FISHERY BULLETIN: VOL. 80, NO. 4 



The momentum carried in the vortex sheet 

 will contribute to instantaneous thrust, and if 

 there is no downstream fin to influence the flow, 

 this momentum will contribute to the mean 

 thrust and power of the fish (Wu 1971; Newman 

 and Wu 1973). However, when there is a down- 

 stream re-entrant fin (i.e., a second downstream 

 median fin spanning the flow from the anterior 

 fin) the vortex sheet will impinge on the leading 

 edge of that fin. If the gap between the fins is 

 small, there is little difference between the mo- 

 tion of the incident vortex sheet and the motion of 

 the leading edge of the downstream fin. Then the 

 mean strength of the vortex sheet shed by the 

 downstream fin is determined by the motion of 

 that fin, with no significant contribution from 

 the upstream vortex sheet from the anterior fin, 

 i.e., the upstream fin has no effect on the wake 

 eventually shed by the fish. In this case, the inter- 

 action between median fins does not influence 

 mean thrust. 



Lighthill (1970) showed that a different situa- 

 tion can occur when the gap between median fins 

 is large. Under these circumstances, there may 

 be a large enough phase difference between the 

 motion of the incident vortex sheet and the down- 

 stream fin, so that the momentum shed upstream 

 is not annihilated by the second fin. Then, the 

 work done by the anterior fin against the momen- 

 tum shed by its trailing edge together with that 

 due to an increased incident velocity at the down- 

 stream fin increase total power output and im- 

 prove efficiency (Lighthill 1975:80-84; Sparen- 

 berg and Wiersma 1975). 



The phase difference in the motion of the trail- 

 ing edge of one fin located at a position ai, along 

 the body, and the leading edge of a second more 

 posterior fin at position a 2 , is 27r(a 2 — ai)/X where 

 \ is the length of the propulsive wave. However, 

 the vortex sheet travels downstream at the mean 

 speed, U, of the fish, while the body undulation 

 travels backwards at a speed c, greater than U. 

 Therefore, the phase difference, 4>, in the motions 

 of the vortex sheet shed by the anterior fin and 

 the leading edge of a posterior fin is given by 

 (Lighthill 1975, equation 28): 



♦—("KM- 



(1) 



Sharks typically have three median fins, the 

 first and second dorsal fins and the caudal fin. 

 Thomson and Simanek (1977) have analyzed sev- 

 eral morphological features of 56 species of 



sharks and show that the second dorsal fin is 

 characteristically small compared with the first 

 dorsal fin, especially in pelagic species. In addi- 

 tion, the second dorsal fin would only be partly 

 re-entrant to most of the vortex sheet shed by the 

 upstream fin because of the posterior taper of the 

 body. Therefore, it seems likely that the second 

 dorsal fin has relatively little effect on the flow 

 between the other two fins during steady cruis- 

 ing. Thomson and Simanek's observations also 

 indicate that the caudal fin depth is typically 

 greater than or equal to that of the trailing edge 

 of the first dorsal fin, as required to maximize 

 the interaction. Therefore, <£ was calculated for 

 interactions between the first dorsal fin and the 

 caudal fin of the blacktip, bonnethead, and leop- 

 ard sharks (Table 2; Fig. 7). 4> was close to, or 

 >0.57r, as required for the interaction hypothe- 

 sized by Lighthill (1970). A single record for the 

 dogfish, Acanthias vulgaris, in Gray (1933) also 

 gives a value of <f> = 0.52tt (Webb 1975). For 



0-9 

 0-8 



0-7 



9 0-6|- 

 6 



S °' 5 ~ 



ta 0-4- 

 0-3- 

 0-2 



0-1 



4- 



:.+ 



_L 



12 3 4 



U/L- SPECIFIC SWIMMING SPEED (L.s -1 ) 



Figure 7.— The relationship between the phase difference 

 <j> (see Equation (1)) and specific swimming speed for three 

 species of sharks. Vertical and horizontal bars are ±2SE. The 

 key to symbols is in Figure 1. 



Table 2.— Separation, (a 2 - «i)/L, between the 

 trailing edge of the first dorsal fin (aO and the 

 mean position of the leading edge of the caudal 

 fin («2>, and phase difference ($) between their 

 movements for three species of sharks. <t> was 

 calculated from data in this table and in Figures 

 3 and 4 using Equation (1). 



Species 



(a 2 -a,)/L 



<t> (X±2SE) 

 (n radians) 



Leopard shark 0.48 0.55±0.20 



Bonnethead shark 0.50 0.48±0.08 



Blacktip shark 0.47 0.51 ±0.05 



810 



