Restricted Three-Body Problem: An Approximation of its General Solution – Part One – the Manifold of Symmetric Periodic Solutions

This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x ₀ ∊ [−2,2],C ∊ [−2,5]) of the (x ₀, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x ₀ ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x ₀, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families.     

Seismic rays in an ellipsoidal earth model: theoretical analysis and estimation of deviations due to the ellipticity

Summary. A first-order form of the Euler's equations for rays in an ellipsoidal model of the Earth is obtained. The conditions affecting the velocity law for a monotonic increase, with respect to the arc length, in the angular distance to the epicentre, and in the angle of incidence, are the same in the ellipsoidal and spherical models. It is therefore possible to trace rays and to compute travel times directly in an ellipsoidal earth as in the spherical model. Thus comparison with the rays of the same coordinates in a spherical earth provides an estimate of the various deviations of these rays due to the Earth's flattening, and the corresponding travel-time differences, for mantle P -waves and for shallow earthquakes. All these deviations are functions both of the latitude and of the epicentral distance. The difference in the distance to the Earth's centre at points with the same geocentric latitude on rays in the ellipsoidal and in the spherical model may reach several kilometres. Directly related to the deformation of the isovelocity surfaces, this difference is the only cause of significant perturbation in travel times. Other differences, such as that corresponding to the ray torsion, are of the first order in ellipticity, and may exceed 1 km. They induce only small differences in travel time, less than 0.01s. Thus, we show that the ellipticity correction obtained by Jeffreys (1935) and Bullen (1937) by a perturbational method can be recovered by a direct evaluation of the travel times in an ellipsoidal model of the Earth. Moreover, as stated by Dziewonski & Gilbert (1976), we verify the non-dependence of this correction on the choice of the velocity law.     

密度和压缩系数的散射层析成像法

本文在速度成像的基础上研究了同时对密度和压缩系数成像的散射波层析成像法.对不同散射角度的计算可以得到一系列反演图像,拟合这些图像,从而可以有效地达到对密度和压缩系数（或速度）成像的目的.与单纯的速度成像相比,增加了反演的难度.首先是对资料的方位性要求增加;其次是对资料的利用率下降.即便如此,从对较少量的炮点和检波点资料的数值计算来看,仍取得了满意的成像结果.我们对组成字母“A”的散射体结构进行了成像计算,结果能够同时再现密度和压缩系数,成像清晰,表明了方法的可行性,并能应用于复杂结构的成像问题.     

青藏高原及邻近区域的S波三维速度结构

本文收集了WWSSN台网和我国台网中13个地震台站的长周期地震记录,用140条10-90s瑞利波频散曲线和作者提出的Tarantola-Backus面波频散层析成象方法,作了青藏高原及邻区的速度反演,得出该地区岩石层速度结构的三维图象.结果表明,1.在10-110km深度范围内,速度结构出现与大地构造特征相关的分区性,显示出四个构造单元:青藏块体、柴达木-巴颜喀拉-三江块体、塔里木块体和印度块体.2.高原内部,深度为10-70km内速度较低,莫霍界面呈不对称盆形分布,藏北那曲附近地壳厚度超过70km,高原边缘壳厚为45-50km,90-110km为高速异常,表明高原内部存在上地幔盖层.3.高原北部的班公湖断裂和东部的三江断裂系是该区重要的分界线,是岩石层结构存在明显差异的重要接触部位,可能是冈瓦纳古陆与欧亚古陆的缝合带.4.柴达木-巴颜喀拉-三江块体内部速度分布不均匀,地壳厚度由北向南从45km加深到60km;在深度90-110km存在一低速层.5.塔里木地块内速度随深度均匀增加,从地壳到上地幔110km内没有发现低速层.地壳厚度约50km.     

