404 



is a constant fraction of the energy incident in the solid angle sub- 

 tended by that element. It is also assvmed that in passing from the 

 explosive to a plate element the energy flux density decreases as the 

 inverse square of the distance. £/ Using this assumption and also taking 



2/ The fact that the energy-distance curve in air explosions is ap- 

 proximately inverse square for several charge radii suggests the 

 possibility of an analogous situation existing in underwater ex- 

 plosions. Since the central strain depends critically only on 

 the plate profile near the center (r<2R, say) it would seem 

 that the assvunption of inverse-square attenuation of the energy 

 might be a good approximation. 



account of the oblique angle of incidence on off-center elements, one 

 obtains a surface density of kinetic energy in the plate proportional 

 to (R + r )"-'/, where r is the radius and R is the distance between 

 the charge center and the middle surface of the plate. Since the veloc- 

 ity of sound in water is much greater than the velocity of transverse 

 plastic waves in the plate, it is assumed that the energy can be treated 

 as though it were delivered to all parts of the plate at the same 

 instant. Consequently, londer these assumptions the initial velocity 

 distribution is given by 



ILC,,), c.i^^',,,), ^- 



(53) 



where x = r/R, y = z/R, and t = ct/R, the quantities z and t being the 

 vertical displaceinent and the time, respectively. The quantity c is, 

 as before, the velocity of transverse plastic waves in the plate. The 

 expansion parameter OC will be assigned a value on the basis of the 

 amount of energy delivered to the plate. Equating the fraction J[oS the 



85 



