Example: An analytic exairple has been selected in oi"der that the 

 approximate method of solution for p outlined above may be conpared to an 

 exact solution. Assume an underwater trough with straight axis (Fig. ^,. 

 lox>rer) along which 



p = 0.1/L^; q = -O.IA/ , (9a,b) 



where L is the deep water wave length. The straight axis is obviously 

 a ray, ° along which p =? p(s) may be determined. Actually, the dinension- 

 less distance s/L is used. From (9a,b) and (2c, d), it may be shown 

 that along the axial ray 



c/c^ = e"°-^^^o . Kc = -lA^ (10a,b) 



where c is the deep water wave velocity. The contours (dashed lines in 

 Fig. 5)° have been dratm by extending tangents from the ends of circular 

 arcs each of radius L and each subtending an angle of 60°. Values of 

 c/c and corresoonding values of relative depth h/L are indicated. 



Successive values of AO and Ap have been calculated along the axial 

 ray using Kelvin's method as a basis. Values of the variables are shown 

 in Table 1, The resulting points P,s/L have been plotted (Fig, 5, upper), 

 and a smooth curve drawn through them. The constant values of p and q 

 given in (9a,b) have been used in the computation. However, determina- 

 tions of p and q have also been made using (3a,b) and (it), and the values 

 were within a few percent of the exact values (9a,b). 



For comparison, a second ray has been constructed by the crestless 

 method as close to the axial ray as the scale of the enlargement (L =1 

 inch) permitted. The second ray has not been drawn beyond the point 

 where the contours become straight. The value of K coitputed from measure- 

 ments of the separation of the two rays at c/c = 1.0 and at c/c = 0,^ is 

 0,I).8, which is to be coitpared to the exact value 0,5l, 



The exact solution of (1) along the straight axial ray^with constant 

 values (9a, b) of p and q is well knoxm and, if the initial conditions 

 |3o= 1, (Dp/3s)^ = are used, tal«s the form (Munk and Arthur, 1952) 



p = e"P^/2[cosh (Xs) + ^ sinh (Xs)] , (ll) 



where p/2 = 0.05/L and X = (p/2)2 - q = 0.1025Aq • Exact values of K 

 for certain values of s are tabulated. 



Discussion 



The exairrole indicates the accuracy which may be expected in the 

 determination of refraction factor, K , along a single ray by Kelvin's 

 method. The accuracy, of course, decreases over those sections of the ray 

 for which there is large change in curvature, /^o, of the curve p = p(s) 



