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II. HYDRODYMMICS OF TIE LOAIING OF THE PUTE 

 BY AN UNDERWATER EXPLOSION WAVE 



1. General Theory 



In the last part we have treated the aucLally synmetric motion of 

 a thin plate under an arbitrary axLally symmetric pressure pC**^ t) . We 

 are ultimately interested in the behavior of an initially plane plate at- 

 tacked on the lovrer side by a plane underwater shock wave traveling in the 

 positive z-direction and backed by air (treated as vacuum here) on the upper 

 side. Here we seek the relation between the actual incident pressure p ir,-t) 

 and the free-field pressure p^ Ct} of the shock wave. 



Let this treatment be limited to the approximation corresponding 

 to coefficient of first power of oC , p * Cfj t) , in the oc-exj^ansion of 

 p(*", t^ (Eq. (16) of Part I). It is then appropriate that we use linear- 

 ized hydrodynamics and, in considering the instantaneous geometry of the 

 system, neglect the devintions of the shell from its initial plane configu- 

 ration. The added approximation of treriting the water as an inviscid fluid 

 allows us to use ordinary linear acoustical theory. In this theory we can 

 define a velocity potential "^ related to the pressure p and particle veloc- 

 ity 1? as follows, 



and satisfying the wave equation 



In the^ equations, o^ and c^ are the density in and sound velocity of water 

 at zero pressure. If the notation of Part I were followed, the quantities 



14 



