RAY DIAGRAMS AND INTENSITY CONTOURS 



59 



below the projector, and if the ray has been reflected 

 once by the ocean bottom, the transmission anomaly 

 is given in good approximation by the equation 



A = 10 log 



29(, -\- dh — 60 



where di> is the angle of inclination at the bottom. In 

 the case of multiple bottom reflections the transmis- 

 sion anomaly is given by 



A = 10 log 



*(2(m 



+ 1) 



6b dh/ 



2(n + 1)^6 + e, - do 



where the number of bottom reflections is n + 1 • 



The transmission anomaly for sound propagated 

 through a thermocline (layer effect) is approximately 

 given by 



10 log 



(hA\ 



where hi is the height of the projector above the top 

 of the thermocline, 0o is the inclination of the ray in 

 the overlying isovelocity layer, and dk is the inclina- 

 tion of the ray at the receiver. 



The transmission anomaly for sound propagated 

 through a succession of layers, each of which possesses 

 a constant gradient of velocity, is given by 



Ri 



_ / sin Op sin g^+i y^ 



V R cos^ 60 i=o sin di sin 6 



7) 



where the various terms are defined under "Combi- 

 nation of Linear Gradients" in Section 3.4.2. 



3.5 RAY DIAGRAMS AND INTENSITY 

 CONTOURS 



3.5.1 



Methods 



The differential equations (25) which govern the 

 path of a ray in a medium where the sound velocity 

 depends only on one coordinate cannot be easily 

 integrated if the velocity depends on depth in a very 

 complicated manner. We have seen, however, that 

 the integration can be accomplished, and the path 

 of a ray with a specified initial direction calculated if 

 the depth interval is divided into layers in each of 

 which the velocity gradient is constant. For this 

 reason, rays are traced in practice by replacing the 

 actual velocity-depth curve with a series of straight- 

 line segments, as in Figure 8. 



The bending of sound rays in the ocean is too 

 .slight to be evident in a drawing that u,ses the same 

 scale for range and depth. The deviation from 

 straight-line propagation is never more than a few 

 hundred feet in a mile; although such deviations are 

 extremely important to a surface vessel seeking a 

 submarine with echo-ranging gear, they do not show 

 up well on paper unless the depth scale is expanded. 

 For this reason, ray diagrams cannot be constructed 

 geometrically in practice as the sum of circular arcs 

 through the various layers. Instead, the change in 

 range as the depth increases must be computed alge- 

 braically as described under "Combination of Lin- 

 ear Gradients" in Section 3.4.2, and the results 

 plotted on a graph with suitably chosen scales. 



A special circular 'slide rule has been invented to 

 simplify this calculation.^ This instrument, developed 

 by WHOI early in the war, gives the horizontal range 

 covered by a ray in its passage through a layer with 

 a constant gradient. It does this by using several 

 scales arranged for convenience as concentric circles. 

 The thickness of the layer, the temperature at the 

 beginning and the end of the layer, and the direction 

 of the ray at the projector are given to start with. By 

 use of the slide rule one calculates directly the direc- 

 tion of the ray when it enters and leaves the layer; 

 from the average of the two directions the horizontal 

 range covered in the layer can then be computed by 

 use of equation (36). The instrument exactly dupli- 

 cates the calculations described in Section 3.3 and 

 avoids the necessity of consulting trigonometric 

 tables. Since the direction of the ray as it enters the 

 following layer is given as an intermediate step in 

 calculating the range in the layer, the process may be 

 dupUcated until the ray has been traced through all 

 the layers. This slide rule may also be used to com- 

 pute intensities by integration along each ray since 

 the scales provided may be used for evaluating equa- 

 tion (90), which appears later. Similar mathematical 

 aids were developed by UCDWR, and by other re- 

 search groups doing a large amount of ray tracing.^ 



Another instrument developed by NDRC for the 

 facilitation of ray tracing is the sonic ray plotter,* 

 pictured in Figure 2L The ray plotter is a device 

 which integrates the differential equations (25) me- 

 chanically and exhibits the solution not as an alge- 

 braic function but as a curve denoting the ray path, 

 drawn, of course, with a much expanded depth scale. 

 The ray plotter has one advantage over the .slide rule 

 — it can plot the ray paths for any type of velocity- 

 depth variation, no matter I'.ow complicated. 



