density of the sound power is N E(f), and the rms sound pressure in a band of width 

 Afis(2NAf)%s(f). 



The sound energy tends to be concentrated at the natural frequency / of the 

 bubble, because the denominator in Eq. (8) is a minimum when f — f . However, in 

 many practical situations the fluctuating environmental pressure is such that h{f) differs 

 from zero only at low frequencies, much below the natural frequency / of the bubble. 

 In this case, the instantaneous sound pressure p s {t) can be expressed directly in terms 

 of environmental pressure p e {t') at earlier time t' = t-r/c by 



p.(0 = - (Ro/r) (2TTfo)-*p e (t'). (10) 



The calculation of the sound radiated by motion about a particular body re- 

 quires a knowledge of the environmental pressure at the moving bubble as a function of 

 time. This can be obtained from the known pressure and velocity field around the 

 body. If, for simplicity, it is assumed that the bubble follows a streamline, its position 

 as a function of time is determined by integrating the velocity along a streamline, and 

 the pressure is then determined from the position. This procedure, although straight- 

 forward in principle, usually results in expressions which cannot be expressed in closed 

 form. 



These calculations have been carried out by one of us [7] for the flow past a 

 circular cylinder, with the bubble moving around the surface of the cylinder. The rela- 

 tively complicated result for the sound pressure is 



32p 2 £V£o 3 



Ps(t) = 



2U \ /2Uo 

 sech 2 [ f }-% sech 4 t' 



Rr / \ Re 



(ID 



where U is the free-stream velocity; R c is the radius of the cylinder; t is measured 

 relative to the instant the bubble passes the median plane; and it is assumed that 

 (2U /i? c ) <^/ . The waveform of the sound pressure is shown in Fig. 2. The sound 

 pressure has a negative peak at t' — and two smaller positive peaks at t' — 

 ±0.6R C /U . The peak pressure can be quite high: for the conditions U = 5 meters/ 

 sec, R — Vs cm, R c = 10 cm, the negative peak from an individual bubble is about 

 16 dyne/cm 2 at 1 meter. No experimental work has yet been performed to verify these 

 calculations. 



For bubbles streaming past a cylinder, the sound pressure should increase with 

 the fourth power of the velocity, as indicated by Eq.(ll). In fact, the sound pressure 

 should increase as U 4 for all similar forms of bubble motion, as is apparent from a 

 dimensional consideration of Eq.(10): the environmental pressure fluctuations increase 

 as U 2 and the second time derivative introduces another factor (U /L) 2 , L being a 

 characteristic length. 



Large-amplitude pulsations. — The rapid increase of sound pressure with increas- 

 ing velocity leads to a consideration of the limitations of the linear theory on which 

 the above results are based. 



The large-amplitude free pulsations of gas bubbles have been studied extensively 

 in connection with underwater explosions [8]. It is known that the radiated pressure 

 loses its sinusoidal form with increasing amplitudes, the negative excursion becoming 

 flatter while the positive excursion develops a sharp peak. The same effect occurs in 

 forced large-amplitude pulsations in response to a widely fluctuating environmental 

 pressure. 



As a first step toward a more-exact description of large-amplitude pulsations, the 

 radial pulsation is described by a non-linear equation 



p(RR + %R 2 ) = P - p e (t), (12) 



where R is the instantaneous radius of the bubble, and p e (t) is again the environmental 

 pressure that would exist at the bubble location in the bubble's absence. Also, P is the 

 actual pressure in the water at the bubble wall; for a bubble filled with gas obeying the 



245 



