leo 



DEEP-WATER TRANSMISSION 



law of simple geometrical spreading. A detailed 

 analysis was given in Section 2.5 for the absorption 

 resulting from viscosity. However, it was pointed out 

 in that section that the observed absorption of sound 

 in the ocean is far greater than can be accounted for 

 by viscosity. 



The effect of absorption on underwater sound trans- 

 mission is shown most simply by considering the 

 propagation of sound in an unbounded homogeneous 

 ocean. Let J be the total energy proceeding outward 

 from a sound source in each second. If the ocean were 

 not at all absorbing, this same amount of energy 

 would spread to all ranges. If the source is nondirec- 

 tional, this energy would be spread over an area of 

 AttR- at a range R, and the sound intensity / wo.uld be 

 given by the equation 



I = 



^■kR^ 





(5) 



where F, defined in Chapter 2, is given by 



In the presence of absorption, a constant fraction 

 n of the sound energy is absorbed in each yard of 

 sound travel. Thus in a distance dr, an amount of 

 energy ^.TrnF dr will be absorbed per second, and con- 

 verted into heat energy. The constant n is called the 

 absorption coefficient of the water. The decrease Air dF 

 of the sound energy over this distance will equal the 

 energy absorbed, or 4xnF dr. Thus we have the 

 equation 



Tr - -^^' («) 



which has the familiar exponential solution 



F = Foe 



-nR 



(7) 



The sound intensity is then given by the equation 



Foe-"'^ 



I = 



R' 



(8) 



In terms of decibels, this equation may be written 



10 log / = 10 log Fo - 20 log R - 



aR 

 1000 



(9) 



where a/1000 = lOn logio e = 4.34n; a expresses the 

 absorption in decibels per kiloyard and is called the 

 coefficient of absorption. The transmission loss H, 

 as defined in Chapter 4, is 10 log Fo — 10 log /. The 

 transmission anomaly A is simply H — 20 log R. 



Thus we have the simple equation 



aR 



A = 



1000 



(10) 



Hence, in an unbounded medium, absorption pro- 

 duces a transmission anomaly which increases lin- 

 early with the distance covered by the sound beam. 



Scattering of sound is more complicated than ab- 

 sorption. When scattering rather than absorption is 

 present, equation (10) may still be used to describe 

 the decay of the unscattered sound. However, the 

 sound that has been scattered must also be considered 

 in computing the expected sound intensity. It is 

 shown in Section 5.4.1 that scattering is probably not 

 very important in isothermal water. However, since 

 the exact role of scattering in isothermal water is not 

 certain, and since some forms of scattering may be 

 very important when temperature gradients are 

 present near the surface, it is customary to refer to 

 the combined effects of absorption, scattering, and 

 similar phenomena as attenuation. The quantity a, 

 determined by direct measurement of A and use of 

 equation (10), is then called the coefficient of attenua- 

 tion. Attenuation, as so defined, includes all effects 

 which may produce a transmission anomaly. 



Extensive observations at a number of laboratories 

 indicate that in isothermal water the transmission 

 anomaly does, in fact, increase linearly with increas- 

 ing range, in accordance with equation (10). Thus, the 

 attenuation coefficient a for each frequency is a con- 

 stant for any one run. The data at 24 kc, in a report 

 on attenuation issued by UCDWR, pro\'ide a check 

 of this point. ^^ Of the many runs available in deep 

 water off the coast of southern California and Lower 

 California, at the time reference 13 was written, 65 

 were made when the temperature difference from the 

 surface to a depth of 30 ft was 0.1 F or 0.0 F. For all 

 these runs the graphs of transmission anomaly against 

 range could "reasonably be approximated by straight 

 lines beyond a range of 1,000 yards." Two sample 

 plots of transmission anomaly, with the corresponding 

 temperature-depth records, are shown in Figure 13. 

 Each point represents the average amplitude of five 

 different pings. 



The linearity of the observed points is evident in 

 Figure 13. On the average, about half of the plotted 

 points lay within 2 db of the straight-line curve 

 drawn for each run. Thus it is reasonable to conclude 

 that in water which is isothermal from the surface to 

 30 ft, the transmission anomaly increases linearly 

 with range from 1,000 to more than 6,000 yd. Since 



