262 



THEORY OF REVERBERATION INTENSITY 



UV, the second factor in the expression (35), we 

 notice in Figure 7 that if UH Ues in S[ and if the ping 

 length is sufficiently short, then UWH is very nearly 

 a right triangle with the right angle at H. Thus, 



COS a 



_ c't 



4 cos a' 



since the perpendicular distance from S'l to S3 is 

 c't/4. With our assumptions, 



UV 

 2 

 Thus 



c't 



UW = 



UV = 



2 cos t 



Substitution of this expression for UV and of ex- 

 pression (36) for p into equation (35) gives 



'^' , c'^COSa' 



pdp = 



Jsi 



C T 



2c' 2 COS a' 



4 ' 



(37) 



In equation (37), c, the average sound velocity, can 

 be replaced with little error in any actual situation 

 by Co, the sound velocity at the transducer. Substi- 

 tuting expression (37), modified by this replacement, 

 in equation (34) gives 



Git) = '^m'h^'^ 

 4ir 4 



■T T" 



- h{e,<t>)h'{e, 



: Jo 



<j>)d<t>. (38) 



The subsequent procedure is similar to that 

 adopted following equation (17); it is convenient to 

 rewrite the theoretical expression (38) for reverbera- 

 tion in terms of decibels as a function of range. As 

 before we define the range r of the reverberation as 

 Co//2; this differs negligibly from the distance along 

 the ray path OW of Figure 7. Proceeding as in Sec- 

 tion 12.2, we find 



10 log Git) = 10 log yjj + 10 log iF-F') 



+ 10 log {~j - 30 log r + JM - 2A, (39) 

 where 



JM = 10 



^'^ilo 



bie,,t>)b'ie,<t>)d<t> (40) 



and 2A is the two-way transmission anomaly along 

 the ray path. 



Equation (38) indicates that the intensity of sur- 



face reverberation, like that of volume reverberation, 

 should be proportional to the ping length if the as- 

 sumptions used to derive equation (38) are satisfied. 

 If these assumptions are not valid in a particular 

 situation, however, the surface reverberation inten- 

 sity may not be proportional to the ping length. 

 Frequentlj'^, these assumptions are not satisfied; thus 

 the proportional dependence on ping length pre- 

 dicted in equation (38) is not as general as the same 

 dependence for volume reverberation predicted by 

 equation (14). For example, if refraction is sharply 

 downward, surface reverberation from ranges near 

 where the limiting ray leaves the surface will not 

 obey the theoretical law (38). Qualitatively, it can 

 be seen from Figure 6 that at a range Co</2 somewhat 

 greater than the limiting range, a halving of the ping 

 length may lead to more than a halving of the sur- 

 face reverberation intensity, since most or all of the 

 shorter ping may be too far from the surface to be 

 effective in scattering. It will be recalled that this 

 situation in which the proportional dependence on 

 ping length predicted by equation (38) is invalid is 

 just the type of situation which had to be ruled out in 

 order to obtain equation (31), which in turn led to 

 the result (38). 



In deriving equation (38), surface reflections were 

 explicitly neglected. As explained in Section 12.2, the 

 surface reflections make for alternative ray paths 

 from the transducer to the scatterers. It is shown in 

 Section 12.5.6 that these alternative paths usually 

 cause the value of 10 log m', computed from measured 

 surface reverberation intensities and transmission 

 anomalies by means of equation (39), to be about 

 6 db greater than the actual value of the backward 

 scattering coefficient of the surface scatterers. 



The quantity J»(S) in equation (39) is called the 

 surface reverberation index. In general, this index de- 

 pends on the orientation of the projector relative to 

 the vertical and on the range of the reverberation, 

 and is difficult to calculate for arbitrary transducer 

 orientations. When the transducer beam is nearly 

 horizontal, however, the expression (40) can be evalu- 

 ated approximately. It is shown * that if the trans- 

 ducer is a large rectangular piston in an infinite 

 baffle then, approximately, 



£'m,W(e.*)<i* - ^"-^'°;?-^'°W , (4.) 



where 



cos 8 



/»2i 



Q(0) = biO,<t>)b'iO,4>)d,f>, (41a) 



Jo 