论裸眼井中的“共振纵波”和“共振横波”

一、引言 文献[1]和[2]指出,充流体裸眼井中存在频率确定、波速恒定、且沿传播方向不衰减的“共振纵波”和“共振横波”。本文的目的在于论证并指出,这两种波是不存在的。 二、论证 设一点源位于z=0的井轴上,则井轴上z处的声波频率响应为     

Structural elements of the Makran region, Oman sea and their potential relevance to tsunamigenisis

The character of convergence along the Arabian–Iranian plate boundary changes radically eastward from the Zagros ranges to the Makran region. This appears to be due to collision of continental crust in the west, in contrast to subduction of oceanic crust in the east. The Makran subduction zone with a length of about 900 km display progressively older and highly deformed sedimentary units northward from the coast, together with an increase in elevation of the ranges. North of the Makran ranges are large subsiding basins, flanked to the north by active volcanoes. Based on 2D seismic reflection data obtained in this study, the main structural provinces and elements in the Gulf of Oman include: (i) the structural elements on the northeastern part of the Arabian Plate and, (ii) the Offshore Makran Accretionary Complex. Based on detailed analysis of these data on the northeastern part of the Arabian Plate five structural provinces and elements—the Musendam High, the Musendam Peneplain, the Musendam Slope, the Dibba Zone, and the Abyssal Plain have been identified. Further, the Offshore Makran Accretionary Complex shown is to consist Accretionary Prism and the For-Arc Basin, while the Accretionary Prism has been subdivided into the Accretionary Wedge and the Accreted/Colored Mélange. Lastly, it is important to note that the Makran subduction zone lacks the trench. The identification of these structural elements should help in better understanding the seismicity of the Makran region in general and the subduction zone in particular. The 1945 magnitude 8.1 tsunamigenic earthquake of the Makran and some other historical events are illustrative of the coastal region’s vulnerability to future tsunami in the area, and such data should be of value to the developing Indian Ocean Tsunami Warning System.     

基于Poulos弹性理论的被动桩改进算法

传统的Poulos弹性理论仅适合于均质土中土体侧向位移时桩的性状分析,无法考虑土的层状特性。通过引入层状地基中作用一水平集中力的广义Mindlin解和地面作用有竖向荷载时的应力和位移通解,对Poulos方法进行了改进,使之扩展到多层土中,还用于研究堆载条件下的被动桩变形和受力响应。算例分析表明,改进弹性理论要比Poulos方法更为严密、合理,提高了计算精度,应用范围也更广。     

斜向波对直立式防波堤堤头作用的试验研究

直立式防波堤堤头在斜向波浪作用下波浪力的计算在现行的技术规范中尚属空白。通过物理模型试验给出了堤头结构设计所需的波压分布图，指出了波浪入射方向和地形对堤头所受波浪力的贡献有时可以超过波高和波周期，根据现行规范按波浪正向入射计算堤头所受的波浪力偏于安全。     

Control strategies for the Clam Wave Energy Device

A promising wave energy device being currently investigated is the ‘clam’. The clam extracts energy by pumping air through a specially designed (Wells) turbine. Although operation of the Wells turbine does not require a rectified air flow, some additional control will be necessary to optimize the phase of the clam motion for good efficiencies. An examination of the equation of motion in the time domain suggests the possibility of phase control by mechanical, power take-off, or pneumatic latching. Latching can be shown to increase the efficiency of the device in the longer wavelengths of the wave spectrum, i.e. those of high incident wave power. Equivalently latching could be used to keep the device efficiency high while reducing its size, possibly resulting in cheaper power extraction.     

在各类天气系统影响下石臼港沿岸波浪场的分布

本文查阅了从１９７３－１９８５，１３年的历史资料，对石臼港沿岸波浪场进行了详细的分类、统计分析和计算，提出了各类较强天气系统影响石（即指石臼港）的极端区域，同时获得了石港沿岸各类天气系统影响下的波浪分布特点以及波高计算式，为保障石港的生产及港口建设为石港的波了提供了有益的分析数据和预报手段。