III-16 



Co -i 



(1/9) (ka)^ 



[(1/9) (ka)^ + gh^ -|(ka)^ J 



The bubbles therefore have a resonant wave number ko (or resonant frequencies 



f ) such that ko = — = — — • If we write the scattered wave as p (r,t) = 



Cq Co ^^ 



+ikr-iU)t 

 e 

 p — , then the amplitude of this wave can be expressed in terms of the 



resonant wave number or the resonant angular frequency as: 



ap. (ka)^ ap 

 P.. = = ^-^ ^. (III-21) 



sc 



(ko a)^ - (ka)= - i(ka)= 



(7)--fe) 



The resonant angular frequency may be given in terms of the physical variables 

 describing the situation according to: 



2 



c^ 3 p' c' 3 Yp 



uu= = k^ c^ = 3 gh^ 4 — ^^ - -^ (III-22) 



° ° a^ p^ a^ p^ a^ 



At resonance, (III-21) becomes purely imaginary so that the scattered wave is 

 90° out of phase with the incident wave. It will be shown later that the imaginary 

 term in the denominator corresponds to the energy that is removed from the in- 

 cident wave and, in fact, is radiated out to infinity as the scattered wave. The 

 real part of the denominator of (III-21) corresponds to the exchange of energy be- 

 tween the bubble and the incident wave. During part of each cycle the bubble is 

 compressed and stores up energy which it returns during the remainder of the 

 cycle. Thus, in this model there is no dissipative mechanism within the bubble 

 itself; the only way it can remove energy from the incident wave is by radiating 

 this energy in the form of an outgoing scattered wave. 



Mechanical Analogue 



This behavior of the bubble is often compared to a simple mechanical 

 analogue. Consider a mass M attached to a spring with spring constant K moving 

 with a displacement x(t) in and out of a dashpot which offers a resistance force 

 D X proportional to the velocity of the mass. If this configuration is subjected to 

 an harmonic exciting force F e'^^^ , the equation of motion becomes that of the 

 damped harmonic oscillator: 



artbur a.littlc Jnir. 



S-7001-0307 



