Abstract: In this paper, a new proof of D'Alembert formula for Cauchy problems of wave equation is proposed. And some D'Alembert-type formulas for Cauchy problem with discontinuous coefficients on semi-infinite initial boundary value problem are proposed. Based on above discussion, the characteristics methods for the direct and inverse problems of the stress wave equation for uniform layerad elastic medium are investigated. The aim of this paper is to stress the significance and application of the characteristics concept and approach.