CONCEPT OF WAKE STRENGTH 



519 



words, with markedly oblique incidence of the sound 

 beam the concept of wake loses its usefulness for 

 short signals. The theoretical derivation of the echo 

 profile for short-pulse echoes obtained with an 

 obliquely trained transducer would require extensive 

 numerical integrations of equation (27), taking into 

 account the continual variations of the boundaries 

 <f>a and ^6. 



Table 1. Conditions for constant wake index. 

 Effective half-width of sound beam <^' = 6° 



I Sound beam trained 

 perpendicularly at 

 wake /3 = 0° 



II Sound beam trained 

 obliquely at wake 

 /3 = 60° 



Condition (35) 

 D <. 200r„ 



Condition (35) 

 D < 1.66ro 



Summing up the discussion of short-pulse echoes, it 

 should be remembered that the analysis, for mathe- 

 matical reasons, had to be restricted to wakes having 

 a constant bubble density in the transverse direction. 

 According to the varying transverse structure of 

 actual wakes, their echo profile will deviate somewhat 

 from the exponential shape given by equation (27), 

 even if the sound beam is trained perpendicularly at 

 the wake. 



33.1.3 Definition of Wake Strength 



The concept of wake strength has been introduced 

 in Section 33.1.1 in the hope of arriving at a wake 

 characteristic that is a function solely of the geomet- 

 ric dimensions and physical parameters of the wake. 

 But the detailed analysis in Section 33.1.2 showed 

 that this aim defies complete realization. The strength 

 of wake echoes depends on the signal length and on 

 the directivity pattern of the transducer in a rather 

 complicated manner; this dependence cannot be for- 

 mulated mathematically in a simple way without in- 

 troducing various approximations. These effects are 

 small compared with those resulting from the varia- 

 bility of echo strength with range; this large variation 

 with range can be eliminated from the wake strength 

 by a suitable definition of this quantity. 



We now define wake strength by the equation 



W = E - S + 2H - lOlogr - -^ , (36) 



where E is the echo level, S the source level, H the 

 one-way transmission loss from the transducer to the 

 wake, r the range in yards of the wake, and ^ is the 

 appropriate wake index. This definition comprises 



both equations (21) and (29), referring to long and 

 short signals, respectively, and implicitly disregards 

 the small difference between the ranges defined as r 

 and ri. For all practical purposes the range to be used 

 in the computation of wake strength may be de- 

 termined from the time interval, purposely recorded 

 on the oscillogram, between midsignal and the in- 

 stant to which the measured echo amplitude refers. 

 When the difference between the transmission loss H 

 and the inverse-square loss 20 log r increases linearly 

 with range, an alternative way of writing equation 

 (36) is 

 W = E - S + 2ar - lOlog ix^ + 30 log r - ^, (37) 



where a is the coefficient of attenuation in the ocean 

 expressed in db per kyd, and (^ — 1) is the reflec- 

 tivity of the ocean surface, so that the factor n^ repre- 

 sents the increase of echo intensity caused by the 

 double path resulting from surface reflection. To be 

 consistent with the standard procedure for comput- 

 ing target strengths, ix should be put equal to 1 (see 

 Section 22.2), so that equation (36) reduces to *" 



W = E - S + 2ar + dOlogr - ^. (38) 



Finally, according to equation (9), the target strength 

 of the wake Ty, is given by 



r„ = TF -I- 10 log r -f- ^. (39) 



The wake index "ir was first defined by equation 

 (7), on the assumption that reflection from a wake 

 can be treated like that from a plane strip. This is, 

 indeed, a good approximation provided that the wake 

 is highly opaque, or N{w) » 1, and long signals are 

 used, or ro^ w (for example, r-o > 2w), according to 

 equation (18) which turned out to be identical with 

 equation (7) : 



^co = 10 log r b{<l>)b'{(t>) [cos <^ - tan /3 sin <j>Jd(j>. (40) 



^ All the numerical values of W reported in this chapter 

 have been computed according to the definition given by 

 equation (38). However, when the original publications are 

 consulted, care should be taken in ascertaining what particular 

 definition of wake strength was used by the author. For in- 

 stance, in one paper,' a* is set equal to 2 in correcting echo 

 levels for transmission loss; moreover, a term 10 log (4ir) is 

 also added to equation (37). The net result is that the values 

 of the wake strength in reference 1 are 5.0 db larger than those 

 computed from equation (38). 



Since the reflectivity of a target has been defined, in Section 

 19.1 of this volume, in terms of the sound reflected into a unit 

 solid angle, it seems desirable to maintain the same convention 

 for the reflectivity s of a wake. Accordingly, the term — 10 log 

 (47r) here appears in equations (45) and (46), instead of in 

 equation (37). 



