35 221 



a bubble to vary in time without the occurrence of marked Inequalities of 

 pressure. The effect of a radial motion of the bubble will then be to super- 

 pose upon this incident pressure field an emitted train of spherical waves. 

 If p^ Is the pressure due to these waves, the local pressure at any point 

 near the bubble will be p + p,. 



The average pressure in the water can be written 



P = p,e"°"'cos(j(t ~ f) ~^ P(i f55] 



where p^, u and a are constants and c' is the speed of propagation of the 

 waves through the bubbly water. The factor e'"^ is introduced to allow for 

 damping due to the scattering action of the bubbles. The corresponding par- 

 ticle velocity, obtained by calculating dp/dx, substituting in Equation [53], 

 and integrating with respect to t, is 





co.u>{t-f.)+ ^sinu(t-f)] [56] 



The mean pressure near a bubble at i = will then be 



p = Pj coswt + pj, [57] 



Under the Influence of this pressure, the bubble will execute 

 forced harmonic vibrations. Because of this vibratory motion, it will emit 

 a train of spherical waves which, according to our assumptions are to be re- 

 garded as superposed upon the average pressure represented by Equation [55]- 

 The pressure In the emitted waves at a distance r from the center of the bub- 

 ble can be written 



P,= -^ P2 cosw t \, t = t + b 



in which R^ is the radius of the bubble when undisturbed, c is the speed of 

 sound in water, and p , w and b are constants. The corresponding particle 

 velocity, taken positive outward, is 



^0 r /.. r ~ Ro \ , c . / r - Rg 



I' = — — p„ cos w ( "- H sin w t - ^ 



per ^ I \ c I ur \ c 



as can be verified from Equation [3] on page 38 of TMB Report 480 (U). At 

 the surface of the bubble, where r can be set equal to R^ in constructing a 

 first-order theory, 



p = p„ cos Cjf 



V, = -; — = — - I coswt' + — :fr- sinwf 

 ' at pc \ uKg 



