CONCEPT OF WAKE STRENGTH 



515 



and c is the sound velocity. This region, which travels 

 outward at a velocity c/2, may for purposes of dis- 

 cussion be referred to as the volume occupied by the 

 echo-ranging signal or pulse. As the pulse travels out- 

 ward, it will cross the wake, which has roughly the 

 shape of a cylinder of infinite length. The determina- 

 tion of the echo strength requires integration of equa- 

 tion (11) over the volume in which pulse and wake 

 overlap. In the general case, this volume has a rather 

 complicated shape which, moreover, varies with time 

 as the pulse travels across the wake. It should be 

 noted that in most practical situations the depth of 

 the wake is small compared with the range to the 

 transducer, so that the vertical curvature of the 

 pulse boundaries can be neglected; in other words, the 

 pulse will then be treated rather as a cylindrical shell, 

 with the cylinder axis normal to the ocean surface. 



AXIS 



TRANSDUCER 



Figure 2. Sound striking wake. 



Now there are two simple limiting cases for which 

 the shape of the volume contributing to the echo can 

 readily be visualized, namely when the signal length 

 is either much greater or much smaller than the width 

 of the wake. In the first case, the entire volume of the 

 wake will contribute to the echo for a considerable 

 length of time during which the average echo intensity 

 will be constant. In the second case, the echo will 

 come only from a thin spherical shell "cut out" of the 

 wake, so that the average echo intensity will vary, 

 while the pulse traverses the wake, without ever at- 

 taining a constant value; the mode of this variation 

 is a function of the density distribution n{x) across 

 the wake. 



Long Pxjlses 



The case of very long pulses will be taken up first. 

 If the signal length Tq, which equals ct/2, is much 



greater than what might be called the slant width w' 

 of the wake, which equals w .sec {fi -\- <j>), then the 

 entire volume of the wake will be intersected by the 

 pulse during a finite interval of time. During this 

 period, the average echo intensity is constant, and 

 can be found by integrating equation (11) over the 

 wake volume. The limits of this integration are most 

 readily given if the variable x is substituted for r, 

 since then x varies between and w, while </> varies 

 from (-7r/2)-/3 to (+7r/2)-/3 (see Figure 2). By 

 writing 



r =; I f cos/3 — — -|- a; ) sec (;3 -f- <p), 



and 



-(-^1) 



sec/3, 



(12) 



then dr = sec (/3 -|- <j>)dx , 



and the new constant f is the range, measured on the 

 axis of the transducer, of what might be called mid- 

 wake — the point of intersection between transducer 

 axis and wake axis. As in the preceding section, the 

 variation along the wake of the transmission anomaly 

 in the ocean will be neglected by setting 



■tr\~0.2ar ■tr\—0.2ar 



By substituting equation (12) in equation (11), the 

 integral now reads 



hf, iff-^r ^f^C fbi<t>)b'i<t>) cos^ (/3 + 4>) sec'/3- 



(.= -- -/S x = 



n{x)e 



—2aeN(x) sec 03+0) 



/ 2x — w\' 



\ 2FcosB/ 



- dxd(j>. 



(13) 



2F cos /3> 



Unfortunately, it is impossible to integrate this ex- 

 pression in closed form. An approximation sufficient 

 for all practical purposes will be given. 



Consider first the integral over dx, namely, 



p (i + ^£:Z^y '„(^)e-2«;v(., sec (^ + «^^^ (14) 

 Jo \ 2F cos /3/ 



and apply the theorem of the mean value of an 

 integral* to the inverse cube term in brackets; in 



" This well-known theorem states 



Jf{x)g{x)dx = f(x)Jg{x)dx , 



XI XI 



with _ 



Xi< X <X2 , 



as long as /(x) and g{x) are continuous over the range of 

 integration, and- g{x) is not negative. 



