44 
to the hull For ships, this effect is probably small, but for small-scale 
targets, which usually only simulate a typical part of the ship's structure, 
the bodily motion will be proportionately larger and it is of some 
importance to estimate whether it is likely to affect the damage. 
415. The one theoretical problem for which the bodily motion due to a 
small-amplitude pulse has been completely solved is that of a rigid sphere 
and then only on the assumption (usually justified) that the motion is 
small canpsred with the radius of the sphere. The analysis cannot easily 
be condensed and attention will therefore be concentrated on the general 
conclusionse The bodily motion of the sphere depends on the twin 
processes of reflection and diffraction, the former tending to increase the 
motion whilst diffraction of the pulse round to the back of the sphere tends 
to decrease the bodily motion If a is the radius of the sphere, x is 
the distance of the explosion from the centre of the sphere and c/n is the 
characteristic length of the pulse (assumed exponential), the qualitative 
results depend on the ratios of these three lengths. The terms long 
pulse and short pulse will be used to denote the two extreme cases where 
c/n is large or small, respectively, compared with the radius a of the 
sphere. For long pulses the analysis and conclusions are restricted to 
the case where x/a is also large since otherwise the assumption that the 
pulse is of small amplitude will be violated. Since long pulses correspond 
to large explosions for most practical targets, this condition that x/a is 
large will usually be satisfied if catastrophic damage is to be avoided, 
For short pulses, the theory is not restricted to large values of x/a and 
in practical cases if the damage is not to be negligible the relevant 
values of x will usually be of order 2a 
416. The main conclusion from the analysis for a rigid sphere is that the 
bodily motion of a target is unlikely to affect the damage appreciably in 
the cases of either:- 
(4) a long pulse from a distant large explosion or 
(2) a short pulse from a small explosion fairly near the target, 
for example, x = 1-25 a 
417. For a long plane pulse from a distant large explosion diffraction 
round the sphere (target) will tend to equalise the pressure on the front 
and back and this equalisation will be hastened by the actual bodily motior 
as the sphere moves away from the explosion, Measuring time fron the 
instant at which a plane pulse first strikes the nearest point of the 
sphere, the diffracted pressure will reach the furthest point on the back 
of the sphere in time t = (1+1¥2)a/c = 2.57a/c and the analysis in fact 
indicates that equalisation is virtually complete by this time. 
Subsequent to this time the residual resultant force on the sphere as a 
whole is small and negligible. For long pulses, this time of about 25 a/c 
is small compared with the characteristic time V/n of the pulse and in 
effect only a small initial portion of the pulse produces any appreciable 
bodily motion, In contrast, the effect of the equalisation of pressure 
is that all elements of the surface are subjected to a pressure approximating 
to that in the incident pulse and it is on this relatively long duration 
pressure that the damage to the target will depend. In general, therefore, 
the bodily motion has little effect on the damage due to a long pulse fron 
a large distant explosion. 
418 For a short pulse from a relatively near small explosion the same 
conclusion is reached on somewhat different grounds, Here the most 
relevant fact is that the mass of the target is large in relation to the 
size of charge and only a small velocity of the target can be produced. 
In contrast, most of the effect of the pressure pulse will be concentrated 
on a small portion of the target nearest the explosion and quite large 
local velocities can be produced, For example, it was estimated that for 
