- 3 - 271 



The Equation of Motion of the Plate , 



Trie investigation is based on an equation obt,iined Dy Butterworth and Wi^glesworth, If 

 the piston is at rest, it forms, in conjunction with the baffle, a rigid surface, and the pressure 

 at it is just double that due to the incident «ave above. If the piston is moving with a velocity 

 u{t), thero is a relief pressure dut to this motion, which tends to decelerate the piston. By 

 integrating the usual expression for the retarded potential over the surface of the piston, one 

 gets the relief pressure at any point on the piston, and a second such integration gives the total 

 force Oh the piston due to the reli.if pressure. It is (if U = for t < o) 



p Ctt H^ 



u(t) -— j' fi -^^ ") u(t-f)df] (1) 



ttRo^ur'/ J 



where R is the radius of the piston, and p and C are the density of and velocity of sound in water, 



2R 2R 



For t -* -Q-. 'be integration stops jit -=• . For a reason that will appear later, we shall not be 



concerned with times greater than ■^. Fjr times up t'. this limit, the factor 



is never less than y | , and since U is positive at any stage of the motion with 



(-^;^)' 



which we shall be concerned, it follows that the error in taking this factor equal to unity cannot 

 possibly exceed 11% even in the very worst possible case (and in all ordinary cases it will be very 

 much less than this). We my th.-r^fore assort with some confidence that the equation of motion 

 obtained by Butterworth and Wigglesworth 



h2 rf 2 p C^ t 



ph 2_5 + p C 2i 2_ y = 2 P^ p-0 (2) 



dt"^ ° dt rr R "^ 



R 

 IS a good approximation for t < j. In this equation, y is the displacement of the piston, 



p its density and h its thickness, also p^^^ is the maximum pressure in the incident pulse and 



d its time-constant. We introduce non-dimensional units as in I. 



a = non-dimensional thickness of plate = 1°" 



fS = nofv-dimensional radius of plate 



Unit of displacement 



Unit of time 

 As in I, equations involving those ui 



a ix + 2ji - Z = p-t 

 dt^ dt ^ 



For a large plate, the term- t becomes negligibU, and the theory reduces to Taylor's (2) case 



»s in Taylor's work, the energy expended in stretching the plate has a negligible effect on the 

 early part of the motion. The solution of equation (3), for which y = ^ = at t = is to 

 the first order in - 

 /3 



y ' 



{2a* 3)ii*0.aP) ' (i*/3-a.^) (1 + /3 - a /3)(2 a + /3) ^ ^ 



(H) 



Comparison 



