The difficulty in making quantitative comparisons between acoustic measurements of volume reverberation and 

 the distribution of swimbladder fishes lies in the determination of both n and o,. The determination of the number 

 of fish and the species, size and swimbladder size of each is strictly a biological problem which will not be 

 considered here. However, the difficulty of this problem is not to be minimized. Given the size and swimbladder 

 size of an individual fish, calculation of o is also difficult, due to the complex structure of fish. 



Simplified models have been developed in order to estimate o near resonance for an individual fish. These 

 models are based on the premise that only the swimbladder is a significant contributor to o near resonance. 

 Experimental evidence [15] shows, in fact, that this is so, whereas, at frequencies much higher than the 

 resonant frequency, the swimbladder and the body of the fish contribute about equally to o [16]. 



Experimental evidence indicates that the existing models have some shortcomings. After a review of the 

 models and the experimental data, it will be the purpose of this report to develop an improved model 

 of resonant scattering from an individual swimbladder-bearing fish in order to eliminate or at least decrease 

 the shortcomings of the existing models. 



Existing Models 



A swimbladder is essentially just an air bubble within the fish, so that the simplest acoustic model of a 

 swimbladder fish is an ideal spherical air bubble having the same volume as the swimbladder. The 

 frequency of the fundamental mode of resonance of a small ideal spherical air bubble in water was 

 determined by Minnaert [17] to be 



2a2 = 3Yaf^ ( ,. 6) 



Pow 



where 0) is the circular frequency of resonance, a the equilibrium bubble radius, P the ambient pressure, y a 

 the ratio of specific heats of air, and p^ the density of water. The acoustic cross section of a small ideal 

 spherical air bubble in water is [18] 



4na 2 . , 



where to is the insonifying frequency and c^ is the sound velocity in water. The limiting factor on these 

 equations is the size of the bubble, which is limited to values such that (coa/cJ«l. 



For a small real air bubble in water, the above equations for co and a remain the same, except that the 

 term (o^a 2 /^ 2 ) in equation I-7 is replaced by d 2 [18]. d is an unspecified damping constant which includes 

 the effects of heat conduction, surface tension, viscosity, and other processes. 



Devin [19] studied the damping at resonance of real air bubbles in water. He found that the damping 

 constant at resonance, 5, was the sum of three damping processes: thermal damping, 5 th , viscous 

 damping, 5 vis , and radiation damping, 5 rad , 



5 = 5 rad + 5 V1S + 5 th . (I-8) 



5 is defined such that, at co = a> , d = 5. Devin determined the values of 5 rad , 6 vis , and 5 th , for bubbles of the 

 size which are of present interest, to be 

 co n a 



Cw 



d-9) 



5 UIS = 4t1s " ■ , (1-10) 



VIS M^ 



6„ 



_3(Ya-1) 



Co-^ 2 V' (HI) 



a V 2co p 0a Cp a / 



where n, Sw is tne shear viscosity of water, K a is the thermal conductivity of air, p 0a is the density of air, and c Pa 



