138 



falls rapidly till it is, say -^tn or ^th of its maximum value and then falls comparatively very 

 slowly. This gradual decay of pressure is shown in most piozo gauge records and was particularly 

 noticeable in Hilliar's early measurements. It must evidently be associated with the motion of the 

 water round the expanding bubble and c^n be treated as associated with radial flow of an incompressible 

 1 iquid. 



In the early stages when the bubble has not risen appreciably under the action of gravity, 

 the formulae derived by neglecting U, U and the variation in z in (5) may be used. If G(a)/w is 

 neglected, i.e. if it is assumed that the whole kinetic energy Is given to the bubble at the instant 

 of the explosion and the work done by the subsequent expansion of the gases is neglected, the equations 

 are then those used by Raleigh in discussing the collapse of a spherical cavity and by Ramsauer and 

 others. In the early stages of such an expansion, i.e. in a time after the explosion which is small 

 compared with the period of the first expansion and contraction of the bubble, the te rm 2/ 3 z" in (5) 

 may be neglectjd compared with (da'/df)^ (5) then assumes the form a*^ i = l/i/Tn. Inserting 

 this value in (30) the leading term (i.e. the term which Is greater than all the others except close 

 to the bubble) is 



P .Id / .2.., - 1 d fa'^'0= 1 ■ a--^'h- = i (32) 



In terms of f this becomes (fee (15)) 



/ 2~7T J 2 </ 2V r' 



^.,- -. _i_ \ I/JZ.X'" V-^'^ - 0.079B r-'" r--' (33) 



^L « 77 r' |_ 5 J 



Restoring the dimensions (33) becomes 



p . p^ = 0.0798 g3/5pLl2/5j-4/5 ^-1 (34) 



Where p is the pressure at the depth of the explosion. 

 Putting g = 981, g'' = 62.37, p = 1, (3t) becomes 



P - P 



= 4.98 L^^^^ 1-"'^ r-^ (35) 



(35) becomes 



n the case corresponding with M= 4.66 lb. T.N.T. where L= 445 cm., l'^ ^ = 2.265 x 10 so that 



p - p^ = 1.131 X 10^ t-^'* r' (36) 



At r = 14 feet ■= 426 cm. this is p - p^ = 2.655 x lo"* t"^ . At time t = 0.005 seconds t = 



69.5 sc that p - p = 1.85 x 10* = 1.85 atmos. ; at time t ■= 0.010, p - p^ = 1.07 atmos. 



» better approximation is found by extending the solution of (5) back step-by-step towards 

 the time when the bubble is srrall, using the expression (ll) for taking account of the gas pressure. 

 The pressure is then given by (30) but It is found that the vertical motion may be neglected so that 

 the simplified expression 



£ . gz = ^aa^ * a^y . i aV (37) 



p r 2 r 



may be used. When this calculation is carried out It is found that the minimum radius a^^ is attained 

 at a time which is actually earlier than the origin of time used In the step-by-step calculations 

 already described. This is because in starting the Calculations the conditions at the beginning of 

 the explosion were not required, all that was required was a knowledge of the energy v. The way 

 in which this energy was communicated to the water to give it radial motion was iinmter\r\] so far 

 as the motion during the first contraction was concerned. Whi?n, however, it is desired to Ciilculate 

 the pressure round the bubble in its early stages the exact instant of explosion has to be determined, 

 because otherwise It Is not possible to get any Idea of the way In which the shock wave pulse Is 

 related to the subsequent pressure distribution due to radial expansion. This question is a difficult 

 one in any Case, If the water were truly Incompressible the pressure would jump to Its maximum value 

 Instantaneously at the moment of explosion. Actually the compressibility delays the first rise in 



pressure 



I 

 I 



