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SUMMARY OF FORMULAE RELEVANT TO REFLECTION OF 

 EXPLOSION PULSES FROM A PISTON IN AN INFINITE 

 FIXED WALL 



E. N. Fox 



Admiralty Undex Worka 



March 1943 



A, 1. Conpressiple flow. 



The Infinite wall is taken as the piano x » o, and ill sources of pulses as lying to the 

 right of this plane (x > o), the total pressure In the Incident pulses arriving at time t at any 

 point on the plane x = o Being denoted by P|. 



From Kirchoff's general solution of the wave equation, the pressure p at time t at any 

 point of the plane x « o satisfies the equation 



p « 2 P| 



2 7T J J r [3xJ^__r 



dS (1) 



c 



where the surface intagral is taken over x = o. 



For a piston In an Infinite rigid wall let 



p = mass density of water 



c • wave veloci ty 



S » area of piston 



s " perimeter of piston 



m « mass/unit area of piston 



U = velocity of piston away from sources 



§ = displacement of piston away from sources 



The boundary condition at x = o i s then 

 ±B . p^ over S 



a X dt 



= c over rest of x « o (2) 



It Is then merely a question of algebra and integration to derive the following formulae, 

 (a) Pressure at any point P within S is 



2 p, - pc U(t) - ^ ♦ 

 ' ^^ J 



U (t - ^) cos V 



£ ds (3) 



where the line integral is taken round the perimeter s, r is the distance of p 

 from any point of the perimeter and v is the angle between the radius vector r 

 and the inward normal to s (Figure 1). 



(b) Pressure at any point on the wall outside the piston is 



or [ 'J (t - ^) cos V 

 p = 2 f - eS. i £ ds (») 



' 27T ] r 



with notation as before (Figure l). 



(c) 



