The phase delay of a continuous sinusoidal elastic wave after transmission through a medium may be used to determine the velocity of propagation of the wave in the medium. The change in path length for a given frequency, or the change in frequency for a given path length, required to change the phase delay by integral multiples of 360 degrees is measured in the laboratory by the use of source and receiver piezoelectric transducers whose signals are applied to the horizontal and vertical deflection circuits of an oscilloscope. The accuracy of the method depends upon the accuracy with which the frequency of the transmitted wave and its path length through the medium (or change in path length) can be determined, provided the effect of extraneous signals (e.g., boundary reflections, multiple reflections, alternate modes of propagation, etc.) is negligible.The phase-delay methods are illustrated and compared with conventional pulse methods by using both to make compressional-velocity measurements in water and compressional- and shear-velocity measurements in a high velocity basalt and in a low velocity dried mud sample. The results of the two methods agree to within a few percent.It is suggested that these phase-delay methods may be especially well-suited for making elastic-wave velocity measurements in media with high attenuation of the waves propagated in them.