Values of p or K determined from measiired values of y^ and Y do 

 not apply to a point on either ray but to some point between the two rays. 

 V/here rays diverge greatly over underwater canyons, it is not always 

 possible to construct rays close enough together in deep water to avoid 

 uncertainty in the value of K in shallow water. On the other hand, two 

 rays may approach so closely over an underwater ridge that minor inaccuracies 

 in ray position will introduce gross errors in the measured separation 

 distance Y . 



The difficulties are avoided if the adjacent ray is imagined to be as 

 close as desired to the given ray, i.e., if § is regarded as the ratio 



between two infinitesimal distances Y and '^■ 



With this interpretation 



it is no longer possible or necessary to construct the adjacent ray and to 

 measure y and Yq • The two rays will still diverge or converge according 

 to the Orientation, spacing and curvature of the depth contours in a 

 manner similar to rays which are separated by a finite distance. The de- 

 pendence of p on the ray path and contours has been evolved. 



The present paper illustrates a method for calculating [3, and there- 

 fore K, in terms of distance s measured along the ray and its infinitesimally 

 close neighbor. The basic relationship is a second order differential 

 equation. The coefficients of the eC|uation must be computed from the 

 spacing, orientation, and curvature of the contours traversed by the ray. 

 The problem, then is to calculate the value of p as a function of s along 

 the ray from the differential equation by use of a graphical or numerical 

 method. 



Results from Theory 



It is assumed that the wave 

 velocity, c, is known from the depth 

 contours and that a ray has been con- 

 structed by the crestless method 

 (Johnson, O'Brien, and Isaacfe, 19ii8; 

 Saville and Kaplan, 19^2; Arthur, 

 Miink and Isaacs, 19^2). Let s denote 

 the arc length along the ray and a 

 the angle between the ray and the 

 X-axis (Fig, 2), By proceeding 

 from basic principles of refraction, 

 it may be shown (Munk and Arthur, 

 19^2) that p = p(s) is a solution 

 of the ordinary second order 

 differential equation 



H 



Fig. 2 



^^P^^qP=0, 



(1) 



-3- 



