequency space includes two elliptic-type vector partial differential equations with unknown displacement vectors in the solid and liquid phases. Considering the Biot equations, alongside with Pride’s equations, the key approaches to the theoretical study of the elastic waves in the two-phase fluid-saturated media, the author suggests an analytical solution for the inhomogeneous Biot equations in the frequency space, which is reduced to finding its fundamental solution (Green’s function). The solution of this problem consists of solutions for two systems of Biot equations. In the first system, only the first equation is inhomogeneous, while in the second system, only the second equation is inhomogeneous and, as it is shown, its right-hand side is exclusively a potential function. The fundamental solution of the full system of inhomogeneous Biot equations (in which both equations are inhomogeneous) is represented in the form of Green’s matrix-tensor, for the scalar elements of which the analytical relations are presented. The obtained formulas describing the elastic displacements of both the solid and liquid phases reflect three wave types, namely, compressional waves of the first and the second kind (the fast and the slow waves, respectively) and shear waves. Similar terms (those describing the same type of the elastic waves in the solid and liquid phases) in the expressions for Green’s functions are linked with each other through the coefficient that links the components of the displacement vectors of the solid and liquid phases corresponding to the given wave type.

     

Near‐field seismic effects in a homogeneous medium and their removal in vertical seismic profile attenuation estimates

To better understand (and correct for) the factors affecting the estimation of attenuation (Q), we simulate subsurface wave propagation with the Weyl/Sommerfeld integral. The complete spherical wavefield emanating from a P‐wave point source surrounded by a homogeneous, isotropic and attenuative medium is thus computed. In a resulting synthetic vertical seismic profile, we observe near‐field and far‐field responses and a 90° phase rotation between them. Depth dependence of the magnitude spectra in these two depth regions is distinctly different. The logarithm of the magnitude spectra shows a linear dependence on frequency in the far‐field but not in those depth regions where the near‐field becomes significant. Near‐field effects are one possible explanation for large positive and even negative Q‐factors in the shallow section that may be estimated from real vertical seismic profile data when applying the spectral ratio method. We outline a near‐field compensation technique that can reduce errors in the resultant Q estimates.     

均匀多边形截面二度体重力异常计算公式的改进

本文基于格林公式,给出了均匀二度体重力异常线积分表达式的严格数学推导过程,其结果与前人推导出的结果完全相同,但我们的方法在数学上更简单,也更严谨.在此基础之上,我们详细讨论了多边形截面二度体重力异常计算公式中所包含的数值不稳定性,并作了适当的修正.新的计算方法在数值上是稳定的,且适用于包括场源内部以及边界上的任意空间位置.为了对新算法进行验证,我们以一个均匀无限长水平圆柱体及其内接正多边形截面棱柱体为模型,分别利用解析公式和本文所提供的算法计算了二者在不同深度的两条水平观测线上所产生的重力异常,其中一条观测线穿过了模型的内部.通过两种途径计算得到的重力异常在两条测线上都是吻合的,这表明新算法是正确的.     

Love wave dispersion: A comparison of results for a semi-infinite medium with inhomogeneous layers and for its approximation by homogeneous layers

Love wave dispersion in various semi-infinite media consisting of inhomogeneous layers is discussed. The phase and group velocities are computed when shear wave velocity and density in each inhomogeneous layer are varying exponentially with depth. At the beginning one or two inhomogeneous layers over a homogeneous semi-infinite medium are considered. The dispersion results for these structures are compared with those for their approximations with homogeneous layers. Comparisons show that differences of phase and group velocities for the original models from those for their approximated models (i) increase with the increase of wave number and (ii) are larger for group velocity than for phase velocity. The difference is approximately proportional to the rate of change of parameters in the layers. Finally, dispersion curves are obtained for model IP3MC, which consists of many inhomogeneous and homogeneous layers over a homogeneous semi-infinite medium. The results are compared with the observed group velocity data across the Indian Peninsula.     

The gravitational field of a homogeneous hemisphere

Summary After remarking that the homogeneous hemisphere might be adopted as a model to represent various geological bodies in the quantitative interpretation of gravity anomalies, a finite expression for its gravitational potential is derived.

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Riassunto Dopo aver rilevato che la semisfera omogenea potrebbe essere adottata quale modello per rappresentare var corpi geologici nell'interpretazione quantitativa di anomalie di gravità, si deriva una espressione finita per il potenziale gravitazionale di tale figura.

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This work has been presented at the XIV General Meeting of the Italian Society of Geophysics and Meteorology (Genova, April 14–15, 1966).     

Slow hydromagnetic oscillations in a rotating spheroidal shell

Abstract

The ray method is used to study slow hydromagnetic waves in an incompressible, inviscid, perfectly conducting fluid of constant density in the presence of a constant toroidal magnetic field. The fluid is bounded below by a rigid sphere and above by a rigid spheroidal surface, and the mean fluid layer thickness is assumed to be small. Both the general time-dependent and time-harmonic (free oscillation) problems are studied and dispersion relations and conservation laws are derived. These results are applied to free oscillations with constant azimuthal wave number in a spherical shell and then compared to those of previous authors. Such oscillations propagate to the east and are trapped between circles of constant latitude. Wave propagation in axisymmetric shells is then studied with emphasis on the relationship between shell shape and direction of propagation, and it is found that such shells can sustain westward propagating modes wherever the shell thickness decreases sufficiently rapidly from a maximum at the poles to zero at the equator; no shells exist which can sustain westward propagation at the equator.     

