128 



DEEP-WATER TRANSMISSION 



on whether the sound was scattered close to the pro- 

 jector or some distance out. Sound scattered from 

 points to the right of the point B will suffer a two- 

 way absorption loss, and may be neglected. There- 

 fore, to calculate the sound intensity at the hydro- 

 phone, taking absorption into account, we must first 

 integrate equation (17) with the infinite upper limit 

 replaced by L, obtaining approximately mJ/Sd; then 

 we must multiply this value by some factor to take 

 the absorption into account. Because we are neglect- 

 ing the sound scattered to the right of B, we may con- 

 sider that the sound reaching the hydrophone has 

 traveled a total path length, on the average, equal to 

 R, the range from the projector to the hydrophone. 

 If a is the attenuation coefficient in decibels per yard, 

 the intensity Is is therefore given by 



_ !!^iri-aR/10 



(19) 



Equation (19) was derived without considering 

 the possibility that sound could be scattered up to 

 the surface by points to the left of B and reflected 

 back to the hydrophone. We can allow for this extra 

 intensity due to surface reflection by multiplying the 

 expression (19) by 2. Our final result is 



7 _ "''^lo-gR/lO 

 4rf 



(20) 



It is convenient to restate equation (20) in decibels: 

 10 log /. = 10 log m 4- 10 log J - 10 log (4d) - aR. 



(21) 



The total emitted power J is related to F, the 

 power output per unit solid angle in the direction of 

 the projector axis, by the formula 



10 log J = 10 log (4TrF) + D, (22) 



where D is the directivity index of the projector. 

 Combining equations (22) and (21) gives 



10 log /, = 10 log m -I- 10 log F -I- D - 10 log d 



- aR+ 10 log X. (23) 



The transmission anomaly A of the scattered radia- 

 tion is given by 



A = 10 log F - 20 log R - 10 log /,. 



If R sin d is substituted for d, from Figure 46, and 7s 

 is taken from equation (23), the transmission anom- 

 aly A becomes 



A = - 10 log fl - 10 log 7n - D + aR - 10 log tt 



4- 10 log (sin 6>). (24) 



For a total temperature decrease of about 20 de- 

 grees in the thermocline, the limiting ray PSC in 

 Figure 46 bends downward below the thermocline at 

 an angle of 12 degrees. Thus, a typical value of d is 

 12 degrees. For a directional transducer of the type 

 normally used in echo ranging, the directivity index 

 Z) is — 23 db. If an absorption coefficient a of 4 db per 

 kyd is used in equation (24), and D and 6 are set 

 equal to —23 db and 12 degrees, respectively, then 

 A is nearly constant from 1,000 to 3,000 yd, and 

 equals — 10 log w — 14. Since A is observed to lie 

 between 40 and 60 for transmission to points well in- 

 side the shadow zone, 10 log m is between —54 and 

 — 74. Because the receiver is directional in a vertical 

 plane, these values for 10 log m must be increased 

 somewhat to take account of the directivity pattern 

 of the receiver. An examination of the receiver pat- 

 terns in reference 34 indicates that this correction 

 should be about 6 db. Thus, we finally have for 

 10 log m a value between —48 and —68 db. 



This result is in general agreement with the value 

 of — 60 + 10 db for the scattering coefficient of 

 volume reverberation given in Chapter 4 of Part II. 

 A value greater than — 40 db seems definitely ruled 

 out by the observations. Thus one may conclude that 

 the scattering coefficient for sound at angles between 

 roughly 10 and 120 degrees is not more than about 

 10 db greater than for the backward scattering which 

 gives rise to reverberation. It is possible that the 

 scattering of sound by the volume of the sea is the 

 same in all directions. More exact conclusions would 

 require simultaneous determinations of reverberation 

 and sound scattered into the shadow zone. In addi- 

 tion, the change of scattering coefficient with depth, 

 frequently observed in the deep scattering layers dis- 

 cussed in Chapter 14, would demand consideration. 

 The present very rough analysis is adequate, how- 

 ever, to indicate that the attenuation observed in 

 deep isothermal water is not the result of scattering, 

 unless one makes the improbable hypothesis that 

 scattering in the isothermal layer is very much greater 

 than the scattering in the thermocline. If an attenua- 

 tion coefficient of 4 db per kyd or 4 X lO'^db per yd is 

 attributed entirely to scattering, the scattering coef- 

 ficient m would be 10 logio e times a or 1.7 X 10~^, 

 giving more than —20 db for 10 log m. This is 20 db 

 greater than the maximum possible value of m con- 

 sistent with the low intensity of sound observed in 

 the shadow zone. If not all the sound in the shadow 

 zone is due to scattering, the disparity becomes even 

 greater. 



