516 



HBRSEY AND BACKUS 



[chap. 13 



The pressure and the normal component of velocity must be continuous at the 

 boundary of the bubble. The pressure inside the bubble is given by (9), while 

 the pressure outside is the sum of the incident pressure and jps taken at R. 

 Since the treatment depends on the assumption R<^X, we can express ^5 and 

 its derivative dps/dr at R approximately by 



^^=^-R-—^ 



dr 



g2vift^ 



ir 



Now for continuity of pressure 



Vi = po+Ps, (12) 



where po^Ae^^^f* is the instantaneous pressure of the incident wave. Sub- 

 stituting (9) and (11) in (12) and cancelling the common factor e^"*/', 



, B 27Ti ^ , 

 A+-- — B-Ai = 0. 



(13) 



dp 



The normal component of velocity vr is related to the pressure gradient -^ by 



dvR dp 



P^^-d^ ^'^^ 



(recalling that p = pcvR and taking appropriate derivatives of p—po e27r<(/(-r/A)j 

 Substituting the value of dpsjdr from (11) in (13), setting r= R, and integrating 

 over time give 



Vr = -{Bil27TfpR^)e^-ifK (15) 



The particle velocity of the incident wave makes no contribution to this 

 equation since the bubble radius is assumed small compared to the wave- 

 length ; hence it is uniform across the bubble. The effect of the incident wave 

 displacement is to move the bubble bodily to and fro. Continuity of radial 

 velocity can now be satisfied by equating (10) and (15) : 



BjiirfpR^ - 27TfRAil3yPo. (16) 



Eliminating At between (13) and (16) yields 



B = AR 



Now substituting in (17) 



3yPc 



(277/)2pi22 



+ ■ 



27TiR 



2^fr = ^ 



^yPi 



yields 



B = AR 



P , 27tR . 



where fr is the resonant frequency of scattering by the bubble. Or from (6) 

 Maximum as occurs at/=/r. 



■171 



;i8) 



:i9) 



(20) 



