Ill -50 



The values of the damping constant used in these computations were 

 those obtained by Carstensen and Foldy. Since no data was available for the 

 damping constant at off- resonant frequencies it was assumed to depend only on 

 the resonant frequency. In the light of other evidence, the damping constant 

 used in these experiments seems high; however Laird and Kendig report that 

 if at 12 kc 5 is assumed to be 0.06 instead of 0. 12, the theoretical attenuation 

 is increased by only 4%. Thus the attenuation is relatively insensitive to the 

 damping constant. 



Figure III- 21 gives a plot of the computed values of the phase velocity. 

 At low frequencies, this is about one-third that of bubble-free water, while at 

 higher frequencies it approaches the velocity in bubble-free water. Some at- 

 tempts were made to verify these phase velocities experimentally, but the wide 

 fluctuations in phase and amplitude of the transmitted signal made this impos- 

 sible. Measurements at frequencies above 50 kc were possible and showed 

 that at these frequencies the phase velocity in bubbly water is "essentially the 

 same" as that in bubble-free water. 



Comparing Laird and Kendig' s computed phase velocity with the cal- 

 culations and measurements of Meyer and Skudrzyk and of Fox, Curley and 

 Larson indicates that Laird and Kendig' s peak phase velocity is much too high. 



Fox, Curley, and Larson measured the behavior of the phase velocity 

 and the absorption in bubbling water for various frequencies. A bubble screen 

 was created by forcing air through a porcelain bacteriological filter. It was 

 found that the large majority of the bubbles had radii in the range of 0. 012 ± 

 0.004 cm, and would, therefore, have resonant frequencies of approximately 

 60 kc. Figure III-22 shows a distribution of the air volume contained in bub- 

 bles of various sizes. The average volume was found to be (2 ± 0.5) (10 ~ ) cc 

 of air per cc of water. In order to eliminate any boundary effects, the trans- 

 ducers were located in the bubble screen and separated by 2-1/2 cm. 



In the analysis of the data, the damping constant 6 was taken as 0.5, 

 although theoretically it should be 0.12. Such a large value of 6 was required 

 in order to make the theoretical predictions correspond at all with experimental 

 results . 



The experiments are summarized in Figure III-23. The dashed lines 

 represent the theoretical prediction when all bubbles are assumed to be of the 

 same size, while the solid lines correspond to the integral forms of the theory 

 and take into account the distribution of bubble sizes. The precision of phase 

 velocity measurements is estimated to be within 5%, while the precision of at- 

 tenuation experiments is within 2 db/cm. Each point represents an average 

 reading. 



artburSl.iattyj^.^ 



