Explosion-Generated Water Waves 



The first peak is called upper critical depth (u. c.d.); the 

 second one is the lower critical depth (l.c.d,). While there is so 

 far no adequate theoretical explanation for the u.c.d. , the l.c.d, 

 is clearly analogous to the influence of burst depth on crater dimen- 

 sions in solid materials, and is related to the balance between ex- 

 plosion energy going into cratering and that vented to the atmosphere. 



Another interesting feature which has been experimentall ob- 

 served is a change of phase between corresponding waves of trains 

 generated by explosions above and below the upper critical depth. 

 Such a change, in fact, is predicted between theoretical models of 

 wave trains generated by an initial impulse acting on the surface and 

 an initial surface elevation, respectively, suggesting that the impulse 

 model may be more appropriate for explosions above the upper criti- 

 cal depth. The u.c.d, is a rather puzzling aspect of explosive wave 

 generation. Abundant experimental data with HE charges within the 

 range 0.5 - 300 lbs exhibit a large scatter under presumably identi- 

 cal conditions, ^Imax varying between 0,5-2 times that at the l.c.d. 

 Moreover, the scaled wave frequency at "Hmax ^^ uniformly higher, 

 indicating a smaller effective source radius. Lastly, the existence 

 of the u.c.d, is still somewhat in question for large explosions, 

 since several attempts to reproduce it with 10,000 lb HE charges 

 have been unsuccessful. It has been suggested (Kriebel [ 1968]) that 

 the upper critical depth effect is obtained from interference between 

 the direct incident shock wave and its reflected waves, resulting in 

 more effective containment and greater cavity expansion than from 

 deeper or shallower charges. As the detonation depth increases, 

 the pressure impulse on the free surface has less and less effect 

 on the cavity formation and ultimately becomes negligible. This 

 undoubtedly influences the shape of a theoretical cavity, which pro- 

 duces an equivalent system of water waves, since the dimension of 

 this cavity is closely related to both frequency and amplitude of the 

 first envelope. Nevertheless, it appears that the large data scatter 

 obtained under fixed experimental conditions at the upper critical 

 depth are largely due to Taylor instability of the collapsing cavity. 



The mathematical model described later can be adjusted to 

 produce practically any type of wave train desired, by assuming 

 various shapes for the initial cavity. 



II. INPUT CONDITION 



The theoretical formulation of an overall mathematical model 

 for simulating the time history resulting from an underwater detona- 

 tion is an extremely complicated task. However, keeping in mind 

 the main objective of our problem -- the generation process of water 

 waves -- many detailed phenomena, chemical or nuclear, can be ig- 

 nored, retaining only the kinematic and dynamic features. 



For exainple, one can consider only the following phases. 



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