III-17 



-itjut 

 Mx + Kx + Dx=Fe 



It has as its solution: 



F e"'*"^ _ (F/K)e-i'^t , o K 



where uJq - r;; (III- 23) 



K-Muu'^-iDu) 1 — (u 



UJ^ K 



We note the correspondence in the form of the resulting motion with (III-21). The 



only difference is the replacement of (the resistance term) — - 1 by the term 



o ^^ 



1 - — . However, for UJ near uJq , i.e., near resonance, these two terms are 



UJo 



approximately equal . Because of the damping term introduced by the dashpot, we 

 frequently refer to the imaginary term in the denominator as the damping constant 

 ujD ^ UJa 

 K Co 



Equivalent Physical Model 



The expression for the scattering amplitude (III-21) which was deduced 

 from the general analysis of the preceding section could equally well have been ob- 

 tained from a more direct physical model of the bubble. Since only the breathing 

 mode contributes substantially to the scattered wave, it suffices to regard the in- 

 cident sound pressure as spatially uniform near the bubble and varying in time as 

 e^'^'^. In other words, spatial gradients in the incident pressure field may be ig- 

 nored when we do a strict breathing mode analysis. Thus, the incident wave 



+ikr-i(Ut 



p. e"^*^^ causes a scattered wave po^ — and an interior wave (also 



^mc ^°^ r 



uniform throughout the bubble) p' e"^^*-. The pressure at the boundary of the bub- 

 ble must be continuous, and this requires that: 



+ ^£ gika ^ (III-24) 



mc a 



Arthur B.HittkJnf. 



S-7001-0307 



