HUNTER and ZWEIFEL: SWIMMING SPEED AND TAIL BEAT FREQUENCY 



ments were merely individual deviations from 

 the general relationship we have described. 



We do not wish to detract from the original 

 and important contribution of Bainbridge 

 (1958), by emphasis on the differences between 

 his and our conclusions. His basic conclusions 

 and equations were not greatly different from 

 our own. We were able to examine more closely 

 the form of the relationships he described be- 

 cause of a larger sample size made possible by 

 the availability of automatic film analysis equip- 

 ment and because of the existence of his data 

 in the literature. 



The question of species-specific size effects re- 

 mains unresolved. In our general model a good 

 fit was obtained in seven species when the min- 

 imum stalling speed was proportional to L~^^, the 

 frequency at this minimum speed was propor- 

 tional to L-'^^ and the slope coefficient was pro- 

 portional to W. A com])arative study on speed- 

 related size effects in fishes would certainly be 

 of value. 



It also remains to be resolved whether or not 

 it was appropriate to apply the minimum swim- 

 ming speed equation developed by Magnuson 

 (1970) for Euthijnnus affinis, a fish that lacks 

 a swim bladder, to such a broad assortment of 

 species. The equation implies a functional re- 

 lationship between minimum speed and hydro- 

 static equilibrium and implies existence of neg- 

 ative buoyancy at minimum speeds. We do not 

 know if these relationships exist in all species; 

 nevertheless his equation did provide a reason- 

 able estimate for minimum speed and it func- 

 tioned well in our equation. 



The relationship between swimming speed 

 and tail beat frequency we have described could 

 be used in any application where it is necessary 

 to measure swimming speeds of fish. For ex- 

 ample, a sonic internal tag could be developed 

 that telemetered tail beat frequency and thus 

 the speed of free-swimming fish could be mon- 

 itored continuously over extended periods. 



The tail beat frequency-speed relationship 

 could be used for size or species identification 

 using Continuous Transmission Frequency Mod- 

 ulated sonar as suggested by Hester (1967). 

 The increase of speed with frequency (our K 



value) varied from species to species and thus 

 might be used for identification. If size were 

 known, the minimum observed velocity would 

 provide additional information for identification. 

 Alternatively, if the species were known, min- 

 imum (or maximum) speed would provide an 

 indication of size. The equation could also be 

 used to estimate size from tail beat amplitude, 

 but caution should be exercised because in our 

 study amplitude was not modulated and conse- 

 quently, we do not know whether or not speed 

 and tail beat amplitude are linearly related 

 within an individual. 



ACKNOWLEDGMENTS 



We wish to thank David Holts and George 

 W. Rommel of the NMFS Fishery-Oceanography 

 Center, La Jolla, Calif., who assisted in con- 

 ducting the experiments and in calibration of 

 the apparatus. Albert Good wrote the program 

 for digitizing the photographic data, and John 

 J. Magnuson, Department of Zoology, University 

 of Wisconsin, Madison, Wis., reviewed the man- 

 uscript. 



LITERATURE CITED 



Alexander, R. M. 



1959a. The densities of Cyprinidae. J. Exp. Biol. 



36: 333-340. 

 1959b. The physical properties of the swimblad- 

 ders of fish other than Cypriniformes. J. Exp. 

 Biol. 36: 347-355. 

 Bainbridge, R. 



1958. The speed of swimming of fish as related to 

 size and to the frequency and amplitude of the 

 tail beat. J. Exp. Biol. 35: 109-133. 

 1960. Speed and stamina in three fish. J. Exp. 

 Biol. 37: 129-153. 

 Beamish, F. W. H. 



1966. Swimming endurance of some northwest At- 

 lantic fishes. J. Fish. Res. Bd. Can. 23: 341-347. 

 Conway, G. R., N. R. Glass, and J C. Wilcox. 



1970. Fitting nonlinear models to biological data by 

 Marquardt's algorithm. Ecology 51 : 503-507. 



FlERSTINE, H. L., and V. WALTERS. 



1968. studies in locomotion and anatomy of scom- 

 broid fishes. Mem. S. Calif. Acad. Sci. 6: 1-29. 

 Gray, J. 



1968. Animal locomotion. Weidenfeld, London, 

 479 p. 



265 



