SCATTERING AND ABSORPTION BY AN ACTUAL BUBBLE 



465 



By proceeding exactly as in Section 28.1, the fol- 

 lowing relation is found instead of equation (19): 



li = 



RA 



($Th-^Mi- 



/3 



1 + 



-v+- 



cpri/ 



•(31) 



If one neglects /3- compared to one and defines 



s = 



p 



equation (31) becomes 



B = 



+ ') + — . 

 cpv 



RA 



(32) 



(33) 



1 + i8if,Ro) 



Substituting this expression into equation (6), the 

 cross section for scattering by an actual bubble can 

 readily be evaluated: 



(34) 



(fO^ 



+ 52 



It will be noted that equation (34) is identical with 

 equation (22), which was derived for an ideal bubble, 

 except that 8^ has replaced rj^ in the denominator. 

 This change affects only the magnitude of the scat- 

 tering cross section near resonance. Thus the fre- 

 quency of resonance and the scattering cross section 

 at frequencies far from resonance are correctly given 

 by equation (22), in agreement with the statements 

 made in the previous section. 



The knowledge of the scattering cross section does 

 not provide all the information that is wanted in the 

 case of an actual bubble, as the incident flux of 

 energy is reduced both by scattering and absorption 

 of sound. Calling the sum of scattered and absorbed 

 energy the extinguished energy, an extinction cross 

 section Ce can be defined by 



0-6 = 7^' (35) 



■to 



where Fe is the total energy extinguished by the 

 bubble per unit interval of time and 7o is the intensity 

 of the incident sound energy. The quantity Fe is equal 

 to the work done, per unit interval of time, on the 

 bubble by the incoming sound beam; this extin- 

 guished energy comprises both absorbed and scat- 

 tered energy. Hence, Fe is equal to 



Po 



dV 

 dt ■ 



(36) 



where the bar means the time average; po is the pres- 

 sure of the incident sound wave, and V is the volume 

 of the bubble. 



To evaluate equation (36) it is simplest to use real 

 quantities. According to equation (2), 



Po = Ae^"^'. 



Since the initial phase may be chosen arbitrarily, let 

 A be real, and let the sound pressure and sound 

 velocity be represented by the real parts of the ex- 

 pressions developed above. Then 



Po = A cos 2Trft. (37) 



From equations (9) and (15) it follows that 



dt fp 



Here again only the real part of the entire expression 

 is to be taken. In order to find this real part, split B 

 into its real part B^ and its imaginary part iB^, and 

 express e^"-^' in terms of its real and imaginary parts : 



(B^+ iB^) (cos 2x/< + i sin 2ir/<) 



dt 



-fp 

 2 

 —- [(5^ cos 2x/< + B^ sin 27r/0 

 ~fp 

 -\- i {B' sin 2ir/« 



(38) 



S'^cos27r/<)]- 



If equation (37) and the corresponding real part of 

 equation (38) are substituted into equation (36), it is 

 found that 



2 A 



Fe= - -^(cos 2TrJt) {B^cos 2Tft + B^ sin 2x/i) ; (39). 

 JP 



where the bar denotes an average over many cycles 

 Since 



and 



1 



COS^ (27r/0 = - 



(cos 2ir/0 (sin 2ir/0 = 0, 



equation (39) becomes finally 



AB^ 



fp 

 According to equation (33), 



RA5 



Fe= - 



B' = 



(^'T 



+ 5^ 



Hence, equation (40) assumes the form 

 RA^ 1 



fp' 



Fe 



(f-0^ 



(40) 



(41) 



(42) 



+ 52 



