Newman 



or from the dispersion relation* 



k = (g/V^) sec^ e. (5) 



Finally, if the ship's velocity V is positive, it follows from (4) (or 

 from an obvious physical argument) that the wave direction must 

 lie in the interval - ir/2 :^ G ^ ir/2. Thus we arrive at the "free- 

 wave" description of the ship-wave system 



C(x,y) = r' de A(e)e'<«/^'^^«^'^^''^°^^*y^*"^^ (6) 



which is the starting point for many analyses of wave resistance. 



Kelvin's ship- wave pattern may be obtained from (6) by noting 

 that If the polar radius R = (x^ + y2)'/2 is large compared to the 

 typical wavelength 2irV^/g, then from the method of stationary phase 

 the dominant contributions to (6) will arise from those angles 

 where the phase function 



(g/V^) sec^ (x cos + y sin 0) (7) 



Is stationary, or 



~ sec^ (x cos + y sin 0) = 0. (8) 



Carrying out the indicated differentiation, it follows that 



X tan + y{2 sec^ - 1) = 0, 

 or that the significant ship waves will be situated at points such that 



I / I sin cos in\ 



- ly/^l = 2 - cos''^ • ^^^ 



The behavior of this function is indicated in Fig. 1 , and the essential 

 features of a Kelvin system are immediately clear: 



1. The waves are confined to a sector |y/x| < 8 ~ 

 tan 19°28'. 



2» On the boundaries of this sector, or cusp line , the waves 

 are oriented at an angle |0| = cot'' 2''' =" 35^16 . 



522 



