The conventional modal methods become inefficient when expanded to the high frequency region. This leads to a large number of degrees of freedom (DOF) even for one-dimension structures. Meanwhile, a short computational time is also the requirement of on-line structural health monitoring. This paper presented an application of B-spline wavelet on interval (BSWI) finite element for one-dimension elastic wave propagation problems. By using central difference method in time domain, numerical results involving wave propagations in crack free rod and beam were obtained and compared with results calculated by classical finite element method. Numerical examples validated the good performance of BSWI. BSWI could provide a accurate and efficient solution for wave propagation problems. The proposed method provides a good reference for structural health monitoring.