from point P to adjacent contours >7ith wave velocities c and c, , 

 ly (see Fig. 2). a b 



respective- 



If the contours are straight, the second term on the right-hand side 

 of (3b) is zero {X,c ~ ^^ a^^ ^ ^ varies linearly along y ,the first 

 term is zero. The curvature of the contour is approximated from a measure- 

 ment of the angle, h0 , included by two tangents to the contour (Fig, 3), 

 and of the length, m, of the chord connecting the points of tangency, since 



Ke* 



A(2f 

 m/2 



ih) 



The derivation of {h.) assumes that 

 the tangents include a section of 

 the contomr of constant curvature. 



By application of (3ajb) and 

 (Li), the values of p and q at 

 points of intersection of the ray 

 and successive contours are ob- 

 tained. 



Solution for p by Kelvin's Method 



The integral curve p = p(s) 

 for (1) can be constructed in a p, 

 s-diagram, where the arc length, 

 s, measured along the ray is laid 

 out rectilinearly (Fig, h) . The 



U 



tangents 



Fig. 3 



basis of Kelvin's method (VJillers, 19li8) is the assumption that over small 

 intervals of s the integral curve p = p(s) has constant curvature. If 

 the angle beti^een the integral curve and the s-axLs is Q , then the 

 of the curve is Dp/Ds = tan , and the curvature, using (l),.is 

 2 2 

 .^ - ^ ^^^ 3/2 = - (P tan e + qp) cos^Q 



K.= 



[l + (Dp/Ds)^]' 



slope 



(5) 



The construction is begun at a point on the ray where p and Dp/i)s are 

 known. Usually this is a point in deep water, aay where the ratio of depth 

 h to deep water wave length L is h/L = 0,5. Along the ray to this point 

 the effects of refraction are usually negligible, so that p = 1 and dp/t)s = 0. 

 The construction may be started at any point, havsevev, if the value of p 

 and the curvature, /<^ , of the wave crest are known since 



p Ds '^ • 



(6) 



It is convenient to take s = at the initial point. The values of 

 coefficients p and q are computed at the initial point by the method of the 

 previous section using (3a,b). The value of (Ko) at the initial point 



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