RAY PATHS FOR VERTICAL VELOCITY GRADIENTS 



49 



Consider now the cliord P1P2 converting the end 

 points of the circiilnr arc. The direction 6 of this 

 chord is by simple plane geometry '2(^1 + ^2); and 

 its length is therefore given by 



PxP'' = . ./ r- (35) 



sm Udi + S2) 



The increase in horizontal range due to the passa,^e 

 of the ray through the layer is P1P2 cos d, or 



— ^ Range in layer = h cot §(^1 + ^2). (36) 



This result may be applied to the following prob- 

 lem. Suppose we have a sum of layers of the sort 

 shown in Figure 10; and we wish to find the horizontal 



VELOCITY 



PROJECTOR DEPTH 



Figure 10. Succession of linear gradients. 



range attained by the time the ray reaches the depth 

 H below the projector. We let the bottom layer ex- 

 tend just to the depth H; suppose this is the third 

 layer below the projector. We know do and we cal- 

 culate 01,^2,^3 by the relations (34). Then the hori- 

 zontal range to the depth H will be the sum of terms 

 of the form (36) : 



Horizontal range to H = hi cot i{6(, + di) + 



hi cot UOi + 62) + h^ cot ^{62 +Bi). (37) 



The inverse problem is a little more complicated. 

 Suppose we wish to find the depth reached by a ray 

 of initial direction So by the time it has traveled a 

 horizontal distance i? in a stratified medium that 

 consists of layers of thickness hi,h2,hs, etc. We cal- 

 culate the range Ri in the first layer, R2 in the second 

 layer, and so on, until the sum of these partial ranges 

 is greater than R : 



Ri -{- R2 -\- Rz 'C R 

 Ri -]- Ri -\- R3 -\- Ri > R. 

 Then the depth the ray reaches at range R will be 



greater than hi + hi + hs and less than hi + h^ + 

 hi + hi. Its value may be obtained with sufficient 

 accuracy by interpolation. 



The ray-tracing methods described in this section 

 are too cumbersome to use in practice. A number of 

 devices have been developed to facilitate the plotting 

 of rays bent by known velocity gradients; these de- 

 vices will be discussed in Section 3.5.1. 



3.3.2 Application to Depth Correction 



The ray-tracing methods described in Section 3.3.1 

 have had a valuable application in correcting the 

 depths determined by the use of tilting beam sonar 

 gear. These instruments are used on surface vessels 

 to determine the true depths of submerged sub- 

 marines. They employ a transducer with good verti- 

 cal directivity and tiltable in the vertical plane, which 

 sends out echo-ranging pings at various angles of de- 

 pression. When velocity gradients are absent, the 

 sound rays are straight lines, and the true depth of 

 the target is just the slant range times the sine of the 

 angle of depression at the orientation for which the 

 target returns the loudest echo. The depth finder 

 computes this latter product automatically. When 

 velocity gradients are present, however, this simple 

 method often leads to serious underestimation of the 

 target depth. In this section, we shall describe a 

 method for estimating the error produced. 



For simplicity, we assume that the projector is at 

 the surface, as in Figure 11. Let the apparent target 



PROJECTOR 



TARGET 



Figure 11. Error in target position due to refraction. 



angle be do, and the apparent target depth indicated 

 by the depth finder be Yo; the true depth of the target 

 is designated by Y. Our aim is to derive an expression 

 for Y — Yo in terms of the way the sound velocity 

 c varies with depth. 



Let y represent the actual depth attained by the 

 sound ray at time t, and yo represent the apparent 

 depth reached by the ray. Then 



yo = Co sin dot (38) 



