462 



ACOUSTIC THEORY OF BUBBLES 



and B from the given pressure amplitude of A of the 

 incident sound wave and from the hydrodynamical 

 boundary conditions. These boundary conditions are 

 formulated in Section 2.6.1. They require that the 

 pressure p and the component vr of the particle 

 velocity normal to the surface have no discontinuity 

 at the surface; the continuity of vr is equivalent to 

 the continuity of {l/p)dp/dR, which was required in 

 Section 2.6.1. 



While the pressure inside the bubble is given 

 by equation (10), the outside pressure is the sum of 

 the incident wave, expression (2), and the scattered 

 wave, expression (4). The pressure of the scattered wave 

 Ps at the surface of the bubble is found from equation 

 (4), with r set equal to Ro. Because of assumption (1), 

 namely that the linear size of the bubble is small com- 

 pared with the wavelength X, the term r/\ in the 

 exponent of equation (4) is much smaller than unity 

 in the vicinity of the bubble and, therefore, ps and its 

 derivative can be replaced approximately by 



X / dr 



Then the continuity of the pressure at the surface 

 of the bubble is expressed by 



Pi = P0+ Ps- 



If the expressions (2), (10), and (12) are substituted 

 into this equation, and the common factor e^ can- 

 celed, the following equation is found to apply at the 

 bubble surface during the small oscillations usually 

 encountered in practice : 



p. 



\Ro X / 





(12) 



B 2ri „ 

 A+--—B-A, = 0. 

 tlo X 



(13) 



The. normal component of the velocity must also 

 be continuous at the bubble surface. From equation 

 (11) the normal component of the fluid velocity in- 

 side the bubble is known. Its value outside the bubble 

 can be derived from equation (12). The relation be- 

 tween the fluid velocity and the pressure gradient 

 dp/dr in a certain direction is, according to the argu- 

 ment in Section 2.61, 



dvR dp 



dt 



dr 



(14) 



By substituting for dp/dr the derivative of the pres- 

 sure of the scattered wave from equation (12) with r 

 set equal to Ro and integrating over dt, equation (14) 

 is transformed into 



To this equation the amplitude A of the incident 

 wave does not make any contribution, because the 

 wavelength is much larger than the size of the bubble; 

 since the pressure gradient dpo/dx is uniform in the 

 vicinity of the bubble, the velocity corresponding to 

 this uniform pressure gradient is a motion of the 

 entire bubble to and fro rather than an expansion and 

 contraction of the bubble. The continuity of vr can 

 now be formulated, according to equations (11) and 

 (15), by 



B _ 2TfRoAi 

 2ivfpRl ~ 37P0 



From equations (13) and (16) Ai can be eliminated 



and a relation between A and B obtained; if the 



subscript is omitted from Ra, the bubble radius in 



equilibrium, then 



AR 

 B = ^r-.^ ^^T^ ■ (17) 



(16) 



37P0 



1 



2TTiR 



^■kT-pR'' " ■ X 

 By introducing now the abbreviation /r, defined by 



"^■'W' 



(18) 



37P0 . 

 p 



the physical meaning of which will soon become ap- 

 parent, equations (17), and (18) and (1) may be 

 combined to give the result 



«^ -. (.9, 



B = 



(f)'- 



1.+ rii 



In order to obtain the scattering cross section o-j of 

 the bubble from equation (6), |.A 1^ and |J5|^ have to be 

 computed from 



|A|2 = AA*, \bY = BB*, (20) 



where A* and B* are the complex conjugates of A 

 and B respectively. According to equation (19), 



RA* 



(21) 



B* = 



(fT-- 



Finally, from equations (6), (19), (20), and (21), 



4,ri?2 



(22) 



d-O" 



+ '?= 



For a bubble of a definite radius R, the scattering 

 cross section o-, has its peak value if / equals fr', it is 

 then said that the incoming wave is in 'resonance 

 with the pulsations of the bubble, and hence /r is 



