RAY PATHS FOR VERTICAL VELOCITY GRADIENTS 



47 



and by reason of relations (24) the equations (22) re- 

 duce to 



d{n cos 9) 



0; 



d{n sin 9) 



(In 

 dij 



(25) 



ds dts 



From the first equations it follows that n cos 9 has 

 a constant value along a particular single ray. That 

 is, if P and P' are two points on the ray, then 



— cos 6 = — cos 6 . 

 c c 



If, in particular, P is located at the depth where 

 c{y) = Co, and if 9(, is the direction of the ray at this 

 point, this equation becomes 



^^ = ^^i. (26) 



cos do Co n 



Equation (26) is identical in form with Snell's law in 

 optics. 



The second equation in (24) is used to compute the 

 curvature of the ray at any point. The curvature of a 

 curve at a point on it is defined as d9/ds, the angle 

 through which the tangent turns as one travels along 

 the curve for unit distance. Because of our conven- 

 tions for the sign of the direction angle 6, upward 

 bending is always associated with negative curvature, 

 and downward bending with positive curvature. 



From the relations (25), we have 



dn d{sm 6) . dn 



— = n — h sm 9— 



dy ds ds 



d(sin 9) d9 . dn dy 



= n ; -)- sm e— — 



d9 ds dy ds 



d9 . dn 



= n cos $— -\- sm^ 9 — 



ds dy 



(27) 



since dy/ds = sin 6, from Figure 5. The solution of 

 equation (27) for d9/ds yields 



d(log n) 



de Idn 



— = - — cos 9 



ds ndy 



dy 



cos 0. 



(28) 



Since log n = log Co — log c, equation (28) can be 

 rewritten as 



de d(log c) 



dy 



cos 9. 



(29) 



We can use equation (29) to describe, qualitatively, 

 what happens when a ray travels to a layer just above 

 it {dy < 0) of different sound velocity. If the new 

 layer has higher sound velocity, the curvature dd/ds 

 has a positive sign, and the ray is bent downward. 

 If the layer just above has lower sound velocity, the 

 curvature d9/ds is negative, and the ray is bent up- 

 ward. We get the opposite result if the ray is traveling 



to a layer just below it (dy > 0) of different sound 

 velocity. Thus we can say, in general, that a ray en- 

 tering a layer of higher sound velocity is bent away 

 from the layer, and a ray entering a layer of lower 

 sound velocity is bent into the layer. 



In the open ocean the vertical velocity gradient 

 usually falls into one of two types, depending on the 

 temperature-depth variation. If the temperature does 

 not depend on the depth, the velocity is determined 

 by the pressure, which increases with depth; there- 

 fore, in such isothermal water the sound velocity in- 

 creases gradually with depth, and sound rays should 

 possess slight upward bending. Another common case 

 has the temperature decreasing with depth. Since 

 velocity is much more sensitive to changes in tem- 

 perature than to clianges in pressure, the velocity will 

 also decrease with depth, and the sound rays will 

 bend strongly downward. The water temperature 

 rarely increases with depth; when it does, the sound 

 rays are bent strongly upward. 



We shall now examine, quantitatively, the change 

 of curvature along an individual ray, and derive cer- 

 tain relationships between the range and depth 

 reached at time < by a ray leaving the projector at a 

 certain angle. Assume that the projector is situated 

 at the depth where c = co; thus the ray may be 

 characterized by its initial angle 9o at the projector. 

 Because of equation (26), equation (29) becomes 



d9 dc cos 9o , , 



(30) 



ds dy Co 



The advantage of the representation (30) is that it 



gives the curvature along a single ray as a function 



of dc/dy only, since So is constant for that particular 



ray. 



We consider, in particular, the case where the 



velocity gradient has the constant value a; that is, 



c = Co -I- az/, (31) 



if the origin of coordinates is taken at the projector. 

 At all points on the ray, in view of equation (30), 



dB a cos So 



ds Co 



We see from equation (32) that the curvature is 

 constant along the ray; this means that the ray must 

 be an arc of a circle. As the radius of curvature is the 

 reciprocal of the curvature d9/ds, the radius r of this 

 circle must be given by 



Co 



(32) 



(33) 

 a cos So 



If a is positive, the curvature (32) is negative, and 



