In this paper, the propagation of lateral waves and axial acoustic waves in a drill-string are studied by using a new numerical method and a stability monitoring scheme is proposed. The drill-string is modeled as a linear beam structure under gravitational field effects. Lateral and axial motions are assumed to be decoupled, and the corresponding equations of motion are derived. An iterative wavelet-based spectral finite element method (WSFEM) model is developed to obtain a high fidelity response. Numerical simulations of the lateral impact wave propagation at the bottom-hole-assembly (BHA) are first conducted, and a time-frequency analysis technique is applied to the response in order to identify the relationship between the position of the transition point between positive and negative strain and the dispersive properties of the lateral wave. Next, axial acoustic wave propagation through the upper drill-pipe is studied to explore the banded transmission properties of the drill-string introduced by periodic joints. Based on the results, a new monitoring scheme is proposed to monitor the stability of the drill-string by conducting a combination of lateral impact wave analysis at the BHA and the axial acoustic telemetry technique. The new numerical method used in this study provides a unified approach to study the wave propagation in drill-strings, and the proposed stability monitoring scheme is expected to be applicable in drill-string operations.