damping; and the dots indicate time derivatives.* The volume oscillates sinusoidally 

 with decaying amplitude about its mean value. Accordingly, the instantaneous sound 

 pressure is also a decaying sinusoid represented by 



p.(t) = p<>e-'*V cos (2tt/V - &), (6) 



where f is the natural frequency of pulsation as given by Eq. (3a), and -n-8 is the 

 natural-logarithmic decrement of the oscillation. The amplitude p and phase constant 

 ■& depend on the initial conditions exciting the bubble. (For an air bubble in Water at 

 atmospheric pressure, f R — 330 cm/sec; and ttS = 0.0045 + 0.0014(/ sec) Vi, the 

 two terms resulting from radiation and thermal dissipation, respectively.) 



For a bubble leaving a nozzle, the constants which depend on the initial condi- 

 tions can be evaluated from the radial velocity attained by the growing bubble just 

 before it separates from the nozzle. The sound pressure was calculated in this way by 

 one of us [7] using values of the radial velocity measured on motion pictures of the 

 bubble. The calculated sound pressures agree quite well with the measured values. The 

 frequency and decay rate also agree with the theoretical values. Accordingly, it can be 

 concluded that the generation of sound at bubble formation is well understood. 



When bubbles split or unite, a pulse of sound is generated just as at bubble 

 formation. In this case the excitation is caused by a change in the pressure inside the 

 bubble: the single larger bubble has less surface tension pressure. If a bubble splits into 

 two smaller ones of equal size, the peak sound pressure of the pulse at distance r can 

 be shown to be of the order of 0.5T/r, independently of the bubble size. This value is 

 considerably less than the pressure generated at bubble formation. 



Sound from entrained bubbles. — Probably the most important sounds result from 

 the flow of entrained bubbles past a body in a stream. The bubbles experience a 

 transient pressure as they move through the hydrodynamic pressure field around the 

 body. The transient pressure causes the bubbles to pulsate and radiate sound. 



The pulsations are described by a differential equation like Eq.(5) but with a 

 forcing term on the right side equal to P — P e {t), where p e (t) is the instantaneous 

 environmental pressure, i.e., the pressure that would exist in the liquid at the bubble 

 location if the bubble were absent. The solution of this equation is conveniently ex- 

 pressed in terms of Fourier transforms. 



If a Fourier transform h(f) of the environmental pressure is defined by 



h(f) = r EP-C) - P*]e- 27rift dt, (7 



J —00 



then the transform s(f) of the sound pressure radiated by the bubble can be calculated 

 from 



(Ro/r)e- ikr 



8(f) = h(f); (8) 



Cfo//) 2 - 1 + id 



where the entire fraction is the response characteristic of the bubble. The spectral 

 density E(f) of the sound energy is related to these transforms by 



E(f) = (8ttVpc) |*(/)l 2 , (9) 



E(f) being defined so that the energy radiated by the bubble in a narrow frequency 

 band of width A/ is A/ £(/), and the total sound energy radiated over the entire fre- 

 quency range in all directions is / E(f) df. If a large number of bubbles radiate simi- 

 lar transients at random intervals with average repetition frequency N, the spectral 



* When written in terms of the volume, Eq.(5) gives an approximate description of 

 the pulsations even for a non-spherical bubble, if R is taken as the radius of a spherical 

 bubble of equal volume. Eq.(4), however, is exact regardless of the bubble shape. 



244 



