472 



ACOUSTIC THEORY OF BUBBLES 



INCIDENT 

 SOUND WAVE 



Figure 3. Scattering from a bubble screen. 



d/.=^^^^^^^/(0) 



47rr2 

 exp — (7e(sec i + sec t] 



I n{x)dx • 



(70) 



For the volume element dV within the cone, we have 



dV = r\dQ.dr; 

 since dri is simply sec erfx as before, equation (70) be- 



comes 



n(x)cs sec td^dx 



dL = —~^~ 7(0) 



4x 



■[- 



exp — (rc(sec i + sec e 



I I n{x)dx • 



(71) 



This equation may be integrated over x from to w, 

 yielding 



cos I 

 (Teiw COS i + COS e 



<7-e(sec i + sec «) 



|l - exp I 



I n{x)dx 



\ 



ir ™ 



It is interesting to note that dl, in equation (72) is 

 independent of the distance from the screen to the 

 point P where the scattered sound intensity is 

 measured. This apparent contradiction is resolved 

 when it is realized that with increasing distance a 

 larger area of the screen is intercepted within the 

 solid angle c?Q. 



Equation (72) has two important limiting cases. 

 When the transmission loss across the screen is large, 

 the second term in the brackets is negligibly small, 

 and 



,^ o-srfO cosi ,„„^ 



d/. = — — • (73 



Ce'iir cos I + COS 6 



It may be noted that when e equals i, as is the case for 

 backward scattered sound, cos e equals cos i; if also 

 equation (46) is used for Us/ae, equation (73) yields 







do 



(74) 



On the other hand, when the transmission loss 

 across the wake is small, it is possible to use the ap- 

 proximate relationship 



e^= 1 - a, 

 yielding 



do f" 



d/s = (Ts — sec e I n{x)dx. {lb) 



4x ^0 



In terms of the average density n introduced in the 

 previous section, equation (75) becomes 



do 



als = as — sec f tow 

 47r 



(76) 



Thus when the transmission loss across the wake is 

 small, dis is proportional to a^ and n. But when the 

 transmission loss is great, the scattered sound reaches 

 a constant value, given by equation (73), and is in- 

 sensitive to changes in n or tw. 



When bubbles of different sizes are present, equa- 

 tion (69) for dIs may still be used, provided that n<Te 

 is replaced by Se, the total extinction cross section 

 per cubic centimeter, and wo-, is replaced by S„ the 

 total scattering cross section per cubic centimeter. 

 The quantity Se is discussed in the preceding section; 

 equation (65) gives the relationship between S^ and 

 u{Rr), the bubble density at resonance. A similar 

 analysis, considering bubbles only of near-resonant 

 size, leads to the following equation for the total 

 scattering cross section per cubic centimeter: 



ZvU{Rr) 



Ss = 



25r 



(77) 



The consideration of only those bubbles near the 

 resonant size is usually legitimate even if absorption 

 by nonresonant bubbles is appreciable. Since Cs, the 

 scattering cross section of a single bubble, does not 

 depend on the damping constant 6 for nonresonant 

 bubbles, it is possible to evaluate precisely the con- 

 tribution of bubbles of all sizes. For a single bubble 



