FISHERY BULLETIN: VOL. 69. NO. 



DISCUSSION 



In all previous studies speed was divided by 

 length then related to tail beat frequency. In 

 our data when speed was converted to body 

 lengths per second the relationship between 

 speed and frequency was nearly identical to that 

 given by Bainbridge (1958) for Carasslus, 

 Leuciscus, and Salmo The confidence intervals 

 for the slopes in the speed-frequency relation- 

 ship in Eiifhynntis and Thmnivs (Yuen, 1966) 

 and in Sarda (Magnuson and Prescott, 1966) 

 overlap the slope in the Bainbridge equation and 

 the ones for Trachurus and Scomber when the 

 body length conversion is used. Thus, when 

 speed is in body lengths per second, the relation- 

 ship between it and frequency is about the same 

 in all fish studied to date from goldfish to mack- 

 erel and is adequately described by the Bain- 

 bridge equation V/L = bf. Thus the Bainbridge 

 equation provides a description of the average 

 relationship for fish in general but little sig- 

 nificance can be attached to specific differences 

 in slope. If more than a rough estimate of 

 speed is required or if specific difterences are 

 important, or if estimates are needed near the 

 minimum swimming speed it would be neces- 

 sary to use the equation developed in this study. 



Bainbridge (1958) concluded from his data 

 that the frequency-speed relationship was curvi- 

 linear below a frequency of about 5 beats sec 

 because fish modulated their tail beat amplitude. 

 His evidence for this conclusion was that in 

 some fish amplitude appeared to decrease at low- 

 er frequencies, and that the distance per beat, 

 calculated by dividing speed by frequency, de- 

 clined at frequencies below 5 beats sec but was 

 constant above that frequency. His evidence 

 for amplitude modulation at low speeds was 

 weak. In the three Salmo studied no trend ex- 

 isted; in Cara,'isius he suggested there might be 

 a decrease in amplitude in one of the two fish 

 studied, and in one of the two Leuciscus studied 

 a trend existed slightly stronger than the one 

 in Camssius. In sum, the evidence for a de- 

 cline in amplitude measurements was based on 

 possible trends in two of the seven fish studied. 

 Two fish could easily give a misleading picture 

 of the general trend in the data, especially when 



the variability in ami)litude measurements are 

 considered. In our .studies we measured the tail 

 beat amplitude in every fish at all possible speed 

 levels and no evidence existed for a consistent 

 change in amj^litude with speed. 



In Bainbridge's data the departure of distance 

 traveled per beat from a constant at low fre- 

 quencies was caused by the division of speed 

 liy frequency. Had the line relating frequency 

 to speed passed through the origin, no bias 

 would have existed but because the line inter- 

 sected the abscissa at about 1 beat/sec division 

 by frequency produced an artificial curvilinear 

 trend at lower frequencies. We produced the 

 same trend in distance per beat in our data 

 by dividing speed by frequency but the trend 

 was eliminated when a correction for the in- 

 tercept was used. Thus the curvilinear trend 

 in distance per beat in Bainbridge's data was an 

 artifact caused by the method of calculation 

 and consequently distance per beat was a con- 

 stant at all frequencies. In addition, the appar- 

 ent nonlinearity below 5 beats/sec in his graphs 

 relating speed divided by length to frequency 

 was also the result of the same intercept problem. 

 Therefore, no evidence exists for consistent 

 amplitude modulation at any speed range and 

 speed appears to be related only/ to tail beat 

 frequency and length in the species studied by 

 Bainbridge (1958) as well as in the ones we 

 studied. We concluded that during steady swim- 

 ming at any speed, tail beat amplitude is a 

 constant proportion of body length of the order 

 of 0.21 L. 



That the mean amplitude during steady swim- 

 ming was constant does not mean that amplitude 

 is not modulated under certain conditions. It 

 is widely known that fish modulate tail beat 

 amplitude when they accelerate (Gray. 1968). 

 Further, we had the impi'ession that some of 

 the variability in the speed-frequency relation- 

 shi]) was caused by diflJ'erences in amplitude. 

 These diff'erences were infrequent and irregular 

 in occurrence and consequently we were not able 

 to evaluate them statistically. We are inclined 

 to believe, however, that fish occasionally made 

 minor adjustments in amplitude and frequency 

 over the entire range of sjieeds, but these adjust- 



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