km/sec 



FIG. 1. Canonical sound channel (left) and the corresponding rays for 6 = 12. T (surface limited), 5. 2°, and 0° (axial ray). The 

 contribution to Fj from various parts along the three ray paths is indicated by the vertical extent of the shaded band (plotted loga- 

 rithmically). F] is plotted separately at the bottom of the figure for the surface limited ray, together with F^ and Fj — ReGi, thus 

 indicating the relative apex contributions toward mean-square phase, rate of phase, and intensity. (F, — ReG, applies only to a 

 source at .r = of a receiver at R*. ) 



R'' = R^ + R' = -nB(. 



■1/2 



(1 



^hi^' 



■), 



where 0^ = (C- U)/((.C), and the "optical" path length 

 equals 



We will require 



5 = i?'" 



d(R'' 



■12€(1- 



(86) 



Finally, we can (laboriously) compute A'J for a com- 

 plete loop. We find 



A-'- 



ZB 



■ 4 



/27^'LV 6 ' 3 <t>j 

 2 i,H^\- »') / , 0Y 



'2/2 



It will also be of interest to have the value of this quan- 

 tity at the apex of a ray. This is 



-3B 



4 /27 



1 \ (, 4>' ^^V 



For upward rays, A'^, = x near x = Q and A'^, =R* - x near 

 x = R* (Fig. 2), and A]^, has zeros at the caustics of rays 

 propagating to the right from x = and to the left from 



km 



FIG. 2. j4„ for a 5.2° upward loop (top) and downward loop 

 (bottom), with tte ±5.2° ray itself shown in the center, all 

 plotted as functions of horizontal distance .r. 



222 



