Abstract: The adaptive Open Boundary Condition (OBC) designed by Chen and the author is extended to cover the cases of dispersive waves. The design was based on a kind of dispersive solutions and an "Equivalent Wave Velocity" was introduced. The latter was derived from dispersive relation ω= ω(k) by Taylor expansion locally. A couple of different kinds of OBC are obtained, which can be used to the numerical computation of dispersive waves, especially, dispersive water waves. Two examples which have exact solutions ar...