A longitudinal elastic pulse is generated in a segment of optical fiber. A laser Doppler vibrometer is used to make noncontact measurements of the axial surface velocity at several locations along the optical fiber segment. The measured temporal phase shift and amplitude attenuation of the elastic pulse as it propagates between measurement locations are used to deduce the elastic wave speed and mechanical loss factor for the optical fiber. The knowledge of these properties is important to the design of optical sensors for dynamic phenomena such as vibrations and acoustic pressure. One potential application of the method is nondestructive testing for delamination between an optical fiber's glass cladding and plastic jacket.