-3- * 



The results of this section show that the effect of an explosive wave upon a circular aperture In a 

 rijid wall initially closed by a very weal< diaphragm is to produce an inflow of water into the 

 aperture whose velocity distribution is ^iven by (6) and whose total enerjy is given by (7), The 

 result holds whether the water is regarded as incompressible or not so long as the tine taken by a 

 wave to cross the the aperture is so short that forces depending upon displacement have no time 

 to be developed. 



Aperture blocked by a ■piston . 



Suppose next that the inflow is forced to assume a uniform velocity throughout the aperture 

 say by the introduction of a piston. The relief momentum is now given by 



277 



ds _ 2 jO va £ /X) 

 r 77 ^ 



where v Is the final piston velocity, a the radius of the aperture and the expression holds at a 

 point distance x from the centre. E (i) Is the elliptic integral of the second kind and has the 



value 



when x 



and the value 1 when x = a. The relief momentum therefore diminishes from 



1.57 to 1 as we proceed from the centre outwards. This non-uniformity of distribution need not, 

 however, prevent us from equating the whole r^ilief momentum over the piston to the applied momentum 

 as the translat ional momentum must be balanced. Thus neglecting the mass of the piston 



TT a I = U p va 



X E (2) dx = I p 



so that the final velocity v is given by 



3 7T J_ 



~S" pa 



(8) 



The energy E of the inflowing water is such that 



oE 

 3T 



so that by (8) 



I = L2L J_S = 1.851 1^ a/p 

 16 p 



(9) 



Miscellaneous forms of restriction . 



If more generally we assume that v is restricted to the form 

 ,2 „" „6 



1 * a T * a^ ^ * a^ i^ * 

 a a a 



(10) 



It is Shown in the appendix that 



I = - pa v„ 



iii a + -liit 



45 



1575 " 11025 * 



' = ^^V „!,[^2n«nJ, IT.4^^rX3] 



(11) 



(12) 



