46 



RAY ACOUSTICS 



the projector be ao,/3o,7o, as in Figure 4. Since n de- 

 pends only on y, equations (22) simplify to 

 d(na) „ d(nP) dn d{ny) 

 dy 



= 



= 0. (23) 



con- 



ds ' ds dy ' ds 



Thus, along any itidividual ray we have na 

 stant, ny = constant, which in turn impHes 



T 



-= constant = k 



a 



along the ray. Then, the initial direction of the ray 



is (ao,/3o,Kao). 



'Vq. A.^i 



Figure 4. Change of ray direction between point 

 (d, 0, ) and point {x, y, z). 



The direction at a general point P along the ray 

 will be characterized by the direction cosines a,0,Ka 

 because of the equations (23). It can easily be shown 

 by the methods of analytical geometry that the nor- 

 mal to the plane determined by OF (direction cosines 

 0,1,0) and OA (direction cosines ao,/3o,Kao) has the 

 direction cosines k/V k^ -|- 1, 0, — 1/V k^ + 1. The 

 direction of the ray at P is characterized by the direc- 

 tion cosines a,fi,Ka; thus the ray direction at P is 

 perpendicular to the normal to the plane AOY; hence 

 the segment PB lies in that plane. Since P was any 

 point on the ray, the entire ray must lie in the plane 

 AOY. 



3.3 RAY PATHS FOR VERTICAL 



VELOCITY GRADIENTS 



3.3.1 Derivation of the Equations 

 of Ray Paths 



We now solve the equations (22) for the special 

 case where the sound velocity depends only on the 

 vertical depth coordinate y and discuss this solution 



in detail. It is intuitively obvious that if we carry 

 through the solution for the xy plane, then the ray 

 pattern in any other plane through the vertical (y) 

 axis will be identical in size and shape. 



Since the water depth increases in the downward 

 direction, we shall take the y axis positive downward. 

 We shall denote the angle which a direction in the xy 

 plane makes with the positive x direction by d, as in 

 Figure 5. To avoid ambiguity, we must specify care- 



FiGUBE 5. Change in ray direction over ray element 

 PP'. 



fully the sign of the angle d. We shall be concerned 

 only with rays moving in the direction of increasing 

 X, in other words, to the right in the figure. If the ray 

 is gaining depth with increasing range, we give the 

 angle d a positive sign; while if the ray is losing depth 

 with increasing range, we give 6 a negative sign. These 

 conventions, illustrated in Figure 6, enable us to use 



DESCENDING RAY^ 

 ©POSITIVE 



z 



CLIMBING RAY 

 NEGATIVE 



Figure 6. Conventions fixing sign of B. 



the following relations both for climbing and de- 

 scending rays: 



a = cosd ; ;8 = sin ; 7 = 0. (24) 



Since the sound velocity is assumed to depend 

 only on y, we have 



dn dn 



dx 



dz 



