ECHOES FROM WAKES 



535 



moderately good agreement with the vakies shown 

 in Table 2. 



34.2.2 Wakes of Model Propellers 



A similar computation may be carried out for the 

 wakes of small propellers. Measured values of the 

 absorption across a wake are reported in Section 32.5. 

 The cross section of the wake was about 1.5 yd wide 

 at the point where the measurements were made. The 

 values of u{Rr) were computed by use of equation (1) 

 for the lO-in. propeller at 1,600 rpm and for depths 

 of 10 ft, 20 ft, and 30 ft. Somewhat smaller values 

 are found for the 14-in. propeller at the same rpm, 

 possibly as a result of the narrower blades and lower 

 pitch of this propeller. The corresponding values of 

 u — the relative amount of air present in bubbles of 

 all sizes, found directly from these curves — are 

 given in Table 3. 



Table 3. Fraction of air present, as bubbles in wake 

 of 10-in. model propeller. 



Depth in feet u 



10 

 20 

 30 



3 X 10-« 

 2 X 10-« 

 9 X 10-' 



It is perhaps unexpected that the bubble density in 

 the wake of a 10-in. propeller be from five to ten times 

 as great as the corresponding density in a destroyer 

 wake. Further analysis shows this is not too surpris- 

 ing. The propeller developed 11 hp during operation, 

 with a tip speed of 70 ft per second. When a destroyer 

 is making 15 knots, its two propellers with diameters 

 between 9 and 11 ft, have a comparable tip speed, 

 about 80 ft per second. Moreover the destroyer is 

 moving rapidly, and it is well known that a propeller 

 which is held stationary in the water tends to produce 

 stronger tip vortices than one at the same rpm which 

 pushes itself through the water. Thus the small pro- 

 peller may be expected to cavitate more vigorously 

 than the propeller of a destroyer at 15 knots. The 

 volumes over which the bubbles produced in one 

 second are spread in these two cases are proportional 

 to the total propeller areas. Thus it would not be 

 surprising to find that the bubble density measured 

 behind the small propeller is greater than the cor- 

 responding density in the destroyer wake. 



.34.3 



ECHOES FROM WAKES 



The wake strength W is related to the bubble den- 

 sity in a more complicated way than is the attenua- 



tion coefficient A'„. In addition, W depends both on 

 the detailed geometrical properties of the wake, and 

 on the physical properties of bubbles of different 

 sizes, and cannot therefore be predicted with any 

 exactness for a known distribution of bubbles. Thus, 

 at most, a rather general agreement can be expected 

 between observed and predicted wake strengths. 



The formulas are simplest for long pulses; when 

 bubbles of a single size are present, the wake strength 

 W for long pulses is given by the equation 



/ ha. 



W = 10 log ^^ [1 



V 87r(7„ 



-2<ieN(.w) 



i 



(3) 



taken from equation (48) of Chapter 33. The quanti- 

 ties ds and cTe are the scattering and absorption cross 

 section defined by equations (34) and (43) in Chap- 

 ter 28, while h is the depth of the wake, measured in 

 yards. N{w) is the total number of bubbles in a col- 

 umn one sq cm in cross section extending through the 

 wake in a direction parallel to the sound beam [see 

 equation (54) of Chapter 28], and the product 

 (TeN{w) is 0.23 times Hu,, which is the total transmis- 

 sion loss across the wake measured in decibels. Thus 

 when this transmission loss is high, the exponential 

 term is very small, and W approaches the limiting 

 value 



W = 10 log A -t- 10 log 



(-)■ 



(4) 



Equation (46) of Chapter 3 gives the ratio of c, to 

 <Te in terms of 5, the so-called damping constant, and 

 V, the ratio of bubble circumference to the wave 

 length of the sound which represents the contribution 

 of radiation damping to the damping constant. Values 



Table 4. Observed frequency dependence of ratio of 

 scattering to extinction cross sections. 



of these two quantities have been taken from Figure 2 

 and equation (23) of Chapter 28 and the resulting 

 values of 10 log (ffs/Sira-e) shown in Figure 1 and 

 Table 4 of this chapter. At 24 kc, this quantity is —26 

 db, and the maximum value of W is equal to 



Tf = 10 log ;i - 26. (5) 



For a typical wake 10 yd deep this gives a maximum 

 wake strength of — 16 db. 



