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APPENDIX 1 

 SKPIRICAL RADIUS-TIIffi CDRVE FOR OSCILLATING GAS BUBBLES IN FREE WATER 



M, Arsove 



In any study of the phenomena associated with the bubble formed by the 

 gaseous products of an underwater explosion, it is essential that one know 

 the size and shape of the bubble as a function of time. Assuming that the 

 bubble is always spherical in shape, the bubble phenomena are completely 

 determined if the Ijubble radius is known as a function of time. High-speed 

 motion pictures of the bubble formed by charges which are not greatly 

 elongated show that this is a good approximation for at least the first 

 bubble oscillation. 



To date it has been necessary to integrate numerically the theoretical 

 equations of motion derived by Herring3), Shiffman and Friednan21), Taylor 18), 

 and others in order to obtain a radius-time curve. If one has an analytic 

 expression for the radius as a fvinction of time, one can avoid the laborious 

 numerical integrations required in order to predict such phenomena as 

 migration, bubble-pulse, etc. 



The purpose of this appendix is to find an analytic expression for the 

 experimentally determined radius-time curves in free water and to indicate 

 possible uses for this expression. 



In attempting to find such an analytic expression, it is convenient to 

 reduce the experimental data (for the radius as a function of time) to non- 

 dimensional form. The units of time and length used in this appendix are 

 the first bubble period (T]_) emd the first maximum bubble radius (A^n.) > 

 respectively. These units of time and length were chosen because they cam 

 be computed from the equations in the body of this report* and because they 

 reduce the data to a form which is easily analyzed. 



The radius-time curves used in this analysis are those mentioned in 

 Sec. 8 of this report. The composite cm-ves (Figs. 4-9) in which a/Amt has 

 been plotted against T/Ti for TNT at depths of 300 and 550 ft; tetryl at 

 300 and 600 ftj torpex-2 at 600 ft; blasting gelatin at 500 ft contain 

 7, 7, 9, 4, 8, and 3 individiial radius-time cxirves respectively. 

 Figures 4, 5, 7, 3, and 9 show that there is relatively little scatter in 

 the daTA from shot to shot when a given weight and type of explosive is 

 observed with the standard charge orientation. There is, however, considerable 

 scatter in the neighborhood of the minimum due to the difficulties mentioned 

 previously in Sec. 3. The data for blasting gelatin (Fig. 9) are of 

 particular interest since they clearly indicate the non-sphericity present 

 in the eatrly stages of the oscillation of a bubble formed by a cylindrical 

 charge vzhose length and diameter are approximately equal. There is con- 

 siderable doubt as to the values of the radius for the contracting phase of 

 the oscillation since the bubble was imstable and apparently formed four 



* It should bs pointed out that in preparing the composite curves the observed 

 values of Ajj^ and T^ ware used to reduce each record to non-dimensional form. 



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