SOUND PROPAGATION IN LIQUID CONTAININ(; MANY BUBBLES 



471 



(3), (5), and (G) ; more simply, it may be written down 

 directly, since by definition aja is the rate at which 

 sound is scattered by a single bubble, and since the 

 energy is spread out uniformly in all directions, at the 

 distance )■ it is spread out uniformly over an area 

 4xr-. In a small region of volume dV, the number of 

 bubbles is nd V, where n is the number of bubbles per 

 cubic centimeter. 



Equation (68) must be modified to allow for the 

 fact that the scattered sound will be attenuated on 

 its way from the region to a distance /• away. Over 

 long distances various sources of attenuation must be 

 considered, such as absorption in the water, scat- 

 tering by temperature irregularities, and so forth. 

 Over short distances, most of these effects may be 

 neglected, and the transmission loss taken from equa- 

 tion (52). The basic equation for the scattered sound 

 measured a distance n from the region dV then be- 

 comes 



dl. 



sdV 



47rr? 



le- 



JlMr)dr 



(69) 



where I is the intensity at the region dV. If sound 

 from different directions is incident on the region, / 

 must be averaged over all directions for use in equa- 

 tion (69). 



Computing the scattered sound intensity from 

 equation (69) is a much more complicated problem 

 than computing the total sound attenuation from 

 equation (51). In the latter case, equation (51) could 

 be integrated along a single sound ray, yielding equa- 

 tion (52) directly. The basic difficulty in solving equa- 

 tion (59) is that the sound intensity I at the volume 

 elemei^t dV includes sound scattered in turn from 

 other regions. To consider multiple scattered sound 

 of this type is rather complicated, and leads to inte- 

 gral equations which in general cannot be solved 

 exactly. Methods for treating this problem have been 

 extensively explored in astrophysical literature. "■ " 

 The problem of multiple scattering in wakes could 

 probably be studied with success by methods de- 

 •veloped for the corresponding optical problem.'^ 



Fortunately, bubbles absorb much more sound 

 than they scatter. From equation (46) and Figure 2 

 it is evident that the ratio (ra/<re for resonant bubbles 

 is less than 1 to 10 for frequencies above 15 kc. For 

 this reason, sound scattered several times from 

 resonant bubbles has usually traveled so far that it 

 is very weak. Multiple scattering will therefore be 

 neglected in all subsequent discussions. In simple 

 cases, the error resulting from this approximation 



will be less than half a decibel at frequencies above 

 15 kc. Even at 5 kc, the error will u.sually be less than 

 1 db. For scattering by nonresonant bubbles, multiple 

 scatterings cannot be neglected unless Ce is much 

 greater than as- 

 Even with this approximation, the computation of 

 Is from equation (69) is not simple. The quantity / 

 now becomes the sound intensity incident on the 

 region containing bubbles, and attenuated by its 

 passage through part of the region. However, to 

 compute Is Sit any one point the sound arriving from 

 all parts of the screen must be computed; the total 

 scattered sound must be evaluated by summing up 

 the contributions arriving from all different direc- 

 tions. In any practical situation, the directivity of 

 the receiving hydrophone must also be taken into 

 account in order to find the electrical signal received 

 in the measuring equipment. A detailed considera- 

 tion of these problems in cases of practical impor- 

 tance is given in Chapter 34. 



To give insight into fundamental features of the 

 scattering problem, it is desirable to eliminate these 

 geometrical complications as far as possible. Equa- 

 tion (69) is here applied to scattering from a bubble 

 screen, that is, from a layer of aerated water bounded 

 by two parallel planes a distance w apart. Instead of 

 integrating over all directions, we shall compute 

 simply the scattered sound reaching the point P from 

 all directions within a small cone of solid angle dU; 

 the quantity dQ is simply the area of a cross section 

 of the cone divided by the distance r^ from P to the 

 cross section. The geometry of this situation is shown 

 in Figure 3. 



Let 7(0) be the intensity of sound incident on the 

 screen; the incident sound is assumed to be a plane 

 wave, whose rays are inclined at an angle i with a 

 line perpendicular to the boundary of the screen. 

 Within the screen the intensity falls off exponen- 

 tially; since the path length dr is equal to sec idx, 

 equation (52) gives for the incident sound at a dis- 

 tance X inside the screen 



I(x) = 7(0) exp — o-esec i- I n{x)dx • 



As Figure 3 shows, the scattered sound which we are 

 considering makes an angle e with a line perpendicular 

 to the boundary of the screen. Thus in equation (69) 

 the length dr along the path of the scattered sound is 

 sec tdx. Thus we find for the sound scattered from a 

 small element of volume dV, at a distance ri from the 

 point P 



