60 



RAY ACOUSTICS 



The accurate plotting of many rays is facilitated by 

 use of a method called the method of proportions 

 (see reference 2). When this method is used, a few 

 rays are drawn with the aid of a slide rule; the posi- 

 tions of intervening rays can be estimated rapidly by 

 an interpolating process. A full description of the 

 method of proportions is given in reference 2. 



If the ray plotter or the method of proportions is 

 used, it is not difficult to obtain a ray diagram with 

 many rays drawn for closely adjacent values of do, the 

 ray inclination at the projector. From such a diagram 



Figure 21. The sonic ray plotter. 



the intensity at any point can be determined graphi- 

 cally by measuring the vertical separation between 

 rays at that point. In most situations this is the 

 simplest method for computing approximately the 

 theoretical intensities in the sound field. 



The basic equation used in this graphical procedure 

 for determining the sound intensity may readily be 

 derived from the analysis in Section 3.4.1. By com- 

 bining equations (49) and (50) the following results 

 for the intensity / 



dOo 



I = 27rFcoseoT7,- 

 do 



The area dS is given by 



dS = 2irR cos ddh, 

 where dh is the vertical distance between the two rays 

 at the point where the intensity is measured, and R 

 is the horizontal range. By combining these formulas, 

 and by substituting into equation (65) for the trans- 

 mission anomaly the following equation results 



(cos 6 dh\ , ^ , „ 

 --) - 10 log ft. 

 cos do del 



For most cases of practical importance, cos 6 and 

 cos ^0 may be replaced by one; dh and dQ may be re- 

 placed by finite increments AA and A^. Thus, we 

 have, finally, the simple result 



10 log 



AA\ 



10 log R. 



(90) 



In equation (90), A/i and R must be expressed in 

 yards; while A9 must be given in degrees. Although 

 this equation usually gives sufficiently accurate re- 

 sults, it is difficult to apply practically in regions 

 where the intensity is changing rapidly, such as near 

 the shadow boundary below an isothermal layer. 



The practical application of equation (90) is given 

 in reference 2, which includes a graph giving the 

 theoretical intensity / in terms of the measured ray 

 separation in feet at the range R and the initial 

 angular separation of the rays in degrees. 



3.5.2 Ray Diagrams for Various 

 Temperature-Depth Patterns 



In practice, the ray paths are usually computed 

 not from the velocity-depth curve, but from the 

 temperature-depth curve obtained with a bathy- 

 thermograph. This is done because the sound velocity 

 is very sensitive to changes in temperature of the 

 magnitude usually encountered in the ocean and rela- 

 tively insensitive to changes in pressure and salinity. 

 The effect of pressure, although small, is usually al- 

 lowed for in the drawing of rays because it is con- 

 stant, causing an increase of 0.0182 ft per sec in 

 sound velocity per foot increase of depth. The effect 

 of salinity on the ray paths is usually ignored, except 

 near regions where fresh water is continually mixing 

 with ocean water; in such cases, the velocity-depth 

 pattern must be calculated explicitly by use of both 

 the bathythermograph record and the safinity-depth 

 variation. 



The following paragraphs describe ray diagrams 

 for various commonly observed temperature-depth 

 patterns. A more detailed explanation of ray dia- 

 grams along with explicit diagrams for some 380 

 temperature-depth patterns of the sort found in the 

 ocean is given in a report by WHOL' 



Very Deep Isothermal Water 



In deep isothermal water all the rays show slight 

 upward bending because of the constant effect of 

 pressure. This bending, for a ray leaving the pro- 

 jector in a horizontal direction, amounts to about 



