520 



OBSERVATIONS OF WAKE ECHOES 



However, if the wake is highly transparent, or 

 N{w) <K 1, the echo level E, and also W as computed 

 from ^co will be found to increase with the oblique- 

 ness /3 of the impinging sound beam. In this case, the 

 replacement of '^cd by ^o, according to equation (19), 

 may be expected to give a wake strength independent 

 of 0: 



+ J-3 



-^0 = I01og]sec/3l b{^)b'{(l>) [cos<^-tan/3sin<^]2rf0>. 



"'"' (41) 



For short pulses (?'o « w, say ro <w/2) the appro- 

 priate value of ^ is 



+ *• 



^' = 10 log { fe^'"" '"'"' * - '' b(4>)b'(,i>)dA. (42) 

 -*' 

 This formula is a specialization of equation (28) for 

 /3 = 0; hence D in equation (28) is approximately 

 equal to r. Moreover, according to equation (35), 

 the range from transducer to wake must be less than 

 200 times the pulse length, if equation (42) is to be 

 valid. For a sound beam trained obliquely at the 

 wake — or for /? 7^ — equation (28) loses its useful- 

 ness, as the wake index becomes a quantity varying 

 with time. 



Though their mathematical expressions appear 

 rather different, the numerical discrepancy between 

 *' and ^„ is quite small, probably never exceeding 

 1 db for transducers of high directivity. For signals of 

 intermediate length, of the order of the wake width, 

 the mathematical analysis is difficult; it is suggested 

 simply that ^„ be used. For practical purposes, the 

 differences between the various types of wake indices 

 might be neglected altogether, by writing for perpen- 

 dicular incidence of the sound beam 



SE' = 10 log fb{<l>)b'(<l>)d<t> . 



(43) 



This approximate formula reveals a close relationship 

 between the wake index and the surface reverberation 

 index, defined in a University of California Division 

 of War Research [UCDWR] report,^ as 



Js= lOlog|lj6(.^)6'(0M«^}- 



Hence, 



/, + 8db = ^. 



(44) 



Furthermore, it is of interest to compare the for- 

 mulas giving the wake strength as a function of the 

 physical parameters and of the pulse length: 



Short pulses, ro <K w 

 W = 10 log hs 





1- 



e 



—2trenro I —2<TenCri — D aec p) { 



(45) 



-2<reA'(w) 



}■ 



(46) 



Long pulses, tq^ w 

 W = lOlogfes = lOlogj — -Fl 



It will be noted that the first exponential in equation 

 (45) becomes equal to that in equation (46) for ro = w, 

 because of equation (54) of Chapter 28, reading 



nw = N(w), 

 which is the definition of n. Equation (45) is an ap- 

 proximate formula, because in its derivation the as- 

 sumption n{z) = constant = n had to be made. 

 However, while equation (46) applies to the constant 

 average echo intensity constituting the central part of 

 long-pulse echoes, equation (45) represents the entire 

 average profile of short-pulse echoes. 



So far the entire discussion has been restricted to 

 average echo intensities. But, as described in Section 

 30. 1 .3, peak echo intensities are customarily measured 

 by the San Diego observers. The measurements re- 

 ported in Chapters 11 through 17 of this volume on 

 the "band" or "point" method of reading reverbera- 

 tion records imply that about 6 db must be sub- 

 tracted from the wake strengths computed by equa- 

 tion (38) from peak intensities, in order to express 

 them on the scale of average intensities envisaged in 

 equations (45) and (46). This correction will be ap- 

 plied only in Section 34.3.1, where the interpretation 

 of the observed wake strengths by the acoustic theory 

 of bubbles is discussed. In that context, the wake 

 strength computed from the measured peak ampli- 

 tudes of short-pulse echoes, and then corrected by 

 subtracting 6 db, will be regarded as corresponding 

 to the maximum of the profile (45) , or 

 n — D sec ;3 = 0. 



33.1.4 Decay Rate of Wake Strength 



The decay rate of wake strength, in terms of the 

 physical properties of the wake, is found by differen- 

 tiating equations (45) and (46) with respect to the 

 time. Before doing so, it is advantageous to make the 

 substitutions 



_ Hw 



and 



Niw) = 



4.34(reW 



4.34o 



