Le Ukhautk 



V = r/t. (16) 



The problem now is to choose '^(j^r^ In such a way, depending on 

 water depth and explosion characteristics, that Eq, (13) best fits 

 an observed wave train. 



3.3 Time- Dependent Free Surface Deformation 



Another possible mathematical model includes the kinetic 

 and potential energy transmitted to the water by both the atmospheric 

 overpressure and the gaseous expansion of the bubble. The initial 

 conditions are now time-dependent, and at least one additional 

 paraineter (time) is added to the Initial conditions. 



While Eq, (13) gives the total energy partitioned to water 

 waves as a function of detonation depth,- as well as the distribution of 

 energy amongst frequencies, the Introduction of time-dependence. 

 If properly used, permits a better fit to observations, not only for 

 the first maximum of the wave envelope "n^gj^ (and Its corresponding 

 wave number k»,Qw)» but also to the whole shape of the wave envelope. 



As an example of a time-dependent Input condition, consider 

 for example , 



ilo(ro.'r) = ^(i-q) sln-^ ~ (17) 



where t^ can be considered as the dlmenslonless period of first 

 expansion of the crater cavity from an explosion (Whalln [ 1965]), 



The Initial surface velocity at time t = is 



Zq = 



z = 

 T = t = 



d]Tp 

 dT 



TT 



"It' 



T=0 



%(ro) (18) 



and the resulting wave train Is given by 



ri(r,t)=jL^ - 3lM__ yjVW sln(\r-tVxtanhX), (19) 



' * r-y/tanh \ ^ ' k=X 



It is Interesting to note that Ti(r,t) Is Independent of real-time history 

 of the free surface deformation and depends only upon Its time deriva- 

 tive at time t = 0, 



80 



