163 



THE BEHAVIOUR OF AN UNDERWATER EXPLOSION BUBBL E 



Approximations based on the theory of 

 Professor G. I. Taylor. 



A. R. Bryant 



December 1942. 



Summary 



Approximate solutions are presentea for the equations of motion of the gas bubble produced by 

 an underwater explosion, as given by Professor G. I. Taylor, The equations enable the most important 

 features of the bubble motion to be computed approximately with comparatively little labour. In 

 addition graphs are given which are based upon the approximations, and which ercble most of the quantities 

 to be read directly as functions of the depth for various charge weights. The effect produced on the 

 bubble motion by changing the charge weight or the depth of the charge may be easily seen from these 

 approximations, which should assist In an appreciation of the Velations which arise when the scale of 

 an experiment is changed. 



Introduc tion . 



Professor G. I. Taylor has treated the radial expansion and vertical motfon of the bubble of gas 

 formed when ■< charge is detonated in water*. He has developed equations which require to be integrated 

 step by £,tep numerically and has shown that exact scaling for different charge weights is not possible, 

 so that the numerical Integration must be repeated for different values of the charge weight and the 

 depth. The following note puts forward approxinate solutions to these equations which enable most of 

 the important features of the bubble motion to be computed with comparatively little labour. In order 

 that these equations may be readily available they have been listed at the beginning of the note, with 

 an explanation of the symbols used, but .vith their derivation omitted. Although the numerical constants 

 for T.N.T. have been employed throughout, the methods of approximation used are applicable to any 

 explosive. 



As in the Report "Vertical motion of a spherical bubble and the pressure surrounding it", free 

 or rigid surfaces have been assumed remote enough to cause no disturbance to the motion. Their 

 perturbing effect as given by Conyers Herring is discussed in an Appendix. 



The non-dtmensional form of the Equations . 



The basic equations of motion of ihe bubble are used in the non-dimensional form given by 

 Professor Taylor. In the list of formulae below, and in their subsequent derivation, some of the 

 equations are best left in this form. To convert to real quantities all non-dimensional lengths 

 must be multiplied by the length scale factor L, and all non-domensional times multiplied by/-, g 

 being the acceleration due to gravity. For T.N.T. the value of L is given by 



L (feet) = 10 M* (l) 



where M is the charge weight in lbs. L is plotted against charge weight in Figure 1. 



To avoid confusion non-dlmefisional quant I ties will be denoted by small letters, while capital 

 letters will be used for dimensional quantities (with the exception of the symbols g, and p the density 

 of water). Non-dimensional equations will be labelled as such. 



Part I 

 ertical motion of a spherical bubble and the pressure Surrounding it. G.l. Taylor. 



