300 



DEEP-WATER REVERBERATION 



intensity at AB, the total energy reaching the surface 

 scatterers per unit time is proportional to I{AB) = 

 licor) sin 6. If the simple assumption is made that 

 the energy scattered in all directions is the same as 

 the energy scattered in the backward direction, it 

 follows from the definitions of m and m' [equations 

 (4) and (32) of Chapter 12] that the total energy 



0A= UPPER EDGE OF MAIN BEAM 

 0B= LOWER EDGE OF MAIN BEAM 



Figure 27. Energy reaching surface scatterers. 



scattered per unit time is proportional to m'. Thus, 

 the ratio of the total energy scattered per unit time 

 to the total energy reaching the scatterers per unit 

 time is proportional to m'/sin 6. The former ratio can- 

 not exceed unity; if it is to remain finite at small graz- 

 ing angles, m' must decrease at least as rapidly as sin 6. 

 If the scattering from the surface obeys Lambert's 

 law (the law of scattering of light by rough surfaces'), 

 then the backward scattering coefficient will be pro- 

 portional to sin^ d. At ranges of 100 yd or more, with 

 transducers at 16 ft, sin d = 6 and is inversely pro- 

 portional to the range. Thus, comparing with equa- 

 tion (43) of Chapter 12, if the scattering arises in a 

 thin surface layer, and if we can assume that equal 

 amounts of energy are scattered in all directions, the 

 surface reverberation would be expected to fall off as 

 the fourth power of the range, or faster. Supple- 

 mented by the added loss due to attenuation, such a 

 variation of scattering coefficient with grazing angle 

 could explain the observed dependence on range in 

 Figures 23 and 24. 



However, before we can accept the variation of m' 

 with grazing angle as an explanation of the depend- 

 ence of surface reverberation on range, we must de- 

 termine how thin a scattering layer is required for the 

 argument of the previous paragraph to be valid. 

 Figure 28 is a more exact drawing of the situation 

 pictured in Figure 27, drawn so that the layer has 

 appreciable thickness. In Figure 28 the projector 

 is at depth d, and the scattering layer has thickness h. 

 The scattering volume has a cross section CADE in 

 the plane of the paper, with CD at long range very 

 nearly equal to Cqt. Energy enters the scattering 

 volume through AE (as in Figure 27) or through AC. 

 From Figure 28 it is easy to obtain a simple criterion 



for the validity of the argument of the previous para- 

 graph, if attenuation in the surface layer can be 

 neglected. For, with this approximation, the energies 

 entering through AC and AE are proportional re- 



-—C^T-— 



Figure 28. Expanded view of surface layer. 



spectively to the solid angles formed by rotating <i> 

 and ^ in Figure 28 about a vertical axis (in the plane 

 of the paper) through 0. At long ranges, d small, 

 these solid angles are proportional respectively to 

 angles <j> and \p. Thus, the calculation in the previous 

 paragraph of the energy entering the scattering 

 volume per unit time is incorrect unless the angle <j> 

 is very small compared to ^. At long range we have 

 approximately, with OC = r, 



h = (AC) cos = r<i> cos d 

 AB = li/ = (AE) tan 6 = Cqt tan 6. 



Thus the condition that (j> be very small compared 

 to xj/ becomes 



h 



rcosd 



« 



(cot) tan d 



(7) 



For small 9, cos d = 1, sin 8 = 6 = d/r. Thus equa- 

 tion (7) becomes 



h « (cot)- ■ (8) 



r 



At 1,000 yd, with 100-msec pings and d = 5 yd, 

 equation (8) gives h<K30 in. There are scarcely any 

 data, but it seems likely that the surface layer might 

 frequently be thin enough to satisfy the relation (8). 

 On the other hand, in rough seas it would not be sur- 

 prising to find that the relation (8) is violated. 



Equation (8) was derived neglecting attenuation 

 in the scattering layer; if attenuation is taken into 

 account then it can be shown that the expression (8) 

 must be replaced by 



1 - e-'"-''^'' « (cr)a, (9) 



where the attenuation in the layer, in decibels per 

 yard, is 4.34a. The value of a probably depends pri- 

 marily on the population of bubbles in the surface 



