276 MOTION OF THE GAS SPHERE 



This result can also be expressed in terms of the total energy Y by 

 Eq. 8.6, giving 



yi/3 

 (8.9) ■ T=lMpc"'- — 



P5/6 



This expression, which has been derived by a number of writers and is 

 usually known as the Willis formula (121), shows that the ''bubble 

 period" varies as the cube root of the total energy or weight of a given 

 explosive, and for a given weight, varies as the negative 5/6 power of 

 the hydrostatic pressure. If the pressure at the surface of the water is 

 one atmosphere, or 33 feet of sea water, the period therefore varies as 

 (d + 33)~^/^, where d is the depth in feet. The particular constant 

 multiplying the period formula depends, of course, on the limit of inte- 

 gration chosen for the initial radius. The functional dependence of the 

 period on F, Po, Po must, however, be of the form of Eq. (8.9) from 

 dimensional considerations, if these variables are assumed to be the 

 only ones affecting the motion. It is therefore reasonable to expect 

 that this form should apply more generally than the approximations of 

 the simplified model would permit. The experimental evidence de- 

 scribed in section 8.3 does in fact demonstrate the validity of the 

 formula over a wide range of depths provided the multiplying factor is 

 suitably adjusted. However, this agreement is not found for charges 

 fired near the surface or bottom, which is to be expected as a result of 

 the distortion of the mass flow of water by such boundaries. The 

 modifications of the simple formula obtained by more refined analysis 

 are considered in section 8.10. 



8.3. Comparison of Radius and Period Measurements with 

 Simple Theory 



A. Radius-time curves. The earliest systematic measurements of 

 the motion of gaseous explosion products are those of Ramsauer (88), 

 who employed an ingenious electrolytic probe method to determine the 

 position of the gas bubble boundary. In these experiments, charges of 

 guncotton weighing one or two kilograms were fired at depths up to 

 thirty feet in forty feet of water. A number of electrodes were sup- 

 ported at suitable distances from the charge by a rigid frame, and to- 

 gether with a common electrode formed conducting circuits with the 

 seawater acting as electrolyte. After the charge was fired, the expand- 

 ing gas bubble isolated the electrode circuits successively, and relay cir- 

 cuits were used to make a spark recording of the times of current inter- 

 ruption. In this way a displacement-time curve of the bubble motion 

 could be determined up to its first maximum radius, and Ramsauer 



