Z - I - 



For snaller charges the time is less so that It may be stated that the blow delivered by the direct 

 wave to any structure is wholly spent in a time which is less than 0.01 second. 



Behaviour of thin plate in path of explosion wave . 



Let the water in which the explosion wave is generated be separated from air by a rigid wall. 

 Suppose a circular hole of radius a to be cut In this wall, the aperture being rendered water-tight 

 by a thin plate. Suppose the natural period of oscill.it ion of the plate to be long compared with 

 the duration of the wave and witn the time talitn by an explosion wave to travel across its diameter. 

 When the pressure wave Impinger, upon this plate each element will begin to move inwards but in so 

 doing will send out a secondary wavelet causing relief of pressure on the surrounding elerents so 

 that in determining the motion of the plate we must take into account not only the force required to 

 accelerate the plate elements but also the force required to balance the relief pressure due to motion. 



If u is the velocity of an element ds of the plate at an Instant t then the wavelet thrown 

 out by this element exerts a relief pressure ^ T 3TE ** ''•stance r from ds and at a t Ire He 

 later than t. In this relation p is the density of the water and c the velocity of propagation in 

 water. Hence if v is the ultimate velocity of the element ds after the passage of the wave the 

 whole relief momentum contributc-d at a distance >■ is ^ ^ the only effect of finite velocity of 

 propagation being a retardation of delivery of this momentum by a time r/c. Thus the momentum I 

 delivered to any element of the plate is partially neutralized by the relief momentum contributed by 

 all elements of the plate and the residue is spent in increasing the momentum of the plate element.. 

 It is here assumed that forces depending on displacement have not had time to come into p)ay. 



If the plate is very thin the momentum required for the plate elements may be neglected so 

 that we have, in this case 



In 



(5) 



and this must hold for all elements. Now the right hand side of (5) is the same as the expression 

 for the electrical potential at a point on a plate which has a charge of surface density v and in 

 this case tne condition implied in (5) is that the charge must be such that the plate is jt constant 

 potential. Now for a circular disc it is known that an electric charge distribution must be of the 



form 







j/41 



where v is the charge density at the centre v the charge density at x 



from the centre and a the radius of the plate. 



2 I 

 element ^g ' 7? lOa ^° *''''' 



Inserting this in (5) we find using the central 



2 I 



T 



(6) 



Equation (6) indicates that the velocity is least at the centre and Increases indefinitely at the 

 ed-j6S. The result is a consequence of neglecting all forces due to relative displacement and 

 indicates how water would flow into a circular aperture previously closed by a very weak diaphragm. 

 It also shows that when the diaphragm offers resistance to distortion the greatest stresses iviU be 

 at the edges. This is to be expected as the relief momentum cannot then act as effectively at tne 

 edges as at the centre. 



We may, however, pursue this case further and determine the kinetic energy of the inflowing 

 water inmediately after the blow. 



Thus if a pressure p is applied for a short time T to the plate, the work done on an element 

 ds of the plate is pvrds = vSlds, since |jr represents an increment SE is the corresponding increase 

 of kinetic energy of the water associated with the plate 



dE 

 dl 



xdx 



. TT7 



4la 

 P 



by equation (6) 

 Hence 



E = 2 1-^ a//0 



(7) 



The 



