516 



OBSERVATIONS OF WAKE ECHOES 



other words, put this term in front of the integral, re- 

 placing X by an unspecified, constant, mean value x. 

 With this procedure the integral over dx in equation 

 (14) can be evaluated without further approxima- 

 tion: 



2x — w 



1 +;^ 



1 + i^Z n{x)e 



L 2r cos /3 J Jo 



_ r 2x - w >3 



~ L 2f cos b\ 



2<TeN(x) sec 03 + 0) 



dx 



V 



2r cos /3_ 



2„eN{w) sec (3+«) COs(g -H 0) 



2crc 



(15) 



Since, by definition, x is confined to the interval be- 

 tween and w, the inverse cube correction factor al- 

 ways lies between the following limits: 



[wsec/3~|~' r 2x— w 1~^ f wsec/3l~' 



Since w sec /3/2r is much less than 1, in most practical 

 echo-ranging situations this correction factor is un- 

 important, even at short ranges. In view of the 

 hmited accuracy obtainable with the current tech- 

 niques of measuring echo levels, this correction factor 

 will be omitted. The echo integral, equation (13), 

 thus assumes the form 



- 



rj:32Q0-2ar _ 



Note that 



ha 



f bi4>)b'{4>)L' 



[1 



cos (^ — tan /3 sin ^ 



1nN(w) sec C/S + *)"] Jj, 



(16) 



10 log e'"^<""> = 2Hy, 

 is the two-way transmission loss in db for perpendic- 

 ular transit through the wake, according to equation 

 (8) of Chapter 32. 



It will be observed that for a high acoustic opacity 

 of the wake \_atNiyo) ^1 ] the exponential in the 

 bracket under the integral is very much less than 1. 

 Hence, in the case of infinite acoustic thickness of the 

 wake, there follows the rigorous formula 



aj^QO.Zar _ 



87r(r. 



+i-^ 



■ l6(<^)6'(</))[cos<^- tani3sin0]'d0- 



"-\-^ (17) 



By comparison with equation (5), the value of the 

 wake index ^^, where the subscript has been added 

 to emphasize that this index refers to infinite acoustic 

 thickness, can be identified: 



^» = 10 log I 6(<^)6'(0)[cos - tan ;8 sin <^Jd^. (18) 



Conversely, for highly transparent wakes \_(isN(yo) 

 <5C 1] the second bracket under the integral in equa- 

 tion (16) may be developed into a series and the quad- 

 ratic and higher terms neglected, yielding 



1 _ g-2«A-f») sec (3 + *) _ 2cr,iV(u>) sec (;3 + 0) • 



The formula for the wake strength of highly trans- 

 parent wakes then reads 



h. j=3ioo.2- = J^^Ar(„,) sec ^ [k^WW ■ 

 io 47r J IT 



-2-ff 



[cos <t> — tan |3 sin 0jd0, 



*^o = 10 log S sec i3 



3 fbWb' 



{<t>) [cos 



— tan /S sin 0]^d0 



(19) 



Numerically, the difference between ^co in equation 

 (18) and ^o in equation (19) is negligible for direc- 

 tional transducers as long as is small. To a very 

 good approximation the general equation (16) can, 

 therefore, be written -as 



^V-3io»-^-- (if) = ^ [1 - e-^'^^''"' =" ^ ] ■ (20) 



Expressed on a decibel scale, equation (20) becomes 

 E - S + 2H + 10 log f - ^a, 



-2aeN{w) sec / 



= 10 log 



iha 



[1 



= ^}. 



(21) 



Hence, the reflectivity of the wake per unit solid 

 angle is 



g _ _ . fi rj^ _ ^-2aeN(w) seat 

 8ir ae 



= ^. 



(22) 



By this equation, the problem proposed at the outset 

 of this section is solved for sufficiently long pulses 

 (ro > w). However, it should be remembered that 

 equation (21) does not represent the entire echo pro- 

 file, but applies merely to its central part which has 

 a constant average intensity because the pulse over- 

 laps the entire wake. The rise and fall of the average 

 echo profile, when only part of the pulse intersects the 

 wake, cannot be represented by a simple formula, be- 

 cause of mathematical difficulties of the same nature 

 as will become apparent presently in the discussion 

 of short-pulse echoes. 



Short Pulses 



For pulse lengths smaller than the wake width 

 (r-o < w), equation (11) has to be integrated over the 



