HUNTER and ZWEIFEL: SWIMMING SPEED AND TAIL BEAT FREQUENCY 



was linear throughout the rang-e of test speeds, 

 but the slope and intercept of the regression 

 lines varied with fish length (Figure 4, Table 1) . 



10 15 



FREQUENCY (tail beafs/sec) 



Figure 4. — Relationship between speed and tail beat fre- 

 quency for six Trachurus, 4.5 to 27.0 cm total length. 

 Equations for regression lines shown in figure are given 

 in Table 1. 



Table 1. — Standard deviation (s„r), intercept (a), and 

 slope (b) for regression of speed (cm/sec) on tail beat 

 frequency and slope and intercept for regression of 

 speed/length on frequency for Trachurus. 



The length-dependent differences in intercept 

 were probably a function of differences in min- 

 imum speed and minimum tail beat frequency. 

 Fish have a minimum tail beat frequency and 

 a minimum swimming speed below which they 

 cannot swim by movement of the caudal fin and 

 these minima were a function of body length. 



In the past, speed was scaled directly to length; 

 that is, speed was divided by length and re- 

 gressed on frequency (Bainbridge, 1958; Mag- 

 nuson and Prescott, 1966; Yuen, 1966). Our 

 data suggest, however, that division of speed 

 by length would introduce bias because of the 

 existence of a minimum speed and tail beat 

 frequency different from zero, the dependence 

 of minimum speed on length, and possible length- 

 dependent differences in the slope of the regres- 

 sion of speed on frequency. For example, when 

 we divided speed by length, size-dependent dif- 

 ferences in intercept and possibly the slojie still 

 existed (Table 1, last two columns) . In addition 

 a curvilinear trend is introduced at low speeds 

 in the combined data because of the lack of an 

 intercept (minimum speed) correction. Thus 

 an equation that relates speed to tail beat fre- 

 quency for fish of different length must include 

 an adjustment for size-dependent differences in 

 minimum swimming speed, minimum tail beat 

 frequency, and perhaps also for size-dependent 

 differences in the slope coefficient. 



The existence of size-dependent variables in- 

 troduces certain problems in the interpretation 

 of these data because of the possibility that spe- 

 cific differences in size dependency may exist. 

 For example, differences exist among species 

 in the coeflicients used to relate size to vaiious 

 swimming characteristics such as burst and sus- 

 tained speeds (Bainbridge, 1960) but it is un- 

 certain whether or not these differences reflect 

 real specific differences or if they are simply 

 differing estimates of a common coefficient. 

 Owing to the great variability inherent in swim- 

 ming speed studies and because of the sensitivity 

 of the length coefficients to the size range of an- 

 imals in the sample, these two alternatives are 

 equally plausible. In addition, specific differ- 

 ences in the relationship between size and swim- 

 ming functions may also depend on the partic- 

 ular function considered (Bainbridge, 1960). 

 For example, the coefficient relating size to max- 

 imum sustained speed may be different from 

 the one that relates size to beat frequency or 

 burst speed. Thus, species may differ from one 

 another in the way each swimming function is 

 related to size. If such specific differences exist 



257 



