514 



OBSERVATIONS OF WAKE ECHOES 



strength W and the target strength of a wake T„ 

 becomes 



T^ = W +I0logf + ^, (9) 



where f is the range to the wake, measured along the 

 transducer axis, and ^ is the wake index defined by 

 equation (7). The physical meaning of equation (9) 

 has already been noted in the first paragraph of this 

 section. 



33.1.2 Dependence of Wake Strength 

 on Physical Parameters 



The fundamental definition of wake strength, 

 given in the preceding section, was facilitated by 

 treating the wake as if it were a plane strip with the 

 coefficient of reflection s. In effect, this approach 

 neglected the wake structure along the transverse x 

 axis. Moreover, that analysis tacitly assumed the 

 use of a continuous signal for measuring the wake 

 strength. But if short sound pulses are beamed at a 

 diffusely reflecting plane, the echo profile on the 

 oscillogram, in general, will not reproduce the shape 

 (usually square-topped) of the signal, and the de- 

 pendence of echo intensity on signal length must be 

 investigated. For a brief theoretical demonstration of 

 this fact see Section 19.3. 



On inspection of the sample oscillograms repro- 

 duced in Figure 7 of Chapter 30, it will be observed 

 that reflection from a wake also alters the shape of 

 square-topped sound pulses. However, the explana- 

 tion of this effect is more complicated than that of the 

 variation with pulse length of the echo intensity re- 

 turned by a plane target. In fact, the echo profile de- 

 pends on the transverse structure of the wake, which 

 therefore must form an integral part of a compre- 

 hensive theory of wake echoes. 



According to the working hypothesis adopted in 

 Section 26.3 wake echoes are composed of a multitude 

 of reflections originating throughout the entire wake. 

 Superposition of these scattered waves leads to con- 

 structive and destructive interference, because their 

 phases are distributed at random. Consequently the 

 echo intensity measured at any instant will not equal 

 the average value. The difference between the in- 

 stantaneous and average echo intensity is a rapidly 

 fluctuating quantity, evidently beyond the reach of 

 theoretical analysis, because it depends on the micro- 

 structure of the entire wake. Physical significance can 

 be attributed only to the average of many echo pro- 

 files recorded in rapid succession. Such averaging is 

 also necessary in order to minimize the effect upon 



the echoes of the rapid fluctuations of the transmis- 

 sion loss in the ocean at large, which were discussed in 

 Chapter 7. Accordingly, the theoretical analysis 

 about to be presented refers to "average" echo in- 

 tensities throughout. 



The problem now is to evaluate the relation be- 

 tween the total number, arrangement and physical 

 parameters of the bubbles and the overall reflectivity 

 s of the wake, filling a volume of constant depth h 

 and width w and of infinite length (— oo <_ y < -|-ao). 

 Let n(x) be the number of bubbles per unit volume 

 at the distance x from the nearest boundary of the 

 wake {0 < X <w); n{x) is supposed not to vary 

 appreciably along the wake axis over a distance of 

 the order of the width of the sound beam, or dn/dy 

 <<C dn/dx. With as and ae representing the cross 

 sections for scattering and extinction defined by 

 equations (34) and (43) of Chapter 28, the echo re- 

 turned by an individual bubble has the intensity 



■I r\—0.2ar 



7, = -' /o '-^^ 6(<^)6'(0) e-^""^'' '^ '^ + *'. (10) 

 4x r* 



The fraction o-s/4ir of the incoming sound energy is 

 scattered into the unit solid angle, and the term 



10" 



""/r* represents the two-way transmission loss in 



the ocean. The sound beam is trained obliquely at the 

 wake, so that the axis of the transducer and the nor- 

 mal to the wake include the angle /3. It is the oblique 

 path of the sound beam traversing the wake, which 

 accounts for the factor sec (|3 -|- <^) in the exponent 

 expressing the two-way transmission loss inside the 

 wake, which is based on equation (52) of Chapter 

 28. The geometry of the situation is illustrated in 

 Figure 2. The echo returned by a volume element of 

 the wake is found by multiplying equation (10) by 

 n(x)hrdrd<j> 



ale= -— ^0 — T 

 4ir r 



(11) 

 on the assumption that all bubbles have the same 

 size, so that the cross sections as and ae are constant 

 throughout the wake. Finally, the echo returned by 

 the entire wake can be evaluated by integration be- 

 tween the appropriate boundaries, which must be 

 chosen carefully; these boundaries are essentially de- 

 termined by the width of the wake and by the signal 

 length. 



When an echo-ranging signal is sent out into the 

 water, the volume from which echoes can be received 

 at anyone time fills a spherical shell of thickness cr/2, 

 where t is the duration of the square-topped signal 



