The usual solutions describing the propagation of continuous waves in solid plates and cylinders fail to predict the observed velocity and attenuation, and secondary signals, in the propagation of a narrow‐band pulse of compressional waves of carrier frequency of the order of 10 Mc/sec. It is shown that all these factors may be accounted for if a slightly different form of solution is chosen, to fit more closely the conditions under which the propagation takes place. Both plates and cylinders are treated theoretically, and methods of obtaining approximate solutions of the modified frequency equations are discussed.The results of this analysis agree closely with those derived by other, less exact methods, which commence with the assumption that the solid medium behaves very similarly to a fluid.The relation of this work to the propagation of pulses at low frequencies is also discussed.A method of analysis similar to that employed here is also necessary to predict experimental results in the propagation of pulses in multilayered wave guides of any configuration, mechanical or electromagnetic.