AbstractA local transmitting boundary is presented in a compact form, which can be directly incorporated into finite elements. The accuracy of the boundary is studied thoroughly for a one‐dimensional model in order to clarify numerical instabilities introduced by the boundary. Discretization of the model and reflection from the boundary are rigorously considered in the study, and the mechanism of the instability is then illuminated in the frequency domain by the amplification of reflection from the boundary and the multi‐reflection of wave motion in a finite computational region. Typical characteristics of the instability in the time domain are illustrated by numerical results of the simple model and explained completely by the mechanism. On the basis of this understanding of the mechanism, a modified transmitting boundary is devised and its stability criterion is given for the one‐dimensional