536 



ROLE OF BUBBLES IN ACOUSTIC WAKES 



4 



i 



2 -2? 



10 20 30 40 M 60 70 



FREQUENCY IN KC 



Figure 1 . Frequency dependence of ratio of scattering 

 to extinction cross sections. 



Considering the systematic difference between the 

 observed and theoretical values of 5, as evident in 

 Figure 2 of Chapter 28, it appears highly probable 

 that at 60 kc the damping constant will not be smaller 

 than its theoretically predicted value, since the actual 

 damping by dissipative effects should not be less than 



Ua/ae- However, the scattering and absorption cross 

 sections of a resonant bubble are so much greater 

 than those of other sizes that it seems unUkely that 

 bubbles other than those near resonance can contrib- 

 ute appreciably to either the scattering or the ab- 

 sorption. Thus equation (5) may be used for actual 

 wakes, provided that a value of appropriate to a 

 resonant bubble is taken. 



On the other hand, when both the product aj^{w) 

 and the transmission across the wake are negligible, 

 equation (3) gives for bubbles all of the same size the 

 equation 



'hN{w)a,\ ,„, fhwii.o 



(6) 



PF = 10 log (^^') = 10 log (^)+ 19.6, 



where w is the width of the wake in yards and n is the 

 average number of bubbles per cubic centimeter. 

 Since N{w) is the number of bubbles per square centi- 

 meter appearing in projection on a plane perpendicu- 

 lar to the sound beam, the equivalent product nw in 

 equation (6) must have the same units — that is, 

 square centimeters. It is customary to measure the 

 wake width w in yards, or units of 91.5 cm. Hence, in 

 order to keep equation (6) dimensionally correct, a 



that resulting from the flow of heat in and out of the 

 oscillating bubble. This predicted value, derived from 

 the theory given in Section 29.2, happens to be about 

 one-third of the observed value of 5 at 24 kc. Hence 

 the true damping constant for 60-kc sound very 

 hkely is greater than one-third of the observed damp- 

 ing constant at 24 kc. This surmise implies that the 

 theoretically predicted maximum wake strength for 

 60-kc sound should not exceed the observed value of 

 TF at 24 kc by more than 5 db — because t; is inde- 

 pendent of frequency in this range — unless the ef- 

 fective value of the wake depth h is quite different at 

 the two frequencies. 



For the general case of a bubble population com- 

 prising all sizes from the largest to the smallest, the 

 analysis is more complicated. If many bubbles of very 

 large radii are present, they will scatter without much 

 absorbing, and ffs/ffe will be increased. On the other 

 hand if many bubbles of very small radii are present, 

 these \vill absorb without much scattering, decreasing 



term 10 log 91.5, which is equal to +19.6, has been 

 added to the right-hand side of equation (6) since w is 

 measured in yards. When bubbles of varying sizes 

 near resonance are considered, equation (6) is modi- 

 fied by the substitution of S^ for ncs; Ss is a weightc^' 

 mean of o-, for bubbles near resonance, according to 

 equation (77) of Chapter 28, and is equal to 



3iru{Rr) 



Ss 



25r 



(7) 



Equation (6) then may be written 



W = 10logh+ 10 log w -I- 10 log u{Rr) 



+ 10 log 



ih) 



+ 19.6, (8) 



where h and w are the depth and width of the wake, 

 respectively, both measured in yards. Values of 

 10 log 3/8S are shown in Table 5 for resonant bubbles 

 at different frequencies. In principle, equation (8) can 

 be used to determine u(Rr) from the observed value 



