III-29 



where p^^j^^, is the pressure of the incident sound and the bar denotes the time 



average. The radial velocity of the air bubble u^ = ^ satisfies (III-25) and 



r dt 

 using (III-26), we obtain: 



. ika-iu)t 

 i'^Po^r ^ Pscr = Psc (ik - -) f 



Determining u^ from this, calculating the time average, and dropping terms 

 of order (ka)^, we find: 



p. Im p 



F - - inc ^sc 

 ex p f 



This demonstrates our earlier statement that the imaginary part of p deter- 

 mines the rate of energy transfer from the incident to the scattered wave. The 

 extinction cross section now becomes: 



2 c^ Im p, 



p. f 

 inc 



Using the models for Pg^ discussed before, we obtain the scattering and extinc- 

 tion cross sections shown in Table III- 2. The scattering and extinction cross 

 sections according to Model III for the four typical bubble sizes (a = 1 cm, 0. 1 cm, 

 0.01cm, and 0.001 cm) are shown in Figures III- 11 and III- 12. The resonant 

 behavior of the bubbles is clearly demonstrated by the maxima of their cross 

 sections. Below resonance the bubbles tend primarily to absorb energy, while 

 above resonance the bubbles act principally as scatterers. 



:artl)ur M.%\ttU3nt. 



S-7001-0307 



