212 - « - 



assumptions we 'ind tor the predominant terms, 



Elj = i e- "^ Sin m T (36) 



» 77 a p_ m 



•^0 



« - t-e-1^ _ J_ .-kT ,^^^, 



4 q Aj 6 (k - l) v^ 2 X kq 2m 



+ -L (e- "T _ g- kT (.^^ ^ T) (37) 



2k 



= 9 (K - 1) e- ^^ Sin m T ♦ '^ ^-^' (^ - «' ^'1 



6 (k - l) s. 2mk 2X kq 







_ A (1 - e" "^ Cos m T) ♦ -I- (38) 



2k 2kX 



Jiseusston of results , 



(1) Applicability of unoerwater explosions . 



Consioerinj the first funOamental assumption of the analysis that the incident pulse can be 

 treated as of small amplitude it was shown in Appendix 1 of the report 'The reflection of a spherical 

 wave from an infinite plate" that this is reasonably accurate for distances D such that q tf'a > 6 

 about. Thus the pulse can be regarded as of small omplitude even at the nearest point of the 

 sphere if q (X - l) ^ 6, and beyond such a distance, corresponding to about 13 charge diameters, the 

 analysis might be expected to give reasonably accurate predictions. At somewhat closer distances 

 the analysis begins to lose in accuracy but as a first approximation it may still be able to give 

 the order of magnitude of bodily velocity in so far as this is due to the pressure pulse. Thus 

 from the results given Dy "enney, we see that at a radius of three charge diameters the maximum 

 pressure is some 801 greater than would be expected if the pressure had continued to increase 

 inversely as the distance when approaching the explosion. Thus at three charge diameters 

 corresponding to about q (x - l) = l.U the present analysis is likely to be accurate within a 

 factor of two and it can thus be used for these closer distances in casee where it is only dasired 

 to know order of magnitude, e.g. whether the bodily velocity is small or large. 



The second fundamental assumption of the analysis is that the movement of the sphere 

 is small with respect to its radius and for any particular case this is most easily checked a 

 posteriori by using the prssant analysis to estimate the displacement. In this connection it 

 should be noted that while the assumed exponential form is a reasonable representation of an 

 underwater explosion pressure pulse until the pressure becomes snail It is not representative of the 

 subsequent stages during which incompressible flow (kinetic wave) effects ara of equal or greater 

 relative importance. In particular, the last term of equation (27), corresponding as it does to 

 an afterflow effect, must be considered as limited by the condition that q T is not too large when 

 estimating the displacement due to the pressure pulse. 



(2) Direction of motion and diffraction effects . 



G.t, Taylor has shown in a previous paper that a sphere which is just buoyant never reverses 

 in motion when given an initial velocity. A similar conclusion is not necessarily true for the 

 motion produced by an underwater pressure pulse. Thus, for the case of a short pulse from a distant 

 source (X — co) v»e see from equation (.}>i] that the velocity of the sphere dies away as a damped sine 

 wave and thus becomes negative for certain periods; for this case, taking k = 3/2, It should be 

 noted, however, that the maximum negative velocity is only .0043 times the maximum positive velocity 

 so that quantitatively the reverse motion is relatively unimportant. 



Some features of the reflection and diffraction of the incident wave by the sphere can be 

 illustrated by considering the maximum unbalanced impulse acting on the sphere as tabulated for 

 certain cases in Table 1. 



Table 1 



