CONCEPT OF WAKE STRENGTH 



517 



volume in which the wake and the cylindrical shell 

 (thickness Vo, inner radius rO of the pulse intersect. 

 Hence, 



= 5^''-//^''<«^'<*' 



n(x)e-*"^<"'"^"'+*'rf?-rf<^- (23) 

 The limits 0a and 4>b of the integral over d4>, unspeci- 

 fied for the time being, are determined by the relative 

 position of pulse and wake and, therefore, vary with 

 time; their explicit form will be evaluated after the 

 discussion of the integral over dr has been finished. 

 Equation (23) is valid only for rectangular pulses — 

 that is, for echo-ranging signals whose intensity is 

 constant for their duration. 



The variation of n(x) and N{x) across the wake 

 prevents integration of (23) in closed form. In order 

 to gain any insight at all into the behavior of short- 

 pulse echoes, a drastic simplification becomes neces- 

 sary. For this reason the discussion is confined to a 

 wake of constant bubble density in the transverse 

 direction. Putting thus 



n(x) = constant = n = N{w)/w 

 and 



N{x) = nx 



the integral reads 



In ^tr J J 1 



— 0.2ar 



6(0)6'(0)ne-=''^"^'= '^+^^drd,j>- 



(24) 



While in the general case <l>a and 06 vary as r increases 

 from n to Ti -f To, for short pulses this change is quite 

 negUgible. The integration over dr may then be 

 carried out before the integration over d0 without 

 difficulty. On account of the geometric relation, from 

 Figure 2, 



cc sec (;3 -I- 0) = r - D sec (/3 -I- 0), 



the integral over dr in equation (24) becomes 



/lQ-0.2ar 

 — — ne'^'"' Ir- D sec (/3 -|- 0)]rfr • (25) 



After appl3dng the theorem of the mean value of an 

 integral with respect to the factor 10~°'^'"'f^ the inte- 

 gral (25) is transformed into 



*— 3in-0.2ar* 





2aCTiro-j - 2(Ten[n— D sec (^+<^)3 



Since r* differs very Httle from ri (because ri < r* 

 < '"i + ro, by definition), r* can be replaced by n 



without any appreciable loss of accuracy. Thus the 

 echo integral, equation (24), reads 



/o 



rllO" 



ha 



-[1-e 



- 2(renro 



■]/«>(0) 



b'{(t>)e-'""^'''~'^^''^ + *'^d4>. (26) 

 The exponential under the integral measures the 

 transmission loss resulting from absorption and scat- 

 tering inside the wake, since ri — D sec (/3 -|- 0) is the 

 distance, along any ray (0 = constant), from the 

 inner boundary of the pulse to the front of the wake. 

 Now by making the substitution 



ri- D sec {fi + 0) 



= ri - D sec /3 - D [sec (j8 + 0) - sec /3], 

 equation (26) assumes the form 



if^3jQ0.2an 



lo 



h(Ts 



lf}b 



[1 



2o-€nro~| — 2ffenO 



"°]e 



n-Dsecft |{,(^) 



6'(0)e*""^ '^''" *^ + *' ~ ''•= ^^ d0. (27) 



Here the factor (1 — e"^"™") comprises the effect of 

 the pulse length on the echo strength. The transmis- 

 sion loss inside the wake has been split into two 

 factors. The first one, namely e-'''"^^"-'^""'^\ de- 

 pends only on the range ri of the pulse, which in- 

 creases with time, but not on the directivity of the 

 transducer; in fact, this first factor is simply the 

 transmission loss, measured along the sound beam 

 axis, from the boundary of the wake to the pulse. The 

 second factor is independent of time and appears as 

 an exponential under the integral over d<t>. Using the 

 abbreviation 

 tftb 



^'= lOlogj r6(0)6'('/')lO''"'"^'^'™ (^+*>- ='"' ^^dA , (28) 



which defines another wake index appl3Tng to short- 

 pulse echoes, equation (27) may be written on a 

 decibel scale as follows. 



E - S + 2H + \0 log n - ^' 



= 10 log I— [1 - e2«S,-,-]g-2.m(n-DsecO)l (39) 



The quantity (ri — D sec /3) is the distance, meas- 

 ured along the transducer axis, which the rear 

 boundary of the pulse has penetrated into the wake. 

 Hence, for a directional transducer no appreciable 

 echo intensity will be obtained outside the range of 

 penetration which is given by 



Dsecfi <ri< {D + w) sec /3. 



