SOUND PROPAGATION IN LIQUID CONTAINING MANY BUBBLES 



467 



the resonance peak is, the smaller is the damping of 

 the pulsation of the bubble. Therefore, 5^ is com- 

 monly called the damping constant. 



28.2.1 Measurement of Damping 

 Constant 



The simplest, most direct way to determine the 

 damping constant 8r is to measure the sharpness of 

 the resonant peak for a bubble in a sound field. Such 

 measurements have been carried out for bubbles in 

 fresh water. In one case,' the amplitude of oscillation 

 of a single bubble was observed as the sound fre- 

 quency was slowly varied. Since o-j is proportional to 

 the square of the amplitude of oscillation, a plot of 

 these observations yields 5, directly. Values of 5r were 

 found by this method for bubbles of hydrogen and 

 bubbles of oxygen, but no systematic difference was 

 found between these two gases. 



In another case, the transmission loss through a 

 screen of bubbles all of the same size was observed.* 

 To produce this screen, six small microdispersers ar- 

 ranged in a line in a laboratory tank were used to 

 produce a stream of bubbles 10 ft below the surface 

 of the water. These bubbles were normally inter- 

 cepted by a hood, which could, however, be swung 

 to one side for about one second to allow a pulse of 

 bubbles to rise to the surface. Since the larger bubbles 

 arrived at the surface first, and the smaller ones at 

 progressively later times, the bubbles near the surface 

 at any one time were of nearly equal radii. The trans- 

 mission loss in decibels of sound at a constant fre- 

 quency crossing this screen was then proportional to 

 ae for a single bubble; from a plot of the transmission 

 loss against bubble radius, a value of dr could then be 

 determined. A typical set of observed curves ob- 

 tained with this technique is reproduced in Figure 4. 

 In analyzing these data, account was taken of the 

 variation of 5 with bubble radius so that points some 

 distance from resonance could be used as well as 

 those close to resonance. 



The values of dr found by these two methods are 

 plotted in Figure 2. The dashed line curve shows the 

 theoretical value of 5t for air bubbles in water, if Ci is 

 set equal to zero, and values of /3 are taken from 

 reference 5. It is evident that at the higher fre- 

 quencies the observed values are much greater than 

 the theoretical values; this discrepancy has already 

 been noted above. The values of 6r found from a single 

 bubble, which are shown as crosses, are somewhat 

 greater than those determined from the transmission 



loss of sound through a bubble screen, plotted as 

 circles in Figure 2. In the former set of measure- 

 ments, the bubble was not free, but was caught on a 

 small wax sphere fastened to a platinum thread, 

 which oscillated to and fro as the bubble expanded 

 and contracted. Since the damping constant may 

 have been increased in this arrangement over its 

 value for a free bubble, these values cannot be relied 

 upon. Thus, the solid line of best fit shown in Figure 2 

 is based at high frequencies on the values found with 

 the screen of freely rising bubbles. Confirmation of 

 these observed values of 8r is found in the next sec- 

 tion, where the observed data on scattering and 

 absorption of sound by bubble screens are shown to 

 be in moderately good agreement with the theoretical 

 values based on equations (34) and (43) and on the 

 empirical curve of 8r in Figure 2. For comparison with 

 the observed values, the damping constant 3r com- 

 puted for an ideal bubble resonating in water at at- 

 mospheric pressure is shown as a dashed line in the 

 figure; the value plotted is taken from equation (23). 



28.3 SOUND PROPAGATION IN A 

 LIQUID CONTAINING MANY BUBBLES 



The results derived in the preceding sections for a 

 single bubble are only the first step toward the solu- 

 tion of the general problem, the propagation of sound 

 through a medium containing many bubbles. This 

 problem is complicated because the external pressure 

 affecting each bubble is the sum of the pressure in 

 the incident sound wave and the pressures of the 

 sound waves from all the other bubbles. While the 

 mathematics of the problem is complicated, the gen- 

 eral results to be anticipated can be presented simply. 



28.3.1 



General Theory 



First, the presence of the bubbles will affect the 

 nature of the medium through which the sound wave 

 is progressing. If the bubbles are spaced much closer 

 to each other than the wavelength, the sound velocity 

 will be appreciably affected by the presence of the 

 bubbles, which alters the compressibility of the 

 medium. In addition, the sound velocity will have a 

 small imaginary part, resulting from the absorption 

 and scattering of sound, and giving rise to an ex- 

 ponential drop of the sound intensity with increasing 

 distance of travel through the aerated water. Thus a 

 sound wave can be reflected, refracted, and atten- 

 uated as it passes through water containing bubbles. 



