III-19 



We substitute (III-28) and (III-26) in (III-25) and obtain a relation between the 

 amplitude of the scattered wave and the amplitude of the interior bubble pressure: 



p^a gika 



P' = P,, (1-ika) -— (III-29) 



3 Y Po *^sc ' ' ^ 2 



From (III-24) and (III-29), we can solve for p ... .^^...^ v.. ^■ 

 same result as (III-21) if we use the identifications of (III-22). This simple 

 physical model and the associated mathematical approximations are therefore 

 equivalent to the approximations of the breathing mode of a fluid sphere leading 

 to (III-21). 



In the above analysis we have ignored the effect of dissipation; the 

 only loss of energy from the incident wave has been due to the radiation of the 

 scattered wave. If we introduce heat conduction inside the bubble and viscosity 

 in the water, and enclose the bubble in an elastic membrane which corresponds 

 to the surface tension of the water, we can still carry through the calculations 

 that lead to (III-21), using the same type of analysis shown in (III-24) and (III-29). 

 Such a procedure has been carried out by Spitzer* and results in a more com- 

 plicated expression for the scattered amplitude. Nonetheless the expression 

 has the same essential features exhibited by the simple models of a damped 

 oscillator or a dissipation-free bubble. We show all three models for easy com- 

 parison in Table III-l, and catalog them for future reference as: 



Model I - damped harmonic oscillator 



Model II - dissipation- free bubble 



Model III - bubble with conduction, viscosity and surface tension 



We wish next to compare these three models with each other and with 

 the available experimental data. If we compare the resonant frequencies pre- 

 dicted by Models II and III, we find that the effect of the dissipative mechanisms 

 included in Model III is small for frequencies below 20, 000 cps. The two curves 

 for the resonant frequency are shown in Figure III-7 together with some data from 

 a number of experiments which are in generally good agreement with the theoret- 

 ical predictions. The anomalous results reported by Exner and Hampe were ap- 

 parently caused by dust particles in the water. It is not surprising that the resonant 

 frequency predicted for Models II and III should turn out to be so similar. After all, 

 resonance comes about through the matching of the inertial properties of the sys- 

 tem (i.e., the mass of the bubble and a portion of the surrounding water) and the 

 compressibility of the system. 



= L. Spitzer, Ref. Ill- 38. 



artbur m.littlc.Ilnt. 



S-7001-0307 



