210 22 



The scattered waves are hard to treat analytically because they are 

 themselves subject to continual re-scatterlng. It may be of Interest, how- 

 ever, to consider for comparison the scattering by a single bubble. It is 

 found that the bubble should scatter as much energy as is transported in the 

 Incident waves across a certain area A, which may be called the scattering 

 cross section of the bubble. The value of A for waves much longer than the 

 diameter of the bubble, as given in Equation [75]. is 



A - i „jf 2 



kt u) = 0, A = 0; from w = 0.6tjj well up toward such high frequencies that the 

 validity of the formula becomes doubtful, A exceeds nR^^, the actual cross- 

 sectional area of the bubble itself. For air in water, N = 124. For such 

 bubbles, A approximates H^nR^^, the superficial area of the bubble, when w 

 lies within the range 



4wp < (J < 20wg 



An extremely sharp resonance effect occurs. At w = <jp, A = 

 ZOy^OOnRg^; the bubble scatters more than 20,000 times as much energy as 

 would fall on it directly. If w differs from coq by 2 per cent, however, the 

 scattering is only a ninth of its maximum value. The half-value width, or 

 width of the resonance peak between points on the curve at which A has half 

 of its maximum value, is 0.0l4w^. 



The energy scattered by a group of bubbles should be Just the sum 

 of the energies scattered by the individual bubbles, provided the bubbles are 

 distributed at random, and provided differences in the intensity of the in- 

 cident waves may be neglected. If the bubbles are not distributed at random, 

 however, interference effects may occur. The reflected beam from a bubbly 

 region of water having a sharp boundary arises from constructive interference 

 of the waves scattered by the individual bubbles, and it is for this reason 

 that the resonance peak in reflection is so much broader than the peak in the 

 scattering curve for a single bubble. If there is no sharp boundary but the 

 density of bubbles varies gradually, the reflection will be weaker; if the 

 density is nearly uniform within any distance equal to one wave length of the 

 incident waves, the process becomes essentially one of scattering with little 

 resemblance to regular reflection. 



