532 HERSEY AND BACKUS [CHAP. 13 



back-scattering coefficient, m{z,f), measured at depth z and for frequency/, 

 can be expressed as follows, 



MzJ) = IMz)^' (41) 



where ai{f)l4:7T is the scattering cross-section of the i-th type of scatterer at 

 frequency/, and ni{z) is the number of scatterers (per unit volume) of this kind 

 at depth z. Clearly, in order to obtain population densities, ni{z), from measure- 

 ments of volume back-scattering, ni{z), it is necessary to obtain as much 

 information as possible about the scattering properties of each individual. 

 From estimates of individual scattering cross-sections one then uses the above 

 relation to estimate population densities from the measured volume scattering. 

 Thus far, to our knowledge, no one has attempted to cope with estimates other 

 than those based on the assumption of a single kind of scatterer, all members 

 of the scattering layer having identical cross-sections. Euphausids and fishes 

 with swim-bladders can be considered to some profit. As mentioned above, an 

 upper limit for the probable scattering cross-section of a euphausid, ct/477~ 2 x 

 10~i2 jn2^ can be derived from Bridgman's measurement of the compressibility 

 of oil from certain euphausids. From equation (41) the population density 

 implied by m~10-9 to IQ-^ m-i varies from 500 to 500,000 individuals per 

 cubic meter. Available information on the density of euphausids is sum- 

 marized by Gushing and Richardson (1956). This shows that, with the excep- 

 tion of the occasional unusual situation (in which a few hundred individuals 

 per cubic meter have been observed), euphausid populations are much below 

 the above requirements, being more the order of 10-15 per cubic meter of 

 water. Nevertheless, these authors correlate net hauls with echo-sounder 

 recordings of near-surface reverberation to suggest that euphausids (about 

 200 per cubic meter) are the scatterers in their observation. Using the value of 

 CT/47r above it is easy to show that euphausids as densely packed as 200 per 

 cubic meter could easily be recorded at shallow depth by echo-sounders 

 operating at 10 kc/s. 



Assuming that the layers for which we have computed values of m consist 

 of bubble scatterers, we have estimated the number of individuals required 

 for peak values of m. The back-scattering cross-sections at resonance were 

 obtained by computing the resonant-bubble radius, R, for the indicated 

 frequency, taking the theoretical damping constant, §, from Fig. 13, and 

 substituting in the formula 



a/47r = R^I8^. 



Both our results and those quoted in Machlup and Hersey (1955) suggest that 

 the population density in layers showing a resonant frequency is not much 

 greater than the count, made by Johnson et al. (1956), discussed above. The 

 greatest density we have so far computed is 1.7 x 10~2 scatterers per cubic 

 meter. 



