from one interval to the next. Decreasing the interval, as was done in 

 the exairple for c/c<0,5, ajd interpolating contottrs, if necessary, will 

 inprove the acciiracy. 



Since exact values of p and q from (9a,b) were used in the exanple, 

 there is no irdication of the effect of inaccuracies introduced by (3a, b) 

 and (ii). Some indication of the effect of errors in p and q may be ob- 

 tained from (11). If, for c/c^ = 0.2?, the values of p and q differ by 

 10 percent and l5 percent, respectively, from values (9a,b), the value 

 of K changes by less than 1? percent. 



The calculation of refraction factor along a ray has an advantage 

 when the depth and wave velocity are expressed analytically, because any 

 ddsired degree of precision xasQr be achieved. This is not the case if the 

 usual method of constructing an adjacent ray and measuring the separation 

 distance is utilized. As mentioned previously, the latter irethod permits 

 determination only of a sort of "average" value of K appropriate to the 

 interval between the rays. There is always a physical limitation to the 

 proximity at which an adjacent ray may be constructed. In Figure 5, it 

 does not seem useful to continue the adjacent ray farther, because it has 

 reached a region where the curvature and orientation of the contours is 

 entirely different from along the axial ray. 



The calculation of K along a ray is more time consuming than the usual 

 method involving construction of an adjacent ray. For this reason, the 

 usual method would appear to be preferable for complex underwater topography 

 where the depth is not known analytically but only from bat hyme trie charts. 

 An exception occurs where rays converge or diverge rapidly as over under- 

 water ridges or troughs, for under these conditions values of K from the 

 usual method may differ greatly from precise values which apply to a point 

 on a ray. It should be mentioned that it is just these circumstances for 

 which the applicability of ray theory and refraction factors is most 

 questionable. However, the present remarks have been concerned entirely 

 with the determination of refraction factors rather than their applicability. 

 Values of K are correctly calculated by the present method even in the 

 region of a caustic, but the breakdown of the approximations of ray theory 

 prevents valid determinations of wave height from refraction factors under 

 these circumstances. 



-11- 



