CONCEPT OF WAKE STRENGTH 



513 



tion, is then 2irs. By comparison with Section 19.1 it 

 is readily verified that the target strength of one 

 square yard of this wake surface, placed perpendicu- 

 larly to the sound beam, is 10 log s. Since the depth 

 of the wake is h yards, the target strength of a 1-yd 

 length of wake is 10 log hs. In order to relate this 

 wake strength to the observed echo intensity, con- 

 sidering the directivity of the transducer and the 

 scattering of sound from elements of the wake surface 

 not perpendicular to the sound beam, a more detailed 

 exposition of this simple case is required. 



The geometry of this experimental situation is 

 illustrated in Figure 1, from which it is apparent that 



NORMAL TO THE WAKE 



TRANSDUCER AXIS WAKE 



TRANSDUCER Ty 

 Figure 1. Horizontal plane through transducer and 

 wake. 



the surface element of the strip having the area hdy 

 receives from the transducer the power 



- O.lar 



/o 



10' 



-b{<t>) cos (j8 -|- 4>)hdy, 



(2) 



where /o is the output of the transducer on the axis; 

 h{<f) measures the angular variation of the output 

 around the horizontal plane; r is the distance from 

 the transducer to the surface element hdy; 4> is the 

 angle between this ray and the axis of the transducer, 

 which subtends the angle /3 with the normal to the 

 wake so that cos (/3 -|- <t>) measures the geometric 

 foreshortening of the insonified area; and a is the 

 coefficient of attenuation in the ocean. If the trans- 

 mission loss on the return path is taken into account, 

 the equivalent echo intensity le is 



= slj 



■in— 0.2ar 



H<t>)b'{4>) cos (0 + <t>)hdy , (3) 



where b(<^)6'(0) is the composite pattern function of 

 the echo-ranging transducer. The factor 6'(<A) is the 

 ratio between the response of the hydrophone to a 

 signal incident at an angle <t> to the axis and the 

 response to a signal of equal strength incident on the 

 axis. Thus the equivalent echo intensity /« is propor- 



tional to the output voltage of the transducer acting 

 as hydrophone. 



By substitution of the perpendicular range D from 

 transducer to wake, 



D = r cos (j3 + <^) 

 2/ = I> tan (iS -1- <^) , 



equation (3) is transformed into 



6(0)& '(0) 10-^-^"^ ''" ^ + *'cos3(^ + 0) d4>, 



or, to a very good approximation 



- i(f-^-Ds<.<.,j^, ^ ^^ fh{<t>)h'{<l>) cos3(^ + ,i>)d<i, ■ (4) 



This approximation neglects the variation of the 

 transmission anomaly along the wake, which is in- 

 significant because of the narrow beam pattern of the 

 transducers used in practice. By writing 



cos {fi + <i>) = cos /3 (cos (^ — tan /3 sin <fj 

 and D cos /3 = f, 



where f is the range to the wake measured along the 

 transducer axis, equation (4) becomes 



, = + j-. 



- 10"-^"^= = /is / bi<j}W{(t>){cos <p - tan;8 sm <^)'d0. 



« = -2-3 (5) 



By collecting the terms representing the transmission 

 loss, 



af + 20 log f = H, (6) 



and adopting the abbreviation 



10 log / 5(0)6'(*) (cos - tan /3 sin (t>yd4> = ^, (7) 



equation (5) can be expressed, in decibels, as 



E - S + 2H - lOlogr - <^ = lOloghs = W . (8) 



The quantity "ir defined in equation (7) will be called 

 the wake index, analogous to the reverberation index 

 defined in Chapters 11 through 17. The product hs 

 in equation (8) has the dimension of a length. Since 

 the ranges appearing in (8) are customarily measured 

 in yards, the wake strength W is the ratio of hs to one 

 yard, expressed in decibels. Then, by comparing equa- 

 tions (8) and (11), the relation between the wake 



