SCATTERING BY A SINGLE IDEAL AIR BUBBLE 



463 



4 

 3 

 2 



I 







-I 



2 

 3 



0.001 0.002 0.004 0006 a008 0.01 aOZ a04 0.06 



2 7rR/X 



Figure 1. Scattering cross section for an ideal bubble. 



called the resonance frequency for the bubble of 

 radius R. 



A plot of (Ts/tR^ as the function of 77 = 2tR/\ 

 = 2irRJ/c is shown in Figure 1, the outstanding 

 feature of which is a sharp peak. This maximum cor- 

 responds to the resonance value Tjr or according to 

 equation (18) 



2TrRfr 



Vr = 



= 11/3^ = 1.36 X 

 cr p 



10- 



(23) 



if Po is atmospheric pressure and c is the sound 

 velocity in sea water at 60 F. Thus, at resonance, o-j 

 is enormously greater than the geometric cross sec- 

 tion of the bubble; specifically 



= ©" 



2.16 X 10^ 



(24) 



ffs, 

 TtR^ \JJr 



where asr is the value of as at resonance. Equation 



(24) can also be expressed in the form 



X2 



(25) 



While equations (24) and (25) must be considerably 

 modified for an actual bubble, as shown in the next 

 section, the phenomenon of resonance is neverthe- 

 less responsible for the great efficiency of bubbles as 

 scattering agents. Moreover, the resonant frequency 

 found from equation (18) is correct for a wide spread 



of bubble sizes. This equation has been confirmed by 

 observations at low frequencies, between 1,000 and 

 6,000 c per sec, "•' and also at high supersonic fre- 

 quencies, between 20 and 35 kc' In each case a 

 single bubble was placed in the sound field, and the 

 sound frequency determined at which the bubble 

 oscillated most violently. The radius of the bubble 

 was then measured either with a microscope, or for 

 the larger bubbles by measurement of the volume 

 of air in the bubble. The values of the resonant 

 frequency /r found in these measurements for bubbles 

 of air, hydrogen, and oxygen in water at different 

 temperatures' agreed with equation (18) within the 

 experimental error of about 5 per cent. Thus within 

 the range from 1 to 50 kc equation (18) may safely 

 be used to predict the resonant radius of bubbles in 

 water. Values computed from this equation are 

 given in Table 1. 



Table 1. Resonant radius for air bubbles in water. 



For very small bubbles, with radii less than 10"^ 

 cm, surface tension becomes important and the com- 

 pressions and expansions of the gas in the bubble be- 

 come isothermal instead of adiabatic. No observa- 

 tions for such small bubbles are available, but a 

 theoretical analysis ^ shows that equation (22) is 

 still valid provided that fr is defined by the equation 



-w 



where 



27r/, 



g = i + 



3gPo 



AT 

 SRPo 



(26) 



(27) 



the quantity T is the surface tension of the gas- 

 liquid surface, and other quantities have the same 

 meaning as in equation (18). Equations (26) and 

 (27) should not be used for bubbles of radii greater 

 than 10~' cm. 



Equation (22), in addition to predicting the im- 

 portance of resonance, also gives correctly the scat- 



