218 



APPEHDIX C 



Calculations for a buoyant sphere 



Numerical calcjlations of velocity have- been carriea out for the case of a sphere which 

 is just buoyant, i.e. of mean density equal to the density of the water. 



From equation (2l), k = 3/2, m = / 3/2 for a buoyant sphere and insert iny these values in 

 equation (25) the non-dimensional ratio V/v can be calculated as a function of the non-dimensional 

 time T for jiven pairs of values of the parameters q and qX. If we use the term "pulse-length" 

 as applied to an exponential pulse to mean the distance a/q the pulse has travelled while the 

 pressure at any point decays in ntio l/e, then the parameter q is the ratio of sphere radius to 

 "pulse-lenjth". Similarly qX is the ratio of distance Xa; between pulse centre and sphere centre, 

 to the " pulse-lenjth". 



The jsneral shape of the velocity-time curves is similar for all cases calculated and a 

 typical set is shown in Fijure i corresponding to the limiting caseqX-'CO, i.e. a plane incident 

 pulse. 



Denoting the maximum bodily velocity by \/_, values of the ratio V /v are qiven in Table 2, 



'm mo 



and in brackets in the same table we give values of T at which these maximum velocities occur. 



Thence the mean acceleration up to maximum velocity can be evaluated if desired. 



TABLE 2 . 



Values of Vj v^ and (in brackets) of, T ^t 

 time of Tiixtmum velocity , 



Intervals in T of 0.1 for <■ T $ 1 and 0.2 for 1 < T ? 3 were used and the values of V /v 

 in Table 2 are correct to an error of one or two \i\ the third digit whilst tne values of T at 

 ^ " ^m ^""^ correct to about one quarter of an interval, i.e. to 0.05 for T > i and 0.025 for T < 1, 

 Further sub-Jivision of interval would nave been necessary to obtain reasonably accurate values of 

 maximum acceleration from the velocity calculations and this was not attempted. Grapnical 

 estimates were, however, obtained from the maximum slopes of curves such as those in Figure 1 and 

 these indicated that the- naximum acceleration varioa b,^tween about 1.6 to 2.1 times the me.in 

 acceleration up to time of iraximum v.-locity. Approximate estimates of maximum acceleration, correct 

 to order 201 or lees, can therefore be obtained by using Table 2 to derive mean acceleration and 

 then multiplying by a factor 1.8. Accurate r-stimates for any particular case are best obtained 

 by use of equation (23) wnich gives the nat force on sphere at any time. 



For short pulses q-ob, the ratio ^J^^^ o but the product qV /v remains finite and is 

 given in Table 3, the values in which are correct to the third decimal place. 



Tdble 3 



