481 



THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 



VOLUME 20, NUMBER 3 



Secondary Pressure Piilses Due to Gas Globe Oscillation in Underwater Explosions. 

 II. Selection of Adiabatic Parameters in the Theory of Oscillation* 



A. B. Arons 



Department of Physics, Stevens Institute of Technology, Hoboken, New Jersey 



(Received January 13, 1948) 



A summary is given of the theory of pulsation of a stationary gas globe in an infinite incom- 

 pressible fluid. If three parameters appearing in the formulation are selected by fitting the 

 equations to three independent experimental results, it is shown that the theory fits the re- 

 maining experimental results and can be used to predict bubble pulse properties over wide 

 ranges of the independent variables. 



I. INTRODUCTION 



1.1. The theory of the pulsation of gas globes 

 formed in underwater explosions has been treated 

 by several investigators during the past few years 

 and has been summarized by Friedman in a 

 recent report of the New York University Insti- 

 tute for Mathematics and Mechanics.' The theory 

 in this formulation depends upon three parame- 

 ters which cannot be accurately determined by 

 means of a priori calculations, and recourse must 

 be had to experimental information regarding 

 certain properties of the bubble pulsation. 



Experimental results obtained in deep water at 

 the Woods HoleOceanographic Institution' afford 

 the necessary information, and it is the purpose 

 of this report to discuss the selection of parame- 

 ters which make it possible to fit the theory to the 

 experimental results over a wide range of the 

 primary variables. 



n. FORMULATION OF THE THEORY 



2.1. The theory referred to above' has been set 

 up to treat the general case in which the gas 

 bubble migrates as a result of the combined effects 

 of gravity and the presence of neighboring free 

 and rigid surfaces. A brief summary of the formu- 

 lation will be given here, modified for application 

 to the special case of the stationary bubble (i.e., 

 negligible migration). 



Consider a perfect sphere of radius A expanding 



* Contribution of the Woods Hole Oceanographic Insti- 

 tution No. 431. 



' Bernard Friedman, "Theory of underwater explosion 

 bubbles," Institute of Mathematics and Mechanics, New 

 Vork University, Report No. IMM-NYU 166. 



'A. B. Arons, J. P. Slifko, and A. Carter, "Secondary 

 pressure pulses due to gas globe oscillation in underwater 

 explosions. I. Experimental data," this Journal. 



at radial velocity ^' in an infinite incompressive 

 liquid of density p. The total kinetic energy of the 

 fluid external to radius A is given by 



/. 



piR'V- 



47r/?- dR- 



2 



■IirpA'iA'y, (1) 



where the primes denote time derivatives. 

 The potential energy of the system is 



U={i/3)irA'Po+G{A), *(2) 



where the first term represents energy stored 

 against hydrostatic pressure (Fo being the abso- 

 lute hydrostatic pressure at the depth of the 

 bubble center), and the second term represents 

 the internal energy of the gas in the bubble. The 

 zero of internal energy is defined as the infinite 

 limit of adiabatic expansion, thus: 



G{A)- 



r. 



pd V, 



(3) 



where the line integral is taken along an adiabatic 

 and V represents the total volume of gas. 



It is assumed that the gas approximates ideal 

 behavior : 



pv = ku (4) 



where v is specific volume, ki is a constant, and y 

 is the ratio of heat capacities. 

 Combining Eqs. (3) and (4), 



dV kiAfy 

 G{A) =kiMy I — = — — :, (5) 





yy 



(7-l)^'<>-"(47r/3)> 



where M is the mass of gas. 



Denoting the total energy associated with the 

 oscillation by E, 



E = 2TpA''iAr-+ii/3)7rA'Po+G{A). (6) 



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