NSWC/WOL TR 77-90 

 where p is the density of the water and I - /p dt, the applied 



3 



impulse. In order to extend the range of usefulness of this 

 approximation we compute the applied impulse at t = as 



e 



I = /p(t) dt 



= PMAX X e X [1 - e ] 



= 0,6 32 X PMAX X e (3.4.2 



The total energy Y following impulsive loading is given by* 



T 9 p . V. 



Y = - p V vl^ + p.v. + -*• ^ 



2 '^ i i ^i i Y-1 (3.4.3) 



where p. and V. are the initial pressure and volume of the bubble, 

 and Y is the adiabatic exponent (= 1.4 for air). In Equation 3.4.3 

 the first term is the kinetic energy imparted to the surrounding 

 water by the impulsive loading, the second the potential energy of 

 the surrounding water, and the third the internal energy of the 

 air inside the bubble. 



From the total energy Y we calculate the dimensionless bubble 

 oscillation parameter k used to describe the motion 



' ' ^-1 (pk ) 



(3.3.3)** 



CJombining 3.4.1, 3.4.3, and 3.4.4 we can express k in terms of the 

 impulse and initial bubble radius 



* Reference 1, Equation Al 

 ** Reference 1, Equation A13 



3. Kennard, E. H. , 1943, "Radial Motion of Water Surrounding a Sphere of Gas 



in Relation to Pressure Waves," published in Vol. II of "Underwater Explosion 

 Research," Office of Naval Research, 1950. 



14 



