where 



p(s) = -(cos 



q(s) = (sin 



■iii\- 



■[i i} 



(sin 



4^i]' 



2(sin a cos 



(2a,b) 



a)\l ^1 . (cos^a^i -^1 , 

 Lc ax 3J.J ^ L" ^' J 



and where the symbol D/tis denotes differentiation with respect to the arc 

 length s along the ray.. In general, p and q are not known analytically, but 

 thejr can be computed at discrete points along' the ray if the depth contours 

 are labelled in terms of wave velocity c. Equations (2a,b) show that p 

 and q depend upon a and c and the first and second derivatives of c, i.e., 

 upon the properties of the contours mentioned previously. After p and q 

 are de'termined, an approximate solution for p along the ray may then be 

 obtained from (l) by a method due to Kelvin. 



Determination of Coefficients p and q 



Values of p and q at any point P on the ray in terms of a local co- 

 ordinate system (x ,y ) oriented with respect to the contour at P (Fig. 2) 



are (Munk and Arthur, 19?2) 



p(s) = -(sin 



q(s) = (cos' 



■■>&*-] 



2 (sin a' cos 



(2c, d) 



- (sin 



4^%]' 



where Kcis the curvature of the contour at point P. The curvature is 

 positive in sign when the contour is concave as viewed from the positive 

 end of the y -axis. Thus, the curvature of the contour at point P is 

 positive in " Figure 2. If derivatives are conputed from finite differences 

 and ii" it is assumed that contours in the vicinity of P have constant 

 curvature and centers of curvature on the y -axis, then the approximate 

 values are 



(sin a') 



(cos a') 



+ b 

 - c 



■] 



+ b 



K 



(3a,b) 



cP 



sin a' 



In (3a,b), the quantities a and b denote the distances along the y -axis 



-h- 



