NSWC/WOL TR 76-155 



_,_. P'(A) "X V(A) ,,„> 



where E (A) =— —+ — 1 (A2) 



A = Bubble Radius 



dA 

 A = -g^r , the radial velocity 



P (A) = Air Pressure Inside the Bubble 

 V(A) = Bubble Volume = -j^ A 



Y = Exponent in Adiabatic P-V Relationship = 1.4 



(for air) 

 p = Water Density = constant 

 P = Ambient Water Pressure = constant 



Y = Total Energy of Bubble Oscillation = constant 



3. The oscillatory motion described by Equation Al has 

 much in common with the oscillation of a simple mass-spring system — 

 the difference is that for the bubble oscillation the "mass" changes 

 with time and the "spring" is non-linear. The overall behavior of 

 oscillating gas bubbles was worked out in considerable detail by 

 Snay and Christian. 



They rewrote Equation Al in the non-dimensional form 



a 3 a 2 + a 3 + k a" 3 ^" 1 ^! (A3) 



where a is the non-dimensional radius a= A/L and a is the derivative 



of "a" with respect to the non-dimensional time t 1 = t/C . The 

 parameters k and y characterize the bubble motion. "k" can be 

 expressed by 



k = PMm 



Po(Y-D 



i + p^M y 



lm "I 



-i) J 



where P„ is the internal pressure at either extremum of the bubble 



Mm r 



oscillation — M refers to the maximum bubble volume (P = minimum 



pressure) , m refers to the minimum bubble volume (P„ = maximum 

 c m 



pressure) — and P is the ambient hydrostatic pressure. 



8 Snay, H. G. , and E. A. Christian, 1952, "Underwater Explosion Phenomena: 

 The Parameters of a Non-Migrating Bubble Oscillating in an Incompressible 

 Medium", NAVORD Report 21*37 



A- 4 



