206 



2. The spacing cf the bubbles is large relatively to their own diam- 

 eter, but yet small relative to the wave length of the waves in the bubbly 

 water. Let 



c be the speed of sound in homogeneous water, 



c' be the speed of sound in the bubbly water, 



fi be the extinction coefficient, with the significance that 

 the amplitude of the pressure decreases by a factor e"^"^ 

 as the wave traverses in the bubbly water a distance equal 

 to its wave length in homogeneous water, 



/ be the fraction of space occupied by the gas in the bubbles, 



/?o be the equilibrium radius of the bubbles, assumed the same 

 for all, 



ia> = 2tt/T, where T is the period of the waves, 



Wq = 2n/To, where Tq is the natural period of small radial 

 oscillation of a bubble, 



N = ,, p . Here Af^ represents the ratio of pc^, the volume 

 elasticity of water, to ypg, the volume elasticity of the 

 gas contained in the bubbles. 



Then, according to Equations [66] and [67], the analytical treat- 

 ment yields the equations: 



^ -H'-l+fN'- ,.,., ■• 3 ^, 121 



('-^)- 



N' u) 



2pi: = V2fN- .,2° o TTT [22] 



.2,2 



At very low frequency, /8 = approximately and 



Since, as in Equation [l], T,, « /?„, the quantity N is in reality 

 Independent of the size of the bubbles. For air in wa,ter at atmospheric 

 pressure, where c = 58,000 inches per second and by Equation [2a], w^ = 

 2n/To = 2jr X 129/i?o, N = 58,000 KT/2587r= 124. 



