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The dispersion of rayleigh waves in orthotropic layered half-space using matrix method

In this paper, the secular equation of Rayleigh surface waves propagating in an orthotropic layered half-space is derived by the matrix method. All the layers and the halfspace are assumed to have identical principle axes. The explicit form of the matrizant for each layer is obtained by the Sylvester’s theorem. The derived secular equation takes only real values and depends only on the dimensionless variables and dimensionless material parameters. Hence, it is convenient in numerical calculation. | Vietnam Journal of Mechanics, VAST, Vol. 38, No. 1 (2016), pp. 27 – 38 DOI:10.15625/0866-7136/38/1/6191 THE DISPERSION OF RAYLEIGH WAVES IN ORTHOTROPIC LAYERED HALF-SPACE USING MATRIX METHOD Tran Thanh Tuan1,∗ , Tran Ngoc Trung2 University of Science, Hanoi, Vietnam 2 Publishing House for Science and Technology, VAST, Hanoi, Vietnam 1 VNU ∗ E-mail: tranthanhtuan@hus.edu.vn Received May 14, 2015 Abstract. In this paper, the secular equation of Rayleigh surface waves propagating in an orthotropic layered half-space is derived by the matrix method. All the layers and the halfspace are assumed to have identical principle axes. The explicit form of the matrizant for each layer is obtained by the Sylvester’s theorem. The derived secular equation takes only real values and depends only on the dimensionless variables and dimensionless material parameters. Hence, it is convenient in numerical calculation. Keywords: Rayleigh waves, matrix method, Sylvester’s theorem, orthotropy. 1. INTRODUCTION The study of Rayleigh surface waves propagation in layered half-space is of considerable interest in the field of seismology since the Earth’s surface could be considered to consist of several layers overlying a half-space. The first systematic and efficient method to find the secular equation is the propagator matrix method proposed by Thomson [1] and Haskell [2]. This method expresses the relation between the displacements and the stresses at two faces of a layer by a matrix called the layer propagator matrix. The product of these matrices is used to find relation of the displacements and stresses at the free surface and at the top of the half-space to formulate the secular equation using boundary conditions. This method has been modified to be more efficient and stable such as in Knopoff [3], Dunkin [4], Kennett [5] and Chen [6] and has been used widely recently. For the homogeneous isotropic layered half-space, the layer propagator matrix has an explicit form because the .