Reflection Transmission Matrix Research Articles

An Integral Equation Model for a Pile in a Layered Transversely Isotropic Saturated Soil

Objective:In this paper, an integral equation model is established to predict the time-dependent response of a vertically loaded pile embedded in a layered Transversely Isotropic Saturated Soil (TISS).Methods:Based on the fictitious pile method, the pile-soil system is decomposed into an extended saturated half-space and a fictitious pile. The extended half-space is treated as a layered TISS, while the fictitious pile is considered as a 1D bar. The pile-soil compatibility is accomplished by requiring that the axial strain of the fictitious pile be equal to the vertical strain of the extended layered TISS along the axis of the pile. The second kind Fredholm integral equation of the pile is then derived by using the aforementioned compatibility condition and the fundamental solution of the layered TISS, which is equivalent to the solution of the layered TISS subjected to a uniformly-distributed load acting vertically over a circular area with the radius equal to that of the pile. The fundamental solution of the layered TISS is obtainedviathe Reflection-Transmission Matrix (RTM) method for the layered TISS. Applying the Laplace transform to the Fredholm integral equation, and solving the resulting integral equation, the transformed solution is obtained. The time domain solution of the pile-soil system is retrievedviathe inverse Laplace transform.Results and Conclusion:Numerical results of this paper agree with existing solutions very well, validating the proposed pile-soil interaction model. A parametric study is carried out to examine the influence of some parameters on the response of the pile-soil system.

Reflection–transmission matrix model for the axisymmetric deformation of a layered transversely isotropic saturated soil

The reflection–transmission matrix (RTM) method for the layered transversely isotropic saturated soil (TISS) undergoing axisymmetric consolidation is developed in this study. To this aim, a system of partial differential equations is established based on the governing equations for the TISS first. By means of the Hankel–Laplace integral transform method, the system of partial differential equations is reduced to a system of ordinary differential equations. By using the general solution to the system of ordinary differential equations, the static wave vector corresponding to the state vector of the TISS is introduced, and the transfer matrices for the state and wave vectors are defined. The RTMs for the aforementioned static wave vector of the layered TISS are then established. By means of the RTMs for the layered TISS, solutions for the layered TISS subjected to external sources are derived. Numerical results due to the proposed RTM method are compared with some existing results, validating the proposed RTM method. Parametric analyses are conducted to illustrate the influence of some parameters on the consolidation of the layered TISS.

Reflection–transmission matrix method for the consolidation of a multilayered saturated soil

SummaryThe consolidation of the layered saturated soil is an important issue in civil engineering and has been investigated extensively during the past decades. In this study, based on the Biot's theory, the reflection–transmission matrix (RTM) method for treating the layered saturated soil under axisymmetric consolidation is developed. To decouple the governing equations of the Biot's theory, the McNamee displacement functions are introduced, and the general solution for the saturated soil is obtained using the Laplace and Hankel transforms. In order to develop the RTM method for the layered saturated soil, based on the obtained general solution, the static wave vector corresponding to the state vector of the saturated soil and the transform matrix relating the aforementioned two vectors are defined. Also, the transfer matrices corresponding to the two vectors are introduced, and the representations of the RTMs for the static wave vector of the saturated soil are presented. As the state vector, static wave vector, and the transform matrix relating the two vectors are all defined in the global coordinate system, the RTMs obtained in this study thus have a reasonable physical meaning. By using the RTMs for the layered saturated soil, the solutions for the layered saturated soil subjected to external sources are derived. Comparison of results due to the proposed RTM method with some existing results and results due to the transfer matrix method validates the developed RTM method. Some numerical results are obtained based on the proposed RTM method for the layered saturated soil. Copyright © 2016 John Wiley & Sons, Ltd.

Free-mode surface-wave computations

The theory of Love- and Rayleigh-wave dispersion for plane-layered earth models has undergone a number of developments since the initial work of Thomson and Haskell. Most of these were concerned with computational difficulties associated with numerical overflow and loss of precision at high frequencies in the original Thomson-Haskell formalism. Several seemingly distinct approaches have been followed, including the delta matrix, reduced delta matrix, Schwab-Knopoff, fast Schwab-Knopoff, Kennett's Reflection-Transmission Matrix and Abo-Zena methods. This paper analyses all these methods in detail and finds explicit transformations connecting them. It is shown that they are essentially equivalent and, contrary to some claims made, each solves the loss of precision problem equally well. This is demonstrated both theoretically and computationally. By extracting the best computational features of the various methods, we develop a new algorithm (sec Appendix A5), called the fast delta matrix algorithm. To date, this is the simplest and most efficient algorithm for surface-wave dispersion computations (see Fig. 4). The theory given in this paper provides a complete review of the principal methods developed for Love- and Rayleigh-wave dispersion of free modes in plane-layered perfectly elastic, isotropic earth models and puts to rest controversies that have arisen with regard to computational stability.

Statistical theory of rough boundaries for electromagnetic waves I Deterministic Green functions and reflection–transmission matrix

Basic equations of a rough boundary are shown based on an effective boundary condition transferred from the real boundary onto two reference boundary planes chosen such that the entire boundary is enclosed by the two planes. The boundary condition is written in terms of a surface impedance, which is an operator and is obtained exactly for given boundary change. Maxwell’s equations are rewritten in a simple form by a 6 × 6 matrix equation (such as Dirac’s equation) with several algebraic formulas, which help us greatly to derive various equations in a compact form and thereby to get a general version of complicated equations. A surface Green function is introduced associated with the boundary condition, and an exact reflection–transmission matrix is obtained from it in an operator form for a given boundary change. The latter is also obtainable as a solution of an ordinary integral equation perfectly free from any operator. Power conservation is directly connected with the Hermitian condition imposed on the surface impedance and is investigated in detail so optical relations eventually can be obtained from it. Other symmetries inherent among associated quantities are also shown. A set of operator equations for the surface Green function is obtained to the first order of the boundary change, consistently with the power conservation, to be applied in Part II of this work [ J. Opt. Soc. Am. A2, 2260 ( 1985)].