518 



OBSERVATIONS OF WAKE ECHOES 



During the time interval in which rj is confined be- 

 tween the limits stated, equation (27) represents the 

 average profile of short-pulse echoes. Consequently 

 the average echo intensity falls off exponentially from 

 its maximum value, attained immediately after the 

 pulse has fully entered the wake {n = D sec /3), pro- 

 vided that the value of the integral over d<t> does not 

 vary with time. This condition can indeed be realized 

 in special cases. 



'TRANSDUCER 

 Figure 3. Successive positions of ping inside wake. 



Although the integrand in equations (27) and (28) 

 is independent of time, the range of integration is not, 

 in the general case. This fact is illustrated by Figure 

 3, showing successive positions of a short pulse 

 beamed obliquely at the wake. As the pulse crosses 

 the wake, the limits, <t>b and 0a of the integral in equa- 

 tion (28) increase steadily. Thus the wake index is, in 

 principle, a variable quantity, which would seem to 

 impair the usefulness of this concept. However, the 

 directivity of the transducer effectively limits the 

 angular width of the sound beam. If it were possible 

 to place the transducer in such a position that the ef- 

 fective half-width of the sound beam remained 

 smaller at all times than the boundary <t>b, the varia- 

 bility of 06 and 0o would become irrelevant for all 

 practical purposes, and the wake index ^' would 

 actually be a constant. 



In order to formulate this idea quantitatively, note 

 that the boundaries of integration in the wake index 

 are explicitly 



<t>b 



4>a 





•i3 



+ ;8- 



(30) 



These formulas can readily be verified by inspecting 

 Figure 3. If the effective angular width of the sound 

 beam be called 20', so that the effective half-width is 

 0', the conditions under which ^' becomes constant is 



0'^06- (31) 



By substitution of equation (30) in equation (31), it 

 follows that 



r ^ 1 



COS"' 



Ln + roJ 



^ <t>' + fi, 



or 



(32) 



D 



^ COS (/3 -1- 0'). 



n +ra 

 Write 



Ti = D -f- Xi , 

 where, according to Figure 3, Xi is confined between 

 the following limits 



{)<xi<w, (33) 



and substitute in equation (32) ; then 



, , , , w^ ^ cos (/3 -I- 0') 

 1 -H (xi -I- ro)/D 



or approximately, since generally (xi -f ro)/D is much 

 smaller than one, 



xi -|- ro 



1 - 



D 



Hence, 



D ^ 



^ cos (/3 + 0'). 



Xi -I- ro 



(34) 



1 - cos (|3 + 0') 

 This inequality may be put into a more stringent 

 form by setting Xi equal to zero, on account of equa- 

 tion (33), so that the desired condition takes the 

 final form 



To 



D^ 



(35) 



1 - cos (a + 4>') 

 v/hich assumes that right from the moment the pulse 

 has entered the wake, or Xi = 0, the effective half- 

 width of the sound beam is smaller than the variable 

 boundary 0;,. Any less stringent form of the condi- 

 tion, such as xi ^ w, would distort the representa- 

 tion of the entire echo profile indicated by equation 

 (29). 



For numerical evaluation of equation (35), 0' = 6 

 degrees appears to be a reasonable value for trans- 

 ducers of conventional design used in echo ranging. 

 Results for two typical cases are given in Table 1. 

 Since the shortest signals used in practice correspond 

 to ro = 1 msec = 0.8 yd, condition (35) is easy to 

 maintain in ranging perpendicularly at the wake. 

 Accordingly, the upper limit 06 in equation (28) may 

 be replaced by a practically constant value 0' if the 

 range D from the transducer to the nearest point of 

 the wake is chosen so as to comply with the first case 

 in Table 1. 



On the other hand, it is evidently impossible to 

 satisfy the second condition in Table 1 — in other 



