19 



207 



The value of N is so large that at very low frequencies, in the ab- 

 sence of all resonance effects, a small amount of air causes a large decrease 

 in the wave velocity. Thus if / = 0.1 per cent, c' = c / W^i^Tz^'lc'oTrZ^ = 

 0.25 c; if / = 1 per cent, c' = 0.08 c. 



The coefficient of reflection, or fraction of the incident energy 

 that is reflected, is given by Equation [70] or 



K = 



(1 + ?)' - ^' 



At very low frequencies this becomes, approximately. 



K = 



Vl + fN^ - 1 



Vl + fN^ + 1 



24' 



■[25] 



from Equation [25], a formula that is easily obtained from a much simpler 

 calculation. The latter formula gives, for / = 0.1 per cent of air, 



a: = (I) = 0.36 



and for 1 per cent of air. 



^=M^-''' 



Curves are shown in Figures 6 and 7 for 0.1 per cent of air in 

 water, or for N= 124 and / = 0.001. In Figure 6 the ratio of velocities 



Figure 6 - Refractive Index relative to 



Homogeneous Water, c/c , and Extinction 



Coefficient /S for Sinusoidal Waves in 



Water containing 0.1 per cent 



of Air in Fine Bubbles 



c' = wave speed 



c = wave speed in homogeneous water 



w = 2ir times wave frequency 



<j„ = 2ir times natural frequency of radial 

 oscillation of the bubbles 



= extinction coefficient 



Pressure decreases by factor e'^"' as wave 

 progresses a distance A = 2nc/u. 



