Many theories have been proposed for the propagation of stress waves in metals. They are classified into three types by the constitutive equation; that is, the Kármán-type theory, the Malvern-type theory, and the Cristescu-type theory. However, these proposed theories are not sufficient to explain certain facts which are found in the impact test of a thin rod consistently. For example, they fail to explain how the front of the stress wave always travels with the elastic-wave velocity even in material prestressed into the plastic region, and the plateau of the uniform plastic strain remains in the neighborhood of the impact end. Therefore, in this paper, the authors apply the Johnston-Gilman-type constitutive equation to the theory of stress-wave propagation and study the propagation of the stress wave produced in a thin rod by an impact of a long duration. The results of analysis account for the above-mentioned two facts consistently, and, moreover, account for other phenomena which occur during the propagation of the stress wave not only in mild steel but also in other metals. From these results, the following conclusions are obtained. It is proper to use the Johnston-Gilman-type constitutive equation for the theory of the stress-wave propagation in mild steel, and it seems that the forms of the constitutive equations of other metals may bear a close resemblance to that of the Johnston-Gilman-type constitutive equation. Though the Johnston-Gilman-type constitutive equation is based on the microscopic mechanisms of the dislocation theory, the theory of stress-wave propagation in which it is used is essentially Malvern's theory, which has a noninstantaneous plastic response to an increase of the stress.