18 



p> 



\ 



/ 

 / 



i 



When the shock wave arrives, there- 

 fore, each element of the plate will 

 start moving as if It were; part of 

 an Infinite plate acted upon by a 

 plane wave of Infinite lateral ex- 

 tent, as Is suggested by Figure 3, 

 and for a short time the simple the- 

 ory of the one-dimensional case will 

 be applicable. 



At first the Increment of 

 pressure p, due to the incident wave 

 is doubled by reflection; then, as 

 the plate accelerates, a relief ef- 

 fect occurs and the pressure rapidly 

 falls. 

 Let it be assumed that hydrostatic pressure on the face of the 

 plate is balanced by an equal pressure on its back surface. Then the approx- 

 imate equation of motion for each element of the plate during the Initial 

 phase is 



Figure 3 - Illustration of a Plane Wave 



Incident Normally upon a Plane Plate 



of Infinite Lateral Extent 



p is the incident wave pressure, m is the mass per 

 unit area of the plate, aiJd i is the displacement of 

 the plate. The pressure p on the plate is the dif- 

 ference between the incident pressure, doubled by 

 reflection, and a relief term proportional to the 

 velocity t of the plate. 



m 



d(2 



= 2p, - PC 



dz_ 

 dt 



[1] 



where z is the displacement of the element in a direction perpendicular to 

 the face of the plate, p is the density of water in dynamical units, c the 

 speed of sound in it, and their product is the specific Impedance of the wa- 

 ter. The Incident pressure Pj is a function of the time t. See Equation 

 [107] In the Appendix, in which 4> is here equal to 0. The right-hand member 

 of Equation [1] represents the load pressure on the plate; the term in dz/dt 

 represents the relief effect due to the motion of the plate. 



The A-phase of the primary pulse can be represented approximately 

 by 



-at 



Pi = Pr 



[2] 



where p„ and a are constants and the time t Is measured from the Instant of 

 onset of the wave. If pj varies in this manner and the plate starts with z 

 0, dz/dt = at time f = 0, it is found from Equation [T] that 



dz_ 

 dt 



^Pr 



pc —am 

 so that the load pressure on the plate Is 



\e e m j 



.3] 