Subharmonic oscillations of a forced hydromagnetic cavity

Abstract

We study the propagation of nonlinear MHD waves in a highly magnetized plasma cavity. The cavity's moving boundaries generate Alfvén waves, which in turn drive and interact with slow magnetosonic waves. The interacting wave system is analyzed by a Galerkin and multiple-scale analyses leading to simple dynamical equations. When the frequency of the forcing provided by the moving boundaries and that of the fundamental Alfvén eigenmode are close, the cavity behaves like a Duffing oscillator. Application of the Melnikov function theory shows that the Alfvén wave's amplitude undergoes both flip and saddle-node bifurcations as the amplitude and the phase of the boundary forcing vary. Direct numerical integration confirms these results and provides an estimate of the amount of energy dissipated in the bifurcations.     

On a homogeneous representation of seismic history in large regions

The possibility to obtain a more complete and unbiased long-term history of seismic shakings over large territories than is explicitly reported from inhabited localities is discussed in the paper. An approach proposed for this purpose consists in complementing the spatial distribution of the macroseismic effect of earthquakes by calculated intensities at localities where information on felt shakings is absent. The calculated intensity is obtained on the basis of data on the epicentral intensity and location of epicenters provided by earthquake catalogs. This approach is applied to the analysis of the history of seismic shaking in Spain. The calculated intensities are shown to be comparable in accuracy with the ordinary practice of intensity determinations at national seismological centers.     

Three-dimensional numerical modelling of wind-driven circulation in a homogeneous lake

A three-dimensional numerical model has been developed to study wind-induced circulation patterns in a shallow homogeneous lake with a complex bathymetry. The governing equations are the unsteady Reynolds-Averaged Navier–Stokes equations in which the non-hydrostatic pressure distribution has been included. The model was tested against analytical solutions and laboratory data for wind-induced currents and then applied to Esthwaite Water, a small lake in Cumbria, UK. The model was used to study the main model parameters and to generate typical circulation patterns for a variety of conditions in the lake. Simulations showed that a non-hydrostatic pressure distribution did not have any noticeable influence on the overall circulation pattern in the lake. However, comparisons with field data at some measurement stations in the near-shore region with sharply varying bottom topography showed that the hydrodynamic pressure component had some influence on the vertical velocity profile.     

The equatorial dynamics of a deep homogeneous ocean

Abstract

The flow properties of an homogeneous fluid which is bounded by two concentric spheres and two meridional planes which intersect along a diameter of the spheres are investigated. The spheres rotate about this diameter with slightly different angular velocities. As in the axisymmetric case studied by Proudman (1956) and Stewartson (1966) the viscous terms in the equations of motion are important only in boundary layers on the spheres and on the cylinder C which circumscribes the inner sphere and which has generators parallel to the axis of rotation, provided the Ekman number E is small. In the inviscid region the velocities are independent of the coordinate measuring distance along the axis of rotation and are much weaker, by a factor 0(E ^½), than the velocities in the Ekman layer on the driving surface (outer sphere). (It is assumed that the reference frame is fixed in the slower rotating inner sphere.) If the separation of the spheres is small compared to their radii then the asymmetric circulation inside C is characterized by an intense jet along the western wall. Loss of fluid from this jet sustains the eastward and northward flow in the inviscid interior where motion is driven by the suction of the Ekman layer on the outer sphere. (Geophysical conventions have been adopted.) Outside C an intense current is present on the eastern, not western, wall while motion in the inviscid region is westward, and away from the axis of rotation. Though there is no transport across C in the inviscid region, the meridional transport of the Ekman layer on the outer sphere is continuous across C and increases, through suction, as the equator is approached until it drains into an eastward flowing equatorial current of width 0(E ^1/7). The eastern boundary current outside C and shear layers on C carry this fluid to the intersection of C and the western wall where it feeds the western boundary current inside C.

The relation between this study and the experiments of Baker and Robinson (1970) is discussed.     

Analytical solutions of one-dimensional macrodispersion in stratified porous media by the quadrupole method: convergence to an equivalent homogeneous porous medium

In this article, the quadrupole method is implemented in order to simulate the effects of heterogeneities on one dimensional advective and diffusive transport of a passive solute in porous media. Theoretical studies of dispersion in heterogeneous stratified media can bring insight into transport artefacts linked to scale effects and apparent dispersion coefficients. The quadrupole method is an efficient method for the calculation of transient response of linear systems. It is based here on the Laplace transform technique. The analytical solutions that can be derived by this method assists understanding of upscaled parameters relevant to heterogeneous porous media.First, the method is developed for an infinite homogeneous porous medium. Then, it is adapted to a stratified medium where the fluid flow is perpendicular to the interfaces. The first heterogeneous medium studied is composed of two semi-infinite layers perpendicular to the flow direction each having different transport properties. The concentration response of the medium to a Dirac injection is evaluated. The case studied emphasises the importance in the choice of the boundary conditions.In the case of a periodic heterogeneous porous medium, the concentration response of the medium is evaluated for different numbers of unit-cells. When the number of unit cells is great enough, depending on the transport properties of each layer in the unit cell, an equivalent homogeneous behaviour is reached. An exact determination of the transport properties (equivalent dispersion coefficient) of the equivalent homogeneous porous medium is given.     

Spectral analysis of oscillations with a time-variable frequency

Summary The paper presents a method of evaluating oscillations with a time-variable frequency using a computer. The given function is first interpolated at points which are not distributed equidistantly in time, but the digitizing step varies with time according to a known optional regularity. The spectrum of the obtained function is computed for various of these interpolations and tests are run to determine when the interpolation best compensates the time variation of the frequency. The initial and terminal frequency in the given sample is then determined. The usability of this method with respect to various types of oscillations with a variable frequency and its accuracy in comparison to sonagrams are discussed.