In this paper, the general principles are introduced to establish the wave-motion equations by use of two descriptions (Lagrangian and Eulerian descriptions) in the continuum mechanics, when the finite deformation is taken into account. After that the dynamics behaviors and wave properties of finite deformation elastic thin rods are paid more attentions to in Lagrangian coordinate. Using basic equations in continuum mechanics, the wave-motion equation in the geometrical nonlinear elastic rod is ...