III-34 



less homogeneous cooperative behavior in transmitting (and attenuating) a sound 

 wave. In other words, we might hope that the medium, viewed macroscopically, 

 could sustain an over-all wave motion even though microscopically (i.e., on the 

 scale of the individual bubbles) the local field would be anything but a simple at- 

 tenuating plane wave . 



Suppose, therefore, that plane waves of the form e^^"^*^^ could 

 propagate in such a medium. The "wave number" k must now be complex; its 

 real part corresponds to the phase velocity of the motion (which is not going to 

 be Cq ), and its imaginary part gives the attenuation of the wave amplitude. 



There should be a corresponding complex sound velocity c - — . How do we 



determine c? 



Consider an element of volume, V^ , which is fairly large compared 

 to a bubble spacing but fairly small compared to a wavelength (say three bubble 

 spacings as a typical dimension). Let p^ , which is the density of the water, 

 also be the density of the mixture- -the error will be very small when the frac- 

 tional volume of air is approximately 10"^ . Let the bubble density be n bubbles/ 

 cm , so that there are N = nV^ bubbles in V^ . To begin with, let all bubbles 

 have the same radius a; later we shall study the case where there is a distribu- 

 tion of bubble sizes. 



Suppose an (average) pressure wave is passing through the medium. 

 The volume element V^ is sufficiently small that we may think of the (incident) 

 overpressure as homogeneous throughout V^, and represented by Ap(t) = 

 Ap e~^^^. As a result of this macroscopic sound wave, the volume of the ele- 

 ment changes according to AV(t) = AVe"^'^'- and its density according to 

 Ap(t) = Ape"^'^^. Conservation of mass of the element requires that p^ V^ = 

 (p -H Ap ) (V- -H A V) so that to first order 



Ap = - AV 



Ct) 



If the "applied" pressure Ap causes a density change Ap , the sound velocity in 

 the mixture is given by: 



m 



^ = - !^v I - I ""-^°' 



artbur Sl.ILittkJnir. 



S-7001-0307 



