Abstract: For the numerical simulation of elastic and visco-elastic infinite foundation, a novel dynamic model for infinite foundation-the coupling model of finite and mapped dynamic infinite elements is presented in this paper. For the one dimensional wave problem, the conception of generalized spring, generalized dashpot and generalized mass is pointed out and the wave propagation functions for elastic and visco-elastic dynamic infinite elements are derived herein. Meanwhile, the fundamental characteristics of the mapped dynamic infinite elements are studied. At last, a numerical example is given to show the accuracy of this model.