222 



Using bars to denote average values taken over the piston at any time, the mean 

 pressure on the piston Is 



2 p. - pc U(t) + f^^ ^^(t) 



2 

 TT 3 



,2 r f ^ (t - p cos A-j COS \^ 



2 TT 3 



-et- ) ds, I ^- £- i ^ ds, (5) 



r 



where both line integrals are tal<en round s, r is the distance between ds^ and dSj, 

 while \, and \, are the angles between the chord r and the inward normals to s at dSj 

 and dsj respectively (Figure 2). 



(d) For a circular piston of radius a equation (5) reduces to 



p = 2 Pi - fX U(t) t i^ r ^ u(t _ ii^inj ) ^^^2^ ^0 (5) 



the fest two terms of which represent the relief pressure as given By Butterworth 

 in equation (l) of the Appendix of the report "Note on the motion of a finite plate 

 due to an underwater explosion". 



(e) For a rsctanjular piston of sides 23 ana 2b and diameter 2l , equation (5) gives 



p = 2?. - pc U(t) * P^^ (^^ ") g^(t) 

 ' 77 ab 



2 77 ab 

 - £d ['"'''''' ^ (t - i2-H£i) cos e d £> (7) 



77 b 



•' u 



->b/3 



- ed r^"'^'^^(t-£2seLf)cosede 

 ^a Jo 



(f) If the piston is subject to an elastic restoring force the equation of motion is of form 



n, ^ ♦ h j^ = p (8) 



For a circular piston equations (6) and (8) give an intcgro-different ial equation 

 which can be solved step by step as in the report "Note on the motion of a finite 

 plate due to an underwater explosion". 



For a rectangular piston equations (6) and (8) involve the solution of ordinary 

 linear differential equations with constant coefficients, but such solution would 

 have tc be carried out for successive intervals of time such as 



.f t ■$ 2a/c, 2a/c •« t ^ 2b/c 



depending on'tne relative magnitudes of a and b. 



A. 2. Incompressible flow. 



The corresponding formulae for incompressible flow are obtained simply by letting 

 c-'O^ in the preceding cauations. 



For a circular piston subject to no elastic resistance equations (6) and (8) give 



m 2M = 2 p. - ^ 2!J (9) 



dt ' 3 77 dt 



For 



