APPENDIX A 



DERIVATION OF THE EXPRESSION FOR WAVE DIRECTION 



Let the coordinates of nearby gage sites be (x^,y^), i = 1, N; 

 with N = number of gages. The water surface displacement at each site 

 due to the passage of a sinusoidal wave of frequency a, and amplitude 

 A, traveling in direction a (a^ in Fig. 8), is given by: 



n • = A cos{k[x. cos a + y. sin a] - 2-nat - cj)} , (A-1) 



where k = 2ir/L is the wave number, L the wavelength, and ^ the 

 initial phase at the origin. 



The phase difference, ^ij > between locations i and j for the 

 sinusoid considered is: 



^^j = k[(x^ - Xj) cos a + (y^ - yj) sin a] . (A-2) 



Thus, for three noncolinear locations, 



$12 = k[(xi - X2) cos a + (y^ - y2) sin a] , 



■^13 = k[(xi - X3) cos a + (yi - yg) sin a] , (A-3) 



suffice for a unique solution of the direction a. Eliminating first the 

 sin a terms and then the cos a terms, to obtain: 



(Xi - X3) $12 - 



(xi - 



X2) *13 





k[(xi - X3)(yi - y2) 



- C^i 



- X2) (yj - 



■73)] 



(71 - 72) 'f'ls - 



(71 - 



73) $12 





(A- 4) 

 k[(xi - X3)(yi - y2) - (xi - X2)(yi - ys)] 



Since k is always positive, consideration need only be given to the 

 other terms. Letting D stand for the quantity in square brackets, the 

 direction, a, for D / is given by: 



, _1 ([(^1 - X3) *12 - (^1 - X2) *13]/D) 

 a = tan ^ {- > . (A-5) 



([(71 - 72) *13 - (7i - 73) *12]/D| 



A unique value for direction can be obtained by considering the signs of 

 both numerator and denominator. The quantity D differs from zero for 

 all nonlinear arrays as shown below. 



Let xj = yi - 0, D will equal zero for y^l^i - 73/X3; thus, 

 X2, 72 and Xg, y^ will be on a straight line with slope given by 

 this ratio. 



35 



