Using a one-dimensional, time-dependent model, a numerical simulation study is presented on the evolution of high-speed streams in the interplanetary space. The result obtained comes to two conclusions as follows. 1. The leading edge of a high-speed stream evolves into a shock system consisting of a forward fast shock, a forward slow shock, and a reverse fast shock. The low-density feature inherent by the high-speed stream diminishes the forward slow shock remarkably, and prevents any reverse slow shock from occuring. 2. A forward and a reverse rarefaction wave of mainly fast mode forms at the tailing edge of the high-speed stream, and both of them exert a significant influence on the fast shock pair formed at the leading edge of the high-speed stream.