SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 



469 



contributes to the scattered sound and appears to be 

 due to interference between different scattered wave- 

 lets. It is not easy to determine the precise point at 

 which this term becomes important, but it can be 

 shown to be negligible, for resonant bubbles, pro- 

 vided the attenuation per wavelength is less than a 

 few decibels. This is the same condition that must be 

 satisfied if the change which resonant bubbles pro- 

 duce in the sound velocity of the medium is to be 

 relatively small. Since this condition appears to be 

 satisfied in observed wakes, this additional term will 

 therefore be neglected in the following derivation of 

 practical formulas for the attenuation, scattering, and 

 reflection of sound by water containing bubbles. 



28.3.2 



Transmission 



The type of analysis developed in the preceding 

 section will now be appUed to find the transmission 

 loss through a region containing bubbles. It will first 

 be assumed that within this region all the bubbles 

 are of the same size. In each cubic centimeter there 

 are assumed to be n bubbles; n may vary from point 

 to point within the region. If / is the intensity in the 

 incident sound beam, the rate at which sound energy 

 is extinguished from the beam by each bubble will 

 be <rj, according to equation (35). Let 7(0) be the 

 intensity at the point where the beam enters the 

 region containing bubbles and let /(r) be the in- 

 tensity after the beam has penetrated a distance r 

 through the region; r is measured along a sound ray. 

 The increment of I(r) after passing an infinitesimal 

 distance dr is, of course, negative and has the value 



dl = -n{r)<Tj(r)dr. (51) 



By integration of equation (51) over the path fol- 

 lowed by the sound, it is found that at any distance n 



lin) = 1(0)6-'" ^'' "<'■'* = J(0)e-""^'"' , (52) 



where iV(ri) is the total number of bubbles in a 

 column of length n and unit cross section. If ri is set 

 equal to w, the total thickness of the region, equa- 

 tion (52) gives the total extinction produced by the 

 bubble screen, or the attenuation as it is usually called 

 in underwater sound work. Expressing the attenua- 

 tion on a decibel scale, equation (52) is equivalent to 



10 log 



m 



I{w) 



= 10 X 0.434 X n(TeW = K^w, (53) 



where n is the average bubble density in the screen, 

 defined by 



IT"" 1 



n = - \ n{r)dr = - N{w) ■ (54) 



wJ w 



The quantity K„ in equation (53) is usually called 

 coefficient of attenuation, which is conventionally 

 given in units of decibels per yard. Since n and (r« 

 are usually expressed in units of cm~' and cm^, re- 

 spectively, and since there are 91.4 cm to the yard, 

 Ke in decibels per yard becomes 



Ke = 396.8n(r„ 



(55) 



The attenuation coefficient Ke is rather easy to 

 determine by acoustic measurements either of a 

 wake (see Chapter 32) or of a bubble screen produced 

 in the laboratory (see Section 28.2). Since as is known 

 for resonant bubbles from the experimental deter- 

 mination of the damping constant 5r already described 

 in Section 28.2, the bubble density n can be computed 

 from Ke and ae by equation (55), on the assumption 

 that only bubbles of resonant size are present. How- 

 ever, among copious masses of bubbles, as found in 

 wakes, there will usually be a wide dispersion of 

 bubble sizes. It is important, therefore, to evaluate 

 the attenuation produced by such nonhomogeneous 

 bubble populations. 



Let the number of bubbles per cubic centimeter 

 with radii between R and R + dR be denoted by 

 n{R)dR, and define Se as the total extinction cross 

 section per cubic centimeter. From equations (34) 

 and (43), it is then found, by adding up or integrating 

 the cross sections of all bubbles contained in one 

 cubic centimeter, 



4:rfl2„(fl)'^ 



& 



=X' 



(I-; 



dR 



(56) 



-1-52 



Bubbles of near-resonant radius will make a large 

 contribution to Se- If n{R) does not change rapidly 

 for radii near resonance, the integral over the reso- 

 nance peak in equation (56) may readily be evaluated. 

 This procedure gives the correct value for Se pro- 

 vided that absorption by bubbles far from resonance 

 can be neglected. Even if the density of bubbles near 

 resonance is comparable with the bubble density at 

 other radii, resonant bubbles will probably make the 

 major contribution to Se, since Oe is unquestionably 

 much greater for resonant bubbles than for those of 

 other sizes. However, according to what has been 

 said in Section 28.1.2 about the gradual shrinkage of 

 bubbles, a large number of veiy small bubbles are 

 likely to be present which may contribute apprecia- 



