476 



ACOUSTIC THEORY OF BUBBLES 



m 

 o 



-15 



z 



UJ 



o 



-20 



-25 



-30 



UJ 



o 

 u 



z 

 o 



I- 

 o 



UJ 



d -40 



^ -35 



0.015 0.020 0.025 0.030 0.035 



BUBBLE RADIUS IN CM ON ACOUSTIC AXIS 



§ -20 



I- -25 



z 



UJ 



9 -30 



8 -35 



z 



9 -40 



yt -45 



K 0.010 0.015 0.020 0.025 0.030 



BUBBLE RADIUS IN CM ON ACOUSTIC AXIS 



Upper and lower limits of experimental data 



Estimated average intensities 



Figure 6. Scattering and reflection bubble pulse 

 screen. 



observations were made of the number of bubbles of 

 different sizes in the screen, since the operation of the 

 microdispersers producing the bubbles tended to be 

 somewhat erratic. If the physical measurements on 

 the number of bubbles of different sizes in the screen 

 did not give the same results before and after the 

 acoustic measurements, the acoustic data were dis- 

 carded. 



The results of the acoustic measurements on bubble 

 pulses showed moderate agreement with theoretical 

 predictions. Figure 5 illustrates typical results ob- 

 tained for resonant bubbles. The upper diagram 

 shows the total number of bubbles per cubic centi- 

 meter at the level of the transducers at the time when 

 bubbles of each radius reach that level. Since the 

 spread of bubble radii at each time was small com- 

 pared to the width of the resonance peak for a single 

 bubble, all the bubbles at any one time may be as- 

 sumed to be of the same size. In the middle diagram, 

 the continuous curve shows the predicted attenuation 

 through the screen, found by substituting in equation 

 (53) the following quantities: the bubble density 

 taken from the upper diagram; the measured thick- 



ness of the screen; and the value of Ce found from 

 equation (43) with / equal to /,, and with values of 5r 

 taken from Figure 2. The average observed trans- 

 mission losses at each frequency are shown by circles, 

 with vertical lines showing the spread of the observa- 

 tions. These experimental points are essentially the 

 maximum difference in sound level produced by the 

 passage of the bubbles; in the middle diagram of 

 Figure 5, for example, the observed resonant trans- 

 mission loss at 20 kc is about 14 db. 



The lower diagram in Figure 5 shows the intensity 

 of the reflected or scattered sound for resonant 

 bubbles. To compute the reflection to be expected 

 from resonant bubbles, the specular reflection was 

 first found from equation (86), with 6 evaluated for 

 bubbles all of resonant size. To this was then added 

 the scattering to be expected; this scattered sound 

 was found from equation (76), since for resonant 

 bubbles the transmission loss through the screen was 

 always great enough to make this equation appli- 

 cable. The value of tj, at resonance was taken from 

 equation (23), while values of 5, at resonance were 

 again found from Figure 2. In the computation of this 

 scattered sound account must be taken of the size of 

 the screen and its distance from the sound projector 

 and hydrophone. The solid curve in Figure 5 shows 

 the theoretical predictions; at 10 kc, specular reflec- 

 tion is most important, while at 30 kc, scattered 

 sound is dominant. 



A similar comparison between theory and observa- 

 tion may be made for nonresonant bubbles. The 

 transmission loss measurements yield nothing further 

 of interest, since the width of the observed resonance 

 curve has already been used to find values of hr. For 

 bubbles whose size is so far from resonant size that 

 the transmission loss is small, the scattering may be 

 predicted from equation (75), suitably modified to 

 take into account the geometry of the situation. The 

 value of (Ts to be used may be taken from equation 

 (34). Specular reflection from nonresonant bubbles is 

 negligible. Plots of the observed data are shown in 

 Figure 6, where the crosses represent the computed 

 values for nonresonant scattering. The spread of the 

 observations is indicated by the dashed lines, with 

 the solid line showing the estimated average in- 

 tensities. The circles represent the predicted scatter- 

 ing and reflection from resonant bubbles, already dis- 

 cussed. The dotted line gives the sound level measured 

 at the reflection hydrophone when no bubbles were 

 present. 



It is evident that the agreement between theory 



