201 



NOTE ON THE MOTION OF A FINITE PLATE DUE 

 TO AN UNDERWATER EXPLOSION 



S. Butterworth 



August 1942 



In the report "Damage to ship's plates Dy underwater explosions" a theory of the motion 

 of an air backed circular plate in an otherwise rijid wall under the action of a submarine explosive 

 wave was developed assuming that waiter could be regarded as incompressible. 



Recently G.l. Taylor in the report "The pressure and impulse of submarine explosion waves 

 on plates* has put forward a theory of the motion of an infinite plate under the action of an 

 explosive wave propagated at tho velocity of sound in water. This theory gives results wliich are 

 totally different from those of the report "Dam-ige to ship's pl-:.tes by underwatur explosions". 

 In particular it leaas to the conclusion that if water could be made incompressible the plate would 

 suffer no motion r,t all under the action of an explosive wave. Although in the Appendix to the 

 report "Damage to ship's plat'js by undtrwater explosions" qualitative reasons were given for 

 preftrring th5 theory thtre aovanceo, it seems now important to ^'xamine the differences more closely 

 in view of the reviv-il of the old theory. 



Regard the plate as a piston of mss m per unit area and let it be moving inwards with 

 velocity U at time t . 



Each element of the plate senas out v/avelcts into the water which give reaction pressures 

 on the rest of the plate. According to orthodox acoustic theory the reaction pressure at a distance 

 r from an element of area os is - ^^ -~ ^ occurring at a time r/c later than t where p is the 

 density of water and c the velocity of sound in water. 



Thus the reaction pressure at the centre of the plate {assumed circular and of radius R) is:- 



-p p' ^ { J(t - p } dr = pc { u(t) - u(o) } (1) 



Jo 



SO long as t < R/c. When t > R/c the integration stops at R and the reaction pressure is:- 



R 



3T 



^{ U(t -p } dr = -pc{ U(t) - uA} (2) 



If the applied pressure is f^ e~" and if we approximate by assuming that the reaction 

 pressure at the centre is that for the whole plate, the equation of motion is:- 



m ^ = P^ e-^* - PC u 0) 



so long as t < R/c and U = when t = and 



" 31 = ""o ■-■""* - P^ ^ ^f" - u(t - T) } - (») 



when t > R/c ( * T ), 



Equation (3) is that used In the report "The pressure and impulse of submarine explosion 

 waves on plates* and holds for all time for an Infinite plate. 



Equation ,.„. 



