470 



ACOUSTIC THEORY OF BUBBLES 



bly to the total extinction cross section. Since a^ for 

 bubbles of sizes far from resonance size is propor- 

 tional to 5, and since the value of this damping con- 

 stant is unknown for nonresonant bubbles, it is not 

 possible to state the conditions under which non- 

 resonant absorption may become important. In 

 practical applications it is customary to assume that 

 bubbles near resonance provide the dominant source 

 of attenuation in wakes; as shown in Chapter 34, 

 this assumption appears to lead to agreement with 

 experimental results, and is probably correct at 

 supersonic frequencies for the bubble distributions 

 occurring in wakes. 



Then, in order to compute the value of Se resulting 

 from bubbles near resonant size, n(R), ■q{R,}), and 

 5{R,f) in equation (56) may be taken outside the 

 integral and given their values for R equal to the 

 resonant radius Rr- Thus equation (56) is trans- 

 formed into 



^"' (57) 



_ 4wRMRr)Sr r dR^ 



+ 6? 



As the radius of the resonant bubbles struck by a 

 sound beam of the frequency / is Rr, then according 

 to equation (23) 



- '- ^^^« (58) 



(59) 



dR. (60) 



By substituting equation (60) into equation (57), Se 

 can be expressed by an integration over the variable 

 q. Only the values near the peak (near to q equals 1) 

 make a considerable contribution to the value of the 

 integral. Therefore, the transformations (49) and (50) 

 may be used, and from equations (57) and (60) it 

 follows that 



4:TRln{Rr)8r /*+" dq 



■03 



4 2,.^ (61) 



rir v'-=> 4g2 -I- 5r 



The integral has been extended to infinity. This 

 simplification can be made because on this approxi- 

 mation the contributions which are not very near to 

 the peak can be disregarded. Evaluating the integral 

 in equation (61) gives 



/:: 



dq 



-co 4g2 _|_ 52 25, 



and from equations (61) and (62) 



2Trm^rn(Rr) 



Se = 



Vr 



(62) 



(63) 



Let now u{R)dR denote the total volume of air con- 

 tributed by the bubbles with radii between R and 

 R + dR in 1 cu cm of the air-water mixture. Hence, 



4x 



u{R) =—R^n{R), 



o 



and from equations (63) and (64) 



Se = 



Sru^Rr) 

 2r;, 



(64) 



(65) 



The quantity rjr, according to equation (23), has the 

 value 1.36 X 10~^ in sea water at 60 F and at atmos- 

 pheric pressure. Hence, 



Se = 346.5M(i2,). (66) 



In computing the attenuation for a region containing 

 bubbles of many sizes, the equations derived at the 

 beginning of this section may be applied directly. 

 It is necessary only to replace the factor no-e in equa- 

 tion (53) by Se, taken from equation (66). If this 

 substitution is made, the coefficient of attenuation is 



Ke = 396.8 X 346.5 X w(fl,) 

 Ke= lAX 10^ uiRr). 



(67) 



This expression is the generalization of equation (55) 

 for bubbles with a wide dispersion in size. It will be 

 used in Chapter 34 to compute the amount of air in 

 wakes from the observed attenuation coefficients. 



28.3.3 



Scattering 



In accordance with the picture for propagation of 

 sound through a region containing bubbles, as pre- 

 sented in Section 28.3.1, the basic equation for scat- 

 tered sound is very simple. The scattered sound in- 

 tensity from a region is, on the average, simply the 

 sum of the intensities of the waves scattered by each 

 bubble. For a single bubble, the intensity at a dis- 

 tance r is given by the equation 





(68) 



where /o is the intensity of the incident sound at the 

 bubble. This equation may be found from equations 



