SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 



473 



smaller than resonant size, a, falls off as the fourth 

 power of the wavelength; hence such small bubbles 

 are not likely to contribute much to S, unless present 

 in very large numbers. Bubbles larger than resonant 

 size have a scattering cross section about four times 

 their geometrical cross section, but are not likely to 

 be present in greater abundance than smaller bubbles. 

 Thus equation (77) should be valid in a wide range 

 of circumstances. 



Since the ratio of S^/Se is equal to the ratio of 

 (T,/ae at resonance, equation (74) is still valid when 

 the transmission loss across the screen is large; thus 

 the scattered sound in this case is just the same as if 

 all bubbles were of resonant size. When the trans- 

 mission loss across the screen is small, however, equa- 

 tion (76) must be used, with S, substituted in place 

 of 7i<r,. 



28.3.4 Reflection and Refraction 



The presence of bubbles changes the velocity of 

 sound. If the bubble density is sufficiently great, this 

 effect may become practically important, leading to 

 reflection and refraction of the sound beam. Since in 

 ship wakes the number of bubbles present per cubic 

 centimeter is usually not sufficiently great to change 

 the sound velocity very greatly, these effects are not 

 discussed in great detail here. The methods of analysis 

 required to deal with this case are briefly sketched, 

 and the results stated. 



The sound velocity is defined by equations (6), 

 (18), and (26) in Chapter 2 as 



dp 



a7 



(78) 



where p and p are the pressure and density respec- 

 tively of the bubble mixture. If only a volume V of 

 the mixture is considered, equation (78) may be 

 written in the form 



c2 = 



at 



dV 



'Tt 



(79) 



using the relation pdV -\- Vdp = 0. The quantities 

 dp/dt a,nddV/dt maybe evaluated from the equations 

 in Sections 28.1 and 28.2 yielding the basic equation 



X2 C n(R)RdR 



^ 



= 1 



■/; 



(f-O 



(80) 



+ iS 



where Co is the sound velocity when no bubbles are 

 present, and n{R) is the number of bubbles per cubic 

 centimeter with radii between R and RdR. The inte- 

 gral in equation (80) extends over all bubble sizes. 

 It is assumed that all bubbles present have a radius 

 much smaller than the wavelength, and that the 

 average distance between bubbles is larger than their 

 radius. If these two assumptions are not fulfilled, the 

 theory on the preceding pages breaks down. The de- 

 tails of the derivation of this equation are given in 

 references 3 and 4. 



It may be noted that equation (80) is valid only 

 when the density of the liquid-bubble mixture is 

 substantially the same as that of the liquid. Results 

 are given which may be used for any density of 

 bubbles, provided that the bubbles are all much too 

 small to resonate, but much too large for surface 

 tension to become important." 



For frequencies far from resonance, the imaginary 

 term in equation (80) may be neglected. For fre- 

 quencies below resonance, this leads to the equation 



A- 1 I 3" 



2 '2 



C TJr 



(81) 



where u is the total volume of air present as bubbles 

 in 1 cu cm of the liquid-bubble mixture. Thus u is de- 

 fined by the equation 



= \—RMR)dR. 



(82) 



The quantity r\r in equation (81) is the ratio of the 

 bubble circumference 2vR to the wavelength X at 

 resonance, as defined in equation (1). Equation (81) 

 is valid only for bubbles which are sufficiently large 

 that surface tension effects can be neglected; more- 

 over, if the expansion and contraction of the bubble 

 are adiabatic rather than isothermal, the last term in 

 equation (81) must be multiplied by the ratio of the 

 specific heats for the gas in the bubble. It is interest- 

 ing to note that, subject to these limitations, equa- 

 tion (81) is independent of the bubble radius. Even 

 if u is as low as 10~^ parts of air at atmospheric pres- 

 sure to one part of water, c/co is 0.62. 



When the bubbles are all greater than the resonant 

 size, the sound velocity is increased by the presence 

 of the bubbles, and the relation corresponding to 

 equation (81) is 



t 

 = 1-^ l^u(R)dR, (83) 



cl 



if^MR)dR, 



Vr -J R 



where u(R)dR, defined in equation (64), is the volume 

 of the bubbles present in 1 cu cm of liquid-bubble 



