III-33 



Multiple Scattering 



In most regions of the ocean, the density of air bubbles is very low. 

 Usually the spacing between air bubbles is so great that, when a sound wave 

 passes through the medium, the scattered spherical wave from any one bubble 

 is negligible compared to the sound field by the time the scattered wave reaches 

 the bubbles nearest to it. In this situation, each bubble scatters the incident 

 wave, and multiple scattering can be ignored. If there are n bubbles per unit 

 volume, each with extinction cross section Og, the total extinction cross section 

 per unit volume will be n o^. A sound wave progressing in the x-direction with 



intensity I(x) will be attenuated according to — = -no^ I; therefore Kx) = 



dx ^ ' \ / 



l(o)e~^°e^. The scattered energy will be in the form of many incoherent spheri- 

 cal wavelets, and attenuation of the sound wave will be the dominant phenomenon. 



Occasionally there are regions of relatively high air-bubble density, 

 amounting to a fractional volume of air of perhaps up to lO''^. We have already 

 mentioned certain regions of high air-bubble density such as fresh wakes, schools 

 of fish with air bladders, or patches immediately under the ocean surface due to 

 rain and spray. If the fractional volume of air is 10"®, the bubble spacing must 

 average on the order of 16a (where a is, as usual, the bubble radius). We have 

 noted earlier that the sound frequency at which a bubble is resonant corresponds 

 to a wave number k^ such that (k^ a)^ = 3gh^ ^ 1.5 x 10'*, so that the 

 resonant wavelength is approximately \^ ^ 500 a. With a fractional volume of 

 air of 10 , we therefore find that a resonant wavelength measures approximately 

 30 bubble spacings. Under these circumstances, we may expect the sum of the 

 amplitudes of the scattered wavelets arriving at a bubble from its nearest neigh- 

 bors to be of the same order of magnitude as the wavelet the bubble scatters itself-- 

 for a bubble may have some 10 nearest neighbors, which are 10 to 20 bubble radii 

 removed from it and all more or less in phase with it (since there are ± 30 bubble 

 spacings to a wavelength). Near resonance, the amplitude of the scattered wave 



at the bubble surface - P tends to be an order of magnitude greater than 



the amplitude 1 p^^^^ of the "incident" wave. This may be seen from Figure 

 III-9, which gives the damping constant at resonance. At resonance, we see 



from Table III-l that - I p I = I p. I /6 . But, from Figure III-9, we see 



a I sc I I mc I ° 



that 6 is typically between 0.01 and 0.2 for frequencies between 100 cps and 

 10, 000 cps, which proves the above assertion. 



In the immediate neighborhood of each bubble, therefore, the bubble's 

 own scattered wave is of the same order of magnitude as the sum of the scattered 

 waves received from neighboring bubbles, and an order of magnitude greater than 

 the "incident" wave. However, since the bubbles and the bubble spacing are much 

 smaller than a wavelength, we might expect the medium to exhibit some more or 



;artl)ur B.ILittleJnt. 



S-7001-0307 



