474 



ACOUSTIC THEORY OF BUBBLES 



mixture with radii between R and R + dR. Equation 

 (83) has the surprising implication that when u{R) is 

 sufficiently great, (^/c^ becomes negative, the sound 

 velocity becomes purely imaginary on this approxi- 

 mation, and the attenuation becomes very large. 

 Under these circumstances the imaginary term in 

 equation (80) which was neglected in equation (83), 

 determines the wave velocity and the wavelength. 

 For the case of all bubbles with twice the resonant 

 radius Rr, the critical value of u at which c becomes 

 infinite is 2 X 10^^, corresponding to a distance be- 

 tween bubbles of roughly thirty times the bubble 

 radius. 



When resonant bubbles are present, the imagi- 

 nary part of the sound velocity becomes important. 

 If an integration is carried out only over bubbles 

 close to resonance, and if u{R) is not changing rapidly 

 with R in this region, the real part of the integral in 

 equation (80) is small and may be neglected, yielding 



3TriRrU(Rr) 



^=1 



27)r 



(84) 



This imaginary part of the sound velocity leads to an 

 exponential decay of sound intensity with distance x, 

 since the sound intensity falls off as e"-"^-'^''' ' If the 

 second term on the right-hand side of equation (84) 

 is small, as it is in most practical cases, the resulting 

 attenuation is exactly the same as was found in equa- 

 tions (53) and (67) in Section 28.3.2. 



In a region containing bubbles, with any assumed 

 distribution of sizes, and having a sharp boundary, 

 sound incident on this region from bubble-free water 

 will be reflected at the sharp discontinuity. The anal- 

 ysis for this situation is given in Section 2.6.2 where 

 it is shown that the ratio of the amplitude of the 

 reflected and incident waves is given by the equation 



A" Ci — Co (cos e/cos t) 



(85) 



Ci -|- Co (cos e/cos t) 



This is essentially equation (119) of Chapter 2, with 

 Pi set equal to p and subscripts used for the incident 

 sound wave. The quantity co is the sound velocity in 

 the bubble-free medium, while ci is the corresponding 

 quantity across the boundary, where bubbles are 

 present. The energy reflection coefficient je is simply 

 the square oiA"/Aa. The angles t and « are the angles 

 which the incident and refracted sound make with a 

 line perpendicular to the boundary. The ratio of cos e 

 to cos t may be found from Snell's law, yielding 



In most cases of practical importance, ci is nearly 

 equal to Co. Thus, cos t is essentially equal to cos i. 

 By writing equation (81) in the form 



c? 



= 1 + 6 



COS'e 



cos' t 



1 -f- tan^ t 



0-1) 



the energy reflection coefficient je found by squaring 

 A"/A^ in equation (85) becomes 



Te = - > (86) 



16 



as long as h is much smaller than 1. This equation 

 may also be used when h is complex but less than 1, 

 provided that the absolute value of h is used. When 

 h is comparable to or larger than 1, the formulas be- 

 come considerably more complicated.^ 



28.3.5 Observed Acoustic EflFects of 

 Bubbles 



The effect of a known distribution of bubbles on 

 the propagation of sound through water has been 

 investigated in the laboratory at frequencies from 

 10 to 35 kc* The method used for producing a screen 

 of bubbles all of the same size has already been de- 

 scribed in Section 28.2.1. Special measurements were 

 made to determine the number of bubbles per cubic 

 centimeter at various points in the bubble screen. 



The bubble screen was about 17 in. .ong. Its thick- 

 ness varied with the bubble radius; for bubbles 0.034 

 cm in radius, corresponding to a resonant frequency 

 of 10 kc, the thickness was about 3 in., while for 

 bubbles 0.020 cm in radius, corresponding to 17 kc, 

 the thickness was more nearly 5 in. In continuous 

 flow, about 1 cu cm of air per second was fed into the 

 screen, resulting in a total density u of about 10"* 

 parts of air per part of water. When a bubble pulse 

 was formed by turning on the stream of bubbles for 

 1 sec, however, the bubble densities at the level of 

 the acoustic instruments were much less than this, 

 ranging between 10"* and 10~'. 



The acoustic measurements with the bubble pulse 

 consisted in measuring the sound reflected from and 

 transmitted through the screen at a fixed frequency 

 as a function of time since the beginning of the pulse. 

 The transmission loss was measured by reading the 

 sound level in a hydrophone placed on the far side 

 of the bubble screen from the projector. The reflected 

 sound was measured by a hydrophone placed on the 

 same side of the bubble screen as the projector, but 

 separated from the projector by several baffles. The 



