It is hypothesized that steady anguilliform swimming motion of aquatic animals is purely reactive such that no net vortex wake is left downstream.This is versus carangiform and tunniform swimming of fish,where vortex streams are shed from tail,fins,and body.But there the animal movements are such to produce partial vortex cancellation downstream in maximizing propulsive efficiency.In anguilliform swimming characteristic of the eel family,it is argued that the swimming motions are configured by the animal such that vortex shedding does not occur at all. However,the propulsive thrust in this case is higher order in the motion amplitude,so that relatively large coils are needed to produce relatively small thrust; the speeds of anguilliform swimmers are less than the carangiform and tunniform,which develop first order thrusts via lifting processes.Results of experimentation on live lamprey are compared to theoretical prediction which assumes the no-wake hypothesis.Two-dimensional analysis is first performed to set the concept.This is followed by three-dimensional analysis using slender-body theory. Slender-body theory has been applied by others in studying anguilliform swimming,as it is ideally suited to the geometry of the lamprey and other eel-like animals.The agreement between this new approach based on the hypothesis of wakeless swimming and the experiments is remarkably good in spite of the physical complexities.