214 :-8- 



unbalanced impulse. In view o' this it is easily shown that 



(39) 



^ 3 (*f - ^p) 



where the suffixes f ana b denote fixed and buoyant sphere respectively, i is the Impjlse/uni t area 

 at the point of the sphere nearest the source while a as before is the unbalanced impulse. 



Now for very short pulses the maximum value of i^, which occurs initially and will be 

 denoted by i ', is approximately twice tne incident impulse/unit area, whence 



2 X A 



(X - 1) 77 a 

 From equations" (39.) and (to) we thus have 



^ («0) 



3 (x - l) i, - A 



-1 — & («1) 



The maximum value of this expression is also tabulated in Table 1 and we see that for close 

 small explosions it is small rising to order 17i for distance short pulses. 



The present analysis is based on tne assumption that tne surface of the sphere is of itself 

 rigid. Any yielding of the front surface will, however, tend to lessen the bodily motion and where 

 conclusions for any unyielding sphere indicate little effect of such motion tney are not likely to 

 be invalidated Dy yielding. 



We can therefore still conclude, in the two cases of (i) large explosion (long pulse) at 

 large distance and (ii) small explosion (short pulse) at near distance, that fixing has little effect 

 on the impulse acting locally on tne surface, this conclusion being limited in the second case to 

 the part of the sphere near the explosion where in practice tne major damage will generally occur. 



The preceding conclusions regarding little effect of fixing on the magnitude of locally 

 applied impulse can probably in most cases be interpreted as applying also to magnitude of resultant 

 damage since this agrees with such experimental data on fixing as are available. 



For any particular example it is probably best to check such interpretation by using the 

 present analysis to estimate the maximum bodily velocity given to an equivalent buoyant sphere and 

 compare with the maximum velocity given to local areas of the surface on assumptions such as that 

 of the report "The pressure and impulse of suOmarine explosion waves on plates". A numerical 

 example of such comparison is given in Appendix B. 



Appendix A, 



