In an earlier paper dealing with dispersion of waves in composite rods, it was noted that the dispersive properties of one rod were quite different from those of another with different properties. This observation prompted the speculation that the dispersive properties of a composite rod could be controlled through a choice of the geometric and material parameters of the rod. In this paper, we show how the dispersive properties can be controlled; that is, we show the influence of each of the parameters on the dispersive characteristics of a rod. Our previous study had led us to suppose that if we were sufficiently adroit, we could choose the rod parameters so that the resulting composite rod would be nondispersive. Our study here shows that this is impossible and that a rod will always be dispersive, at least if it has a traction‐free surface. We show that, with a very restricted, theoretical exception, the least dispersive composite rod is the degenerate one: the one‐material rod.