Savitsky 



Tr'=T^+T: (7) 



The wave length for waves progressing in the same direction as the 

 current is thus: 



^ 



fl Wl ^4(V^ ZC^T 



(8) 



It can be seen that the effect of a following current is to increase the 

 wave length relative to the water. 



The analytical solution for the refraction of waves traveling 

 through a finite current field is obtained by application of Fermat's 

 principle that waves will travel in a path such that the travel time is 

 a minimum. Applying the method of calculus of varlatl'^ns will lead 

 to a time history of the path of individual wave rays passing through 

 the current. For the purposes of this analysis, It will be assumed 

 that the wake properties do not vary with time. This Is a reasonable 

 assumption since it has been demonstrated that, for a grid- speed of 

 1 ft/sec, the mean wake-velocity Is an order of magnitude less than 

 the wave speeds. Mathematically, the problem Is to determine the 

 minimum time path of a given wave ray through a current region de- 

 fined by a position dependent velocity vector. The magnitude and 

 direction of the current are known as fianctlons of position (Eq, 2). 

 The magnitude of the wave crest velocity relative to the water Is C, 

 given by Eq. (6). The problem is to determine the path of a wave 

 ray such as to mlnlnnlze the time necessary to travel from point A 

 to a point B. Analogous optimization problems for dynamic systems 

 are described by Bryson and Ho [ 1969] . 



The equations of motion are: 



x(t) = - V^(x,y) - C(x,y,a) cos a 



(9) 

 y(t) = - CCx.y,^) sin a 



with initial conditions 



x(0) = Xq 



(10) 



y(0) = yo 



and at end of computation 



x(t^) = X, 



426 



