275 



The Equation of motion of the plate according to this Approximation, 



We take as our expression for the velocity of the plate 



(12) 



Where r is the distance from the centre. 



As the plate is not moving as a rigid body, it wojid not be correct to integrate up the 

 pressure over the plate and then write down the equation of motion directly from this, as we did 

 for the piston. We must first multiply the pressure at each point of the plate by the velocity 

 at that point, and then integrate over the surface of the plate. This gives us the rate of 

 doing work on the plate, which we can equate to the rate of change of kinetic energy of the plate 

 to give the equation of motion. In equation (l2) we have chosen the factor 2 in order that U 

 nay represent the velocity of the "equivalent piston", i.e. the piston moving with a velocity 

 U would sweep out the same volume asthe plate. The total kinetic energy of the plate is given 



by 



i p ^,'p' '^ f f (sir/ 

 2 ° ° '^ •'o R^ 



, H, ^a - 27T „2 ,, 2 „ 2 iujt 

 r dr dc7 = — " ^n P 



(13) 



The relief pressure due to the motion of the plate is given at a point distant r from the centre, 

 by 



1(1* 



2 77 



2 '^o i" 



sLi-d g' '"^ 



d S, 



where the integration is taken over the whole plate, and s is the distance between the point in 

 question and a typical point on the plate. The rate of doing work on the plate is obtained by 

 multiplying this integral by expression (12), and then again by 2 tt rdr and integrating from 

 to R. The multiple integral thus obtained can be evaluated by the method used by Rayleigh (u), 

 the result for the rate at which the relief pressure does work being 



Srrp u„'cp^ '^ 



(J, (2 kR) - i H (2 kR)) /- -1_ + l^") +5 ( (2^1,) _ I „ ^2|^|,),j 

 \^ Rk k^ y k^ ° ° 



17 



3Tr 



The forms which expression (in) takes for kR large and kR small are respectively 



8 " Po "o^ "= >= ' ''^ (7 * ^) '"" '^--g^' 



and 



B^Po%''p'''^' fiiliiii . -J- r' A (,Rs™n) 



° ° \ 315 77 16 y 



(14) 



(15) 



(16) 



These expressions, together with expression (lU), suggest the following equation in place of 

 equation (2). 



large; i ph S-X * I p C 



' 1 rtt * q 



Si 



'Pn^' 



Cd - ■ } dt' 3 " dt R 



and the following equation in place of equation (8) 



J y d t = 2 p - g 



R 



— sma 

 C9 



n^ at' [^ 7T^ 3 i dt^ 2 



- S y 



(17) 



(18) 



where 



