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In this paper, a new semi-analytical method is developed with introducing a new global nonreflecting boundary condition at medium−structure interface, in which the coefficient matrices, as well as dynamic-stiffness matrix are diagonal. In this method, only the boundary of the problem's domain is discretized with higher-order sub-parametric elements, where special shape functions and higher-order Chebyshev mapping functions are employed. Implementing the weighted residual method and using Clenshaw–Curtis quadrature lead to diagonal Bessel's differential equations in the frequency domain. This method is then developed to calculate the dynamic-stiffness matrix throughout the unbounded medium. This method is a semi-analytical method which is based on substructure approach. Solving two first-order ordinary differential equations (i.e., interaction force–displacement relationship and governing differential equation in dynamic stiffness) allows the boundary condition of the medium−structure interface and radiation condition at infinity to be satisfied, respectively. These two differential equations are then diagonalized by implementing the proposed semi-analytical method. The interaction force–displacement relationship may be regarded as a nonreflecting boundary condition for the substructure of bounded domain. Afterwards, this method is extended to calculate the asymptotic expansion of dynamic-stiffness matrix for high frequency and the unit-impulse response coefficient of the unbounded media. Finally, six benchmark problems are solved to illustrate excellent agreements between the results of the present method and analytical solutions and/or other numerical methods available in the literature. [ABSTRACT FROM AUTHOR]