A Review on SH-Wave Propagation for Orthotropic Topographic Features

The wave propagation problem is one of the most important topics studied by numerous researchers. Therefore, in this paper, the background of the researches on the propagation of anti-plane SH-waves in a non-homogeneous linear elastic orthotropic medium is presented based on the topographic features as a case study. In this regard, a brief review is illustrated on the theoretical expression of the elasticity of non-homogeneous materials, scalar wave equation, and the technical literature of the obtained Green's functions to solve the mentioned problems. The researchers have proposed various approaches for seismic analysis of topographic features where their studies are categorized according to the development. In general, these methods can be divided into analytical, semi-analytical, and numerical methods. Despite the high accuracy of analytical methods, their lack of flexibility in modeling and analyzing the complex features in accordance with real paradigms in nature, has forced the researchers to use alternative approaches such as numerical methods. In recent decades, increasing the power of computers besides the development of numerical approaches has made researchers eager to use them for analyzing wave propagation problems as well as predicting the real responses of topographic features more than ever. Based on the formulation, the numerical methods can be usually divided in two general categories known as the domain and boundary methods. The common domain methods are including the Finite Element Method (FEM) and Finite Difference Method (FDM), which require discretization of the whole body including internal parts of the model and its boundaries. Although the simplicity of domain methods makes them favorable for seismic analysis of finite media, the models are complicated because of discretizing the whole body and its boundaries at a considerable distance from the desired zone. In boundary methods that are mostly known today as the Boundary Element Method (BEM), due to the concentration of meshes only around the boundary of the desired features, automatic satisfaction of wave radiation conditions at infinity, reducing the volume of input data and analysis time is remarkably achieved as well.On the other hand, because of the large contribution of analytical processes in solving various problems by BEM, the high accuracy of the obtained results is guaranteed. Therefore, the BEM provides a better manner for analyzing the infinite/semi-infinite problems. The BEM formulation can be formed in two categories, full and half-plane. In full-plane BEM, in addition to truncate the model from a full-space, it is required to discretize all the boundaries of the problem including the interfaces, smooth ground surface, and enclosing boundaries. This leads to approximate the satisfaction of stress-free conditions on the ground surface and makes its results less accurate in some cases. In the half-plane BEM approach, the discretization of smooth surface and definition of fictitious elements for enclosing boundaries are ignored, and the stress-free boundary condition of the surface is satisfied in an exact process. Despite difficult implementation and creating large equations in the half-plane BEM compared to the full-plane case, the mentioned advantages help to make the simple models. According to the appropriateness of the BEM in the analysis of wave propagation problems, especially in the presence of topographic features, this method is expanded in two mediums of isotropic and orthotropic. This paper is recommended as a starting point for all researchers who are interested in the field of seismic analysis of homogeneous and non-homogeneous sites.