CONCEPT OF WAKE STRENGTH 



521 



where Hw is the one-way transmission loss for hori- 

 zontal passage of sound through the wake, as defined 

 in equation (8) of ('hapter 32. Equations (45) and (46) 

 then read 



( h c. 

 If =101og<--7[l 



( h a, 

 W = 101og<^- — [1 - 



^-0.46//,r„/„.-j| (,^«y,) (47) 



- 0.46// » 



']| (ro»M). (48) 



The result of the differentiation is : 

 Short pulses, r^ « w 



dt "- "' 



0.466-°'"''"'°'"' n 



h dt ' i-e-"""""'"/" w 



dH^ 



dt 



w 



dw 

 dt 



^ • (49) 



Long pulses, n » w 



dW_ Idh OAQe-°*^"" 



~dt~^-^'^hdt'^l-e-°''"" 



dH^ 

 dt 



(50) 



The first term in these equations is the same for long 

 and short signals. It represents the effect of the 

 change in depth of the wake and is known to be quite 

 small; for two destroyer wakes, according to data re- 

 ported in Section 31.3.1, {l/h) (dh/dt) was found to be 

 — 0.08 and -1-0.04 db per minute, respectively. The 

 second term seems to be the dominant one. While the 

 factor in front of it, containing the exponential, is ex- 

 ceedingly small for fresh wakes, it grows rapidly and 

 approaches infinity for very old wakes; numerical 

 values of this factor can be read from the graph in 

 Figure 4. 



The differential quotient dHy^/dt is equal to 3 db per 

 minute, for destroyer wakes, and {Hy,/w) (dw/dt) can 

 be estimated from the same data to be of the order of 

 1 to 2 db.per minute. It will be noted that equation 

 (49) would be transformed into equation (50) by 

 setting Ta/w equal to one — except for the term pro- 

 portional to {H,c/w) (dw/dt) which does not appear in 

 equation (50). The physical meaning of this term is 

 interesting: as the wake ages, it spreads laterally, 

 causing dw/dt to be a positive quantity; conse- 

 quently, the factor H^/w is bound to decrease, even 

 if Hw, the total attenuation across the wake, remains 

 constant. According to equations (46) and (48), for 

 long pulses the wake strength is a function of N{w), 

 which is directly proportional to H^ or the total atten- 

 uation, and which is not affected by a mere spreading 

 laterally of the wake without simultaneous disinte- 

 gration of the bubble population. But for short pulses 



Figure 4. Factor appearing in formula (50) for the de- 

 cay rate of wake strength. 



[see equations (45) and (47)] the wake strength is a 

 function. of the product of signal length ro times the 

 average bubble density n which is proportional to the 

 attenuation coefficient. Hence, the decay rate of the 

 wake strength for short pulses is a function of the 

 decay rate d{Hu-/w)/dt of the attenuation coef- 

 ficient which gives origin to a term proportional to 

 (dw/dt) or the lateral spreading, even if dHy,/dt is 

 negligibly small, corresponding to an extremely small 

 physical disintegration of the bubble population. 



Summing up, for short pulses, whose volumes do 

 not intersect the entire wake, there exists a progressive 

 decay of wake strength having a purely geometric 

 origin — namely the lateral spreading of the wake. 

 Naturally, for long pulses, which overlap the entire 

 wake, such an effect cannot arise. If the decay of a 

 wake is followed over a very long period of time, and 

 a constant pulse length is employed, it may well 

 happen that the pulse which was chosen, at zero age 

 of the wake, so as to be long will finally become short 

 with respect to the steadily growing width of the 

 wake. At the moment the critical point w = ro is 

 passed, the term (Hy,/w) (dw/dt) suddenly begins to 

 operate, causing an accelerated decay. In order to 

 avoid all unnecessary complications, it may be ad- 

 visable, therefore, to choose very long signals for the 

 study of the decay rate of wake strength. 



The general significance of equations (49) and (50) 

 is that they establish a relation between the decay 

 rates of the transmission loss and the wake strength 

 which can be tested by observation. 



