488 VIGOUKEUX AND HEKSEY [CHAP. 12 



is proportional to IjR provided the degree of roughness of the bottom is such 

 that the scatter does not depend on the angle of incidence. In many cases back- 

 scatter decreases as the angle of incidence increases, with the result that the 

 ratio of echo intensity to reverberation intensity is roughly independent of 

 range over a considerable portion of the range. 



An argument similar to the one used above indicates that the less serious 

 body or volume scatter causes the echo/reverberation ratio to vary as IjR^ 

 (Horton, 1957). 



Reverberation is thus a factor in limiting the range at which echoes can be 

 detected. In shallow water it may well be the limiting factor, but in the deep 

 ocean, noise originating from the sea usually sets the limit to the range of 

 detection. 



J. Fluctuation of Sound 



Even when the emitter delivers a perfectly steady signal to the water, as 

 measured a few feet away, the signal received at long or moderate range is 

 subject to random fluctuations of intensity. This phenomenon, akin to scintilla- 

 tion of stars or fluctuations of radio signals received via the ionosphere, is due 

 to time variation of inhomogeneities of the medium. The surface, for instance, is 

 in continual motion, swell changes the angle of reflection, and there may be 

 turbulence in the body of the water especially near the surface ; also air bubbles 

 move before disappearing by solution or at the surface, and there are local 

 variations of temperature. All these effects cause the signal to fluctuate. 



An idea of the way in which the fluctuations depend on the range, the wave- 

 length and the mean size of the irregularities can be obtained by supposing that 

 the velocity, c, of propagation of sound waves varies from point to point accord- 

 ing to the law 



c = Co[l+an{x, y, z)], (25) 



where a is the r.m.s. value of the variations of velocity and n{x, y, z) is the 

 variation of velocity whose r.m.s. value has been normalized to unity (Mintzer, 

 1953). The autocorrelation function of the variations is supposed to be of the 

 form exp { — r^ja'^), where a is the mean size or "scale" of the inhomogeneity. 

 If the range and the propagation constant are denoted by R and k, it is found 

 that if kR and ka are both much larger than unity, the variance of the ampli- 

 tudes of short pulses is 



cr2 = TTy^k'-a^aR. (26) 



If, on the other hand, the frequency is so low that ka is much less than unity, 

 although the range is large enough for kR to be still much greater than unity, 



a^ = ny^k^a^a^R. (27) 



For a continuous signal the values above must be multiplied by a factor 

 greater than unity. 



