- 3 



323 



where P is the- peak pressure in IB/square inches, R tht distance of th; DubbU ctntrc in feet, 

 and V the equivalent weijht of T.N.T, in lb.; z^ is the non-dimensional depth of tnt charge., 

 »htn surfaces ar-.. present equations (?) ana (8) und Report A must be used. 



The Attracti on of an Explo sion Bubble to an 

 Infinite Rigid Plane 



Schiffmann's equations for the motion of a bubble close to a rijio plane wall have been 

 integrated in paper O.S.S.O. 3841 for the sp-cial case where the wall is horizontal and below the 

 bubble - i.e. tne case considered is that of a char9e at a dist-incr. d non-dinensional units above 

 a rijid sea bed. Their loproximato equ:ition is found to jjree very closely with the results of 

 their full numeric.il int.jrjtion, ind, wh;n put in Taylor units, jives 



9.012O . 0.00123 . 0.1218 . 0.0157 



~^ ^v v^ -^ 



(non-dimensional) (l) 



Here m is the "momentum constant", which at any instant near to the occurrence of the 



where a is the radius of the bubble and v its linear velocity.. 



,3., 



It may be noted that when the rigid plane is infinitely rrmote - i.e. when d = co, the 

 formula becomes 



(non-dimensional) 



This agrees, within 10 to 15 per cent, with the results of exact numerical integration given in previous 

 reports. 



Equation (l) is somewhat currfcersome, and accordingly values of m were plotted against 

 l/d for a wide ranje of values of z . The following empirical formula has been found to fit the 

 values given by (l) over a very wide rang''' - the error does not exceed 5S of the larger values of m, 

 over the range z = 1.5 to 16, and d = °c to 0. 3. 



0.122 0-°l" '^0- ^' 



2/3 



■. — &JT-7 (non-dimensional) (2) 



Here the momentum is positive upwards - i.e. away from the plane. 



It is interesting to compare this equation with that given in Report A, based on Conyers 

 Herring's first order theory. This gave 



5/3 



(non-aim_nsional) 



Now a^ z = 0.195 - an approximation found to be true within about u» over the range 

 z = 2 to z =8. Insertihy this in the above equation Jives 



^ s 



„^ 0.0138 z^'^ 

 m = ' \\ If. 7 — rm; — (non-dimensional) 



From this it appears that the simple first order theory, valid strictly only for values 

 of d large compared to the maximum bubble radius, is in f^.ct a fair approximation right up to the 

 point where the bubble touches the plan^- at its maximum ridius. 



0.0107 (z - 1) 



If the plan= is not horizontal thun it is necessary to add the momentum < — rn'T. 



<^ % 

 directed towards the plane, vectorially to the momentum ',i ,- upw.rds due to gravity. 



2/3 



