Impulsively Generated Waves Propagating into Shallow Water 



years Kranzer and Keller [ 1959] , Kajiura [ 1963] , Whalin [ 1965a] , 

 and Whalin [ 1965b] have made significant contributions on the sub- 

 ject. The last cited is an extension of the Kajiura work and appears 

 to be the most general treatment on the subject. 



The theory as presented by Whalin relates an initial distri- 

 bution of impulse, surface velocities, and surface deformations to 

 the waves produced at some distance from the source in water of 

 finite but uniform depth. To a certain extent the choice of a source 

 model is arbitrary in that several models may give an adequate fit 

 to experimental data with each having its own particular advantages 

 and disadvantages. No physical reality is assigned to the source 

 model in terms of the dynamics of the explosion; fortunately, how- 

 ever, the dimensions of the source model have been found scalable 

 in terms of explosive yield so that useful predictions can be made. 



A source model that has been found to give good agreement 

 with experiment is a paraboloidal cavity given by: 



nrr) = 2d„[(^/ -i], ;^r„ (1) 



.2 







r > r. 



The collapse of this cavity at time t = generates the wave system. 

 In cases of large explosions where the dimensions of the cavity are 

 not small compared to the water depth, this model yields a poor 

 prediction and a different source model, perhaps utilizing an initial 

 time dependency should be employed. 



According to Whalin the surface elevation, (r,t), for an 

 axially symmetric surface deformation is: 



Ti(r,t) = - \ Ti"(or) cr cos (flit) jQ(ar) do- (2) 



where T|((r) is the Hankel transform of the initial surface deforma- 

 tion r|(r): 



^(cr) = \ ^(r)Jo(o-r) r dr. (3) 



All unprimed quantities in these equations have been non-dimensional- 

 ized using the water depth, h. Primed variables indicate the cor- 

 responding dimensional quantities. 



r^jjdp = dimensionless radius and height of initial surface 

 deformation = r^j'/^' <^o'/h 



243 



