258 . u _ 



(2) Taylor's theory . 



The first stage of the motion has alreaay Oeen treatea in aetail Cy Taylord), so tfat 

 a brief sumcHry of l-.is work is all that is needed, and we take up the problem at the point at 

 which he leaves it. 



Let y be the displaceirent of the plate in tne direction of the incident pulse. 

 1 

 Let p = p e " (t + -) tie the incident pulse cominj from the positive direction, 



p = (t - i) be the reflected pulse (due to the motion of the plate). 



p - mass of plate per unit area, 



p = density of octer. 



6 - the time constant of the pressure pulse (assumed exponential as usual). 



The equation of motion of the plate is now:- 



t 



P. H 



d t' 



P„ e 



° ♦ (E ( 



(1) 



Continuity at the surface of the plate requires that:- 



PnC 



dt 



c? (t) 



(2) 



El iminat ina (^, we obtain the result:- 



PaB ^ ^0=^ = ^Pm^" 



a t 



(?) 



The solution of this, for which y=j|=Oatt=0 is:- 



31 1 _JJq 



dt p^ C (1 - a) 



y . ^^^ 



p„ C (1 - a) 

 Pa 



t t 



(1- 



a + a e 



t 

 Si 



Where we have wri tten a for p q ^ . I n Taylor's notat ion a =• 5 

 ^0 



(«) 



We assure with Taylor the result that, in any practical csose, the effect of elastic and 

 plastic forces on the motion of the plate is negligible during the time while the net pressure 

 on the plate is positive. This fortunate circumstance has enabled us to drop the 'sprinj' term 

 in equation (l). In general also a is snail compared with unity. a= 1 would Imply a plate 

 about i foot thick for a 300 lb. charge, and about -j Inch thick for a 1 oz. charge. The 

 expression for the pressure at the plate is:- 



p = P^c ^ ♦ (^ (t) 



ifm [/ 



a f " J .using equations (2) and (u). 



(5) 



The expression for the pressure in the water is obtained by taking account of tne fact that i^ 

 represents a reflected pulse:- 



P = P„ 



-k - ^ - i + X 



« 5 Pt 4 p e e ETC 



2P™ 



t . X ' + X 



(6) 



For 



