Equivalent Relationships And Solution Methods For Wave Motion In Inhomogeneous Media

With the technique development of the composite materials,wave motion problem in inhomogeneous media has been a hot research topic in the scientific and engineering fields.Because of the advanced approaches for solving problems and the continuously growing computing capacity of the computer,it becomes possible to deal with many complex and large-scale problems.The governing equation of the wave motion in inhomogeneous medium is the wave equation with variable coefficients,but not the wave equation with constant coefficients as the case in homogeneous medium.Hence,many traditional theories and methodologies are hard to solve the wave motion problem in inhomogeneous medium,so that the models,the theories and the analysis techniques for this problem should be promoted or reconstructed.The studies on above areas are significant to the development of this field.This paper contains the work both on the analytical approaches and the numerical schemes for solving the wave motion problem in inhomogeneous medium.The contents includes the equivalent relationships for the wave motions,the transformation method for 1D wave equation with variable coefficients,the radiation boundary condition for 1D wave equation with variable coefficients,the numerical method for the transformation of the wave field and the absorbing boundary condition for the finite difference analysis of the elastodynamic problem in inhomogeneous medium.Based on the equivalent mathematical models,the equivalent relationships are presented among the cylindrical wave in axisymmetrically inhomogeneous medium,the spherical wave in radially inhomogeneous medium and the plane wave in 1D inhomogeneous medium.This paper also gives the equivalent relationship between the plane wave in orthotropic inhomogeneous medium and the cylindrical wave in isotropic inhomogeneous medium.The physical nature of these equivalent relationships is explained through the wave equation and the energy in this paper.The solving formulations of the equation transformations are derived in present paper,where the transformation is from 1D wave equation with variable coefficients to the one with constant coefficients and from 1D wave equation with variable wave velocity to the one with constant wave velocity.Using the variable separation approach,the SH wave problem in orthotropic inhomogeneous medium(including the isotropic case)can be simplified to be theproblem of solving two 1D wave equations with variable coefficients.The necessary conditions of this operation are given in the paper.The radiation boundary condition and the dynamical condition for wave motion in 1D inhomogeneous media can be equivalent on their mathematical model.Based on this idea,a new approach is proposed to find the transformation relationship for 1D wave equation with variable coefficients.Then,the radiation boundary condition and the dynamical condition are presented for 1D wave equation with variable coefficients.The general form of the solution of1 D wave equation with variable coefficients is obtained in this paper,and the implementation process is given as well.By the transformation invariant of finite element equation,the numerical schemes are proposed for the transformation of the wave field.The DIFEM(dynamic inhomogeneous finite element method)is applied to construct the basic theoretical frame and the implementation path for the numerical wave transformation.The linear triangle element and the bilinear quadratic element are utilized to derive the DIFEM formulations under the systematic scale,the element scale and the harmonic-wave condition.The absorbing boundary condition is presented for the wave simulation in the inhomogeneous medium based on the radiation boundary condition given in this paper.In order to implement this methodology,the finite difference formulations of the absorbing boundary condition are derived for simulating the waves in the 1D inhomogeneous medium,the SH waves in the 2D inhomogeneous medium which is able to separating variables and the SH waves in the 2D general inhomogeneous medium.