NOTE Salvado et al.: Dolphin detection avoidance 



419 



arctan 



Vd\ sin a 



Vb l-(VDA^B)cosa 



(7) 



Next we investigate whether Eq. (2) vanishes non- 

 trivially. Computing the derivative with respect to a 

 of Eq. (6), we get 



da ^ VpteLcos g-CVp/VB)] 

 da [l-(VD/VB)cosa]2 



which vanishes only if 



/Vol 

 arccos — = a a. 



Vb 



(8) 



(9) 



Noting the similarity between a and p given respec- 

 tively in Eq. (6) and (7), we can easily and simply ex- 

 press one in terms of the other. Writing 



p = arctan 



V^t 



B^-c 



(10 



the derivative of p with respect to a is given by 



dp cos^ p da 

 da Vstc da 



(11) 



Substituting into Eq. (2) the equivalence of Eq. (11), 

 we find that 



dr 

 da 



= cos p 



1 - 



(a-i-f) sin p cos p 



da 

 d^' 



(12) 



which allows us to conclude that r^jn is extreme at the 

 point a = ao because, as we found in Eq. (8), a is ex- 

 treme at that point. However, as can be appreciated 

 in Eq. (12), there may be other points where rn^^ is ex- 

 treme. We now investigate whether r^^jn is extreme 

 for values of a other than ag. 

 Other extreme values of r^jn may be attained if 



or 



cos /? = 

 (a-i-f) sin p cos p = VBte 



(13) 

 (14) 



are physically realizable. Eq. (13) is satisfied for p = 

 nn/2 (n= 1,3,5,...), and of these only p = n/2 concerns 

 us. It corresponds to the upper limit of the physically 

 realizable range of 0^p<n/2, and represents the un- 

 interesting case Vd = Vb (i.e., ao = 0) which is the 

 trivial special case of this problem: r^^^ is rendered 

 constant when Vd = Vb (i.e., the dolphin swims with 



the same velocity as the vessel), so the vanishing of Eq. 

 (2) is trivially satisfied. 



In Eq. (14) let f=0. This limit will not diminish the 

 generality of the solution. The direction a dolphin 

 should take to maximize its distance to the vessel at 

 the point of closest approach will not depend on how 

 close the dolphin is initially to the projected path of the 

 vessel. So without loss of generality we investigate if 

 there is a physically realizable angle p such that 



a sin p cos p = VBte 



(15) 



is satisfied. With the result from Eq. (10) and the iden- 

 tity tan p = sin p cos" ' p, Eq. (15) can be expressed as 

 the condition sin /3 = ± 1 which is satisfied by the same 

 uninteresting case that satisfies Eq. (13). Therefore, 

 we can conclude that the nontrivial extreme value 

 attained by rmin at a = ao is unique in the interval 

 0<a<7i. 



We only have left to show that the extreme value at- 

 tained by r^in at a = ao is a maximum. Let ao and pQ 

 be the respective values of a and ^ at a = ao . The sec- 

 ond derivative of r^jn with respect to a evaluated at 

 a = ao is given by 



d^r, 



min 



da^ 



"0 



(16) 



cos Po 



1 - 



ao sin /?o cos ^o 



VBt 



d^a 

 da2 



"o" 



Because the second derivative of a with respect to a 

 evaluated at a = ao is 



d^a 

 da2 



-Vnt 



Dl-c 



[1-(Vd/Vb)2]3/2 



<o, 



(17) 



to determine whether Eq. (16) is negative we must 

 show that 



ao sin p cos Pq < VBte 



(18) 



We have shown already that Eq. (15) can only be sat- 

 isfied for an angle /J = n/2. Then Eq. (18) is satisfied for 

 0</}<n/2. We have seen already that /? = n/2 when a = 0, 

 so da/da = at that point also. Therefore, we can also 

 conclude that the nontrivial extreme value achieved by 

 r^in at a = ao, unique in the interval 0<a<n, is a max- 

 imum. Because Eq. (2) vanishes term by term, the same 

 result is achieved by finding the extreme value with 

 respect to a of either /3 or a. 



In conclusion, a dolphin escaping at speed Vp at an 

 angle a relative to the velocity Vb of an approaching 

 vessel will maximize its distance to the vessel at the 

 point of closest approach if a = arccos (Vd/Vb). 



