272 - " - 



Comparison with Rayleigh' s Vork . 



It is interesting to compare equation (l) with the apparently different equation obtained 

 by Rayleigh (u) (see, for example, Theory of Sound. Volume II, paragraph 302) for the reaction 

 pressure on a piston vibrating in an aperture in a rigid vvall with a frequency a)/2 tt and maximum 

 velocity U . The expression is 



F = -77r2 U^p "^ p„C 



( k R k R y 



(5) 



where k = 'j , and J, and H are respectively Bessel and Struve functions of order unity. Since 

 any motion, periodic or not, can be represented by a Fourier Integral, It Is to be expected that 

 equation (5) is equivalent to the more general form of equation (l) when .U(P) Is not zero. This 

 is In fact so, but the fornel proof is rather long and is given in the Appendix. Equation (5) 

 takes especially simple forms when kR is very large or very small. 



(a) kR l^rge: <= --rr r' U p "^' ( pC * £1 12-11 (6) 



kR L^rge: f -n r' Uo '"^'- f pC * 11 ^2.1 ^ 

 '^ ° V. ° V a> R J 



The first of these terms is the radiation damping term, corresponding to the second term on the 

 L.H.S. of equation (2), while the second terra (corresponding to the third term on the L.H.S. of 

 equation (2)) measures the effect of diffraction on to the piston of the pressure-charges at the 

 immobile walls. We thus conclude that equation (2) is likely to be a better approximation for a 

 finite plate than Taylor's (2) infinite plate theory, which takes only the radiation damping term 

 into account. Thiswould have been sufficient justification for investigating the solutions of 

 equation (2), but the argument in paragraph 2 is more precise, and enables us to set an upper 

 1 imit to the error. 



(b) kR small : F-77R^U„p'"^ ( '^Jll + _L o i <u R ) (7) 



o \. 2 3 77 ° y 



The second of these terms is proportional to the acceleration, and, in fact, arises from the "virtu 

 mass" of the water moving with the piston that is obtained from incompressible theory. The first 

 term is proportional to the differential coefficient of the acceleration, and also to r> ^'^ is 

 therefore a correcting term, absent in incompressible theory. This suggests that we try the 

 following equation as a correction to incompressible theory (expressed In norMlimenslonal units). 



Tt • i_flf ' -) '^ 



i^ 'It, , (^ 5 . .1 «^ . P-' . m 



Equation (8) (without the first term) is what would be obtained using Incompressible theory. 

 The correcting term tends to make the deflection smaller. A discussion of the effect of this 

 correction on the final deflections is reserved for another report. 



Regions of Validity of Equations (3) and (8) . 



The position is now as follows. The crude argument that the main contribution to the 

 Fourier integral representing the motion of the plate will be from frequencies corresponding roughly 

 to the inverse time-constant of the explosion, loads us to equation (3) for large plates and 

 equation (8) for smell ones, but gives no indication of tne errors to be expected from their use. 

 The more precise argument of paragraph 2 indicates that equation (3) will be best for short times 

 and enables us to estimate the error. An argument due to Kennard (5) shows that the total impulse 

 due to relief pressure plus diffraction from the edges vanishes for long times so that incompressible 

 theory (or the imprw^d version of it given by equation (8)) should be best for Ijng time, provided 

 that cavitation does not occur. His argument is based on the fact that the total pressure due 

 to the motion of the plate is given by an integral involving the acceleration of the plate, so that 

 the integral of this pressure between zero time and the time at which the plate comes to rest must 

 vanish. We therefore use equation (3) in order to determine our cavitation criterion. Equation (8) 

 could have been obtained directly from equation (l) by expansion of the second term in powers of 

 h. Integrating twice by parts. 



Pressure 



