ECHOES FROM WAKES 



545 



where J, is the surface reverberation index, defined 

 by 



7. = lOlogri-j6(<^)6'(<^M0j, (11) 



where h{<i>)h' {<t>) is the composite pattern function of 

 the echo-ranging transducer. For typical transducers, 

 the surface reverberation index can be computed from 

 the equation 



J, = 10 log y - 23.8 , (12) 



where 2y is the horizontal angular width, measured 

 in degrees, of the sound beam between points down 

 3 db from the axis. 



Thus in this simple case, the target strength of an 

 opaque wake at 24 kc is given by 



T„ = -26 + 10 log A + 10 log r 



+ 101og2/-24 + 8 (13) 



This equation may be used to predict the initial 

 strength of echoes received from the wakes behind 

 surface vessels. 



If the total attenuation Hw across the wake is less 

 than 1 db, the wake strength W may be less than the 

 value found from equation (10). The wake strengths 

 of observed surface wakes are constant for about 2 to 

 5 minutes, and thereafter decay at a rate of 1 to 2 db 

 per minute; wake echoes can be observed, under good 

 conditions, for 20 to 40 minutes after the passage of a 

 vessel. These times are not inconsistent with what is 

 known of the times required for air bubbles initially 

 0.1 cm in radius to disappear by diffusion back into 

 sea water. In this situation, ^ for a directional trans- 

 ducer is 



^ = J, -(- 8 -h 10 log sec /3 (14) 



where /8 is the angle between the projector axis and a 

 line perpendicular to the wake axis. The increase of 

 echo strength with increasing § predicted by equation 

 (14) holds only so long as the ping length is greater 

 than the extension AB oi the wake along the pro- 

 jector axis in Figure 4, and so long as the absorption 

 loss along the path AB is less than 1 db. 



The wake strengths of submerged submarines at 

 45 and 90 ft at speeds of 6 knots are about —30 db. 

 Surfaced submarines appear to have about the same 

 wake strength as that predicted for large surface 

 vessels from equation (10) (see Section 33.1.2). 



35.3.2 



Short Pulses 



4, the preceding equations are less useful. Although it 

 is possible to predict wake strengths by adding to 

 equation (10) a correction term depending on the 

 signal length, the resulting values of W cannot simply 

 be transformed into echo levels, using equation (7), 

 or into target strengths, using equation (9), unless 

 the echo ranging transducer is beamed perpendicu- 

 larly at the wake. 



For short pulses, the wake strength W decreases 

 with the • decreasing ratio of the geometric pulse 

 length ro measured in yards to the geometric width w 

 of the wake, and can be predicted from the following 

 equation 



W = 101ogs+ lOlog/i 



-I- 10 log [1 - 10 



-(.H„/5)/(r„/w) 



] + 6, 



(15) 



where h is the depth of the wake, measured in yards, 

 and 10 log s = A; is the same function of the frequency 

 only as in equation (9), with the values indicated in 

 Table 2. Numerical values of the third term on the 

 right side of equation (15) can be read from Figure 6 



When the pulse length ct/2 is less than the exten- 

 sion AB of the wake along the projector axis in Figure 



Figure 6. Wake strength term as function of attenu- 

 ation and ratio of ping length to wake width. 



as a function of /7u„ the total attenuation across the 

 wake, and of ro/w, the ratio of signal length in yards 

 to wake width. The wake strengths and echo levels 

 computed from equation (15) refer to the peak of 

 the average echo. 



Echo levels E and target strengths T„ predicted 

 for values of W computed on the basis of equations 

 (15) and (11) should be quite satisfactory, provided 

 that the sound is beamed at the wake nearly perpen- 

 dicularly. For lack of anything better, the same pre- 

 dictions may be used in case the sound beam strikes 

 the wake obliquely. The expected discrepancies be- 

 tween observations and predictions, for that case, are 

 believed to be smaller than + 5 db. 



