Each mode has a natural frequency of free oscillation. For volume pulsation, 

 the natural frequency is given by 



/o = (37Pfl/p)*/2*#o, (3a) 



where p is the density of the liquid, and F and y are the mean pressure and specific- 

 heat ratio for the gas in the bubble. The natural frequencies of shape oscillation depend 

 on the surface tension T in accordance with the relation (see Lamb [6] §275) 



/» = [(n 2 - l)(n + 2)T/pR ]i/2TrRo. (3b) 



By substitution of the natural frequencies given by Eqs. (3) into Eq. (2), it can 

 be shown that the sound pressure associated with free oscillations is negligible except 

 for the zeroth mode. For example, at the relatively large amplitude of A n — X A R 

 (with R = Vs cm, P = 1 atmos) the sound pressures are about 3 X 10 4 and only 

 9 X 10" 4 dyne /cm 2 in the zeroth and second mode respectively, at r = 100 cm. Ac- 

 cordingly, only volume pulsations are of importance in calculations of the sound radiated 

 by bubbles. 



For volume pulsation, the instantaneous sound pressure p s (t) and the instan- 

 taneous bubble volume V(t) are related simply by 



p.(0 = pV(f)/4*r, (i) 



where the two dots indicate the second time derivative, and t r — t — r/c. 



Sound of bubble formation. — When a bubble forms on a nozzle in water, a 

 pulse of sound is emitted just as the bubble leaves the nozzle. An oscillogram of a 

 typical sound pulse is shown in Fig. 1, together with synchronized frames from a 



c igure 1. Oscillogram of the sound pulse from an air bubble formed at a nozzle, with synchronized 

 photographs of the bubble. The time each photograph was taken corresponds to the point below 

 the bubble on the oscillogram. 



motion picture of the bubble. Although the oscillations of the bubble shape are appar- 

 ent in the motion picture, the volume pulsations generating the sound are not visible 

 because of their small amplitude. 



If the pulsations are of small amplitude, the instantaneous volume of the bubble 

 pulsates in accordance with the linear differential equation 



( p /4tR )V + DV + (yPo/V ) (V - V ) = 0, (5) 



where V is the mean volume of the bubble; D is a constant which accounts for the 



243 



