Chapter 28 

 ACOUSTIC THEORY OF BUBBLES 



THE RIGOROUS TREATMENT of the acoustic char- 

 acteristics of bubbles, especially of the cumula- 

 tive effects of a multitude of bubbles, requires a 

 great deal of rather advanced mathematics. For a 

 comprehensive exposition of these theories, reference 

 must be made to several monographs on the sub- 

 ject. i~* In this chapter only the principal features 

 of the problem will be sketched, primarily with a 

 view to the later elementary interpretation of the 

 acoustic properties of wakes in Chapter 34. Actual 

 wakes have such a complicated structure that many 

 physical and mathematical refinements incorporated 

 in the rigorous treatment of certain ideal cases have, 

 at present, only academic interest. 



The first two sections of this chapter deal with the 

 acoustic properties of individual bubbles. In the 

 third section, the combined acoustic effects of many 

 bubbles are discussed, and the results are applied to 

 the evaluation, from laboratory experiments, of cer- 

 tain physical constants — acoustic cross sections, 

 damping constants, which cannot as yet be pre- 

 dicted from pure theory. 



28.1 SCATTERING BY A SINGLE IDE4L 

 AIR BUBBLE 



For application to wakes, only those air bubbles 

 need be treated whose radius R is very small com- 

 pared with the wavelength X of sound in water, or 



R«^ 



or n = 



27rK 



»/ = -^ « 1, 



(1) 



where t] is the ratio of the bubble circumference to the 

 wavelength. Then the pressure amphtude of the in- 

 cident sound wave can be regarded as constant in- 

 side the bubble and in its immediate vicinity. De- 

 noting this amplitude by A, the pressure Po of the in- 

 cident wave can be described by 



Po = Ae^^'\ (2) 



Although the effect of the sound wave impinging 

 upon the air bubble is a rather complex one, the re- 



sulting phenomena may be classified under two prin- 

 cipal headings. 



First, the periodically variable pressure of this in- 

 cident sound wave produces a forced vibration of the 

 air inside the bubble, which reacts on the surround- 

 ing water and produces in turn an emission of sound 

 waves from the bubble. This secondary sound wave 

 is spherically S3Tnmetrical for all practical purposes. 

 Such a process of transforming an incident wave into 

 waves of different pressure distributions is commonly 

 known as scattering. The concept of scattering refers 

 solely to the process of redistribution of sound energy 

 — in other words, it is understood that none of the 

 sound energy is converted into other forms of energy. 

 Actually, scattering by a bubble is always accom- 

 panied by conversion of part of the impinging sound 

 into heat by any one of a number of processes. These 

 effects are described together under the name of ab- 

 sorption of sound. In the present section, scattering 

 by a single bubble will be treated as though it were 

 possible to produce this phenomenon apart from ab- 

 sorption; therefore, the term "ideal air bubble" has 

 been used in the title of Section 28.1. 



If the incident sound wave is a plane wave, the 

 rms sound intensity la, or the average rate at which 

 soimd energy crosses a unit area placed perpendicular 

 to the sound beam, is according to equation (56) in 

 Chapter 2 



h = f^> (3) 



2pc 



where A is the complex pressure amplitude, c is the 

 velocity of sound in sea water, and p is the density of 

 sea water. These waves excite vibrations of the air in 

 the bubble and indirectly excite pressure waves in 

 the surrounding water. In order to compute rigor- 

 ously the possible types of vibration, the method of 

 normal modes of vibration would have to be applied 

 (see Section 27.1). It can be shown that of the various 

 modes of vibration only the spherically symmetrical 

 ones are significant in the present analysis, and that 

 the other modes, corresponding to directional pat- 



460 



