365 



Now we wish to obtain an asymptotic egression for Eq, (26) suitable for 

 small values of t . To accomplish this, we make an asymptotic development 

 of the integrand of the integral in (26) valid for large co , We make use 

 of the fact that 



J^(^) - . c ^U-^ ) (27) 



for large u. • From this we immediately obtain 



Substituting (28) into (26) we obtain the result 



where t* = t - Ci-^:^ V^ ( >-7c) 



andh(t) = -L_ \ ^f\ a^ - \ At'\ At" PcCt") ^ t>.o^ 



= 0, t 6 . 



On accoimt of the incompressive approximation we must not take "t 

 too small (i.e. t 6 R«/<^,, where c^ is the velocity of sotind) in using Eq, 

 (28). On the other hand, the asymptotic development breaks down if we take 

 "t too large (i.e. t<^^Cl-v^)''* /o^o » which is the time required for a trsins- 

 verse plastic wave to traverse the radius of the loaded plate). 



The meaning of Eq. (29) is perhaps clearer if we rewrite it in the 

 form 



52 



