A hypothetical case of refraction over straight, parallel contours is 

 shown in Figure 4-» The parametric equations for the orthogonal given by 

 Arthur and Munk (3) are 



^-.= c cosfi; ^^csjn (3; ^^sm(3^ ~ cos (3 ^ 

 dt c^i ^^ Ox dy 



These may be solved by simple separation of variables for a velocity field 

 which is a function of y alone o (Here ^=90°—Or. ) , In particular, for a 

 field Yiiich varies linearly with y, c ■■= co(l -- ay) the solutions for x and 

 y are 



x= — — fcos or- cos (X„ ) 3f?o/ \/= — -^~— . fsin a - sin oc) 



which are the parametric equations of a circle of radius 



The solution for y may be put in the form 



^/r? a- sin (a^-/\a)=fl~ t™ j sm a^ 



° '' which is Snell's law« 



From these, exact values of Aa and x at any point in the field may be 

 found <, 



The values for the particular case chosen are compared with values 

 obtained from Isaacs' equations and the ones contained herein* This cchu- 

 parison is shown in Figure 4.5 and also in Table 1, where both values of the 

 change between contours and cumiolative values are given^ It may be seen 

 that angular values obtained from, the new equations agree aljnost exactly 

 with those determined from Snell's law, and are considerably more precise 

 than those obtained from the original Isaacs' equation ^ Distances along 

 the contoiirs are still somewhat in error if the raid-contour is used as the 

 turning point (as in the Isaacs' case), though not nearly as greatly in 

 error as obtained from the Isaacs' method (see column S-K-I). If the 

 turning point is selected so that the segments of the tangent lines within 

 the contours ar-e equals then the result is almost identical with Shell's 

 Lawo This is shown under "S-ICo 



It is concluded that these new protractors, or overlays represent 



a considerable advance over the original Isaacs' protractor, in 

 that they approach much more closely exact values» 



It may be noted that a paper by Arthtur^^ Munk and Isaacs recently 

 presented at the May 5 meeting of the American Geophysical Uniony but 

 not yet publisbedj has considered the equations of the wave rays, or 

 orthogonalSo Solutions of these equations using essentially the same 

 assumptions as used in the geometrical derivation terein, give for 



Aa = sin j(/ -h jJ^\ sm aT~0(^ sin "V/^-M ) s/n al - a 



32 



