12 239 



The equation of motion for the plate Is 



m j^ = p + p [9] 



where p Is the excess of pressure in the Incident wave above the hydrostatic 

 pressure p^, which is assumed to be the same on both sides of the 

 plate, 



p" is the excess of pressure over p^ in the reflected wave, 



m is the mass of the plate per unit area, 



X is the positional coordinate of the plate in a direction perpendicu- 

 lar to its surface, and 



t is the time. 



Elimination of p " gives 



d X . dx o I 1 nl 



where p is the density of water and c is the speed of sound in it. Compare 

 here Equations [9] and [10] on page 2k of TMB Report 480 (U). 

 The solution of Equation [10] for p = is 



dx '^^ 



jf=u,e - [11] 



With a suitable choice of the constant u^, this solution will represent the 

 motion of the plate after returning to contact with the water if ( represents 

 the time measured from the instant of contact and Wq is the velocity of the 

 plate at that instant. 



To represent the impact of the pressure wave, we set p = for 

 t less than and, for t greater than 0, 



p = p(f) = p^e-"' [12] 



in terms of two constants pg and a. It is assumed that the displacement of 

 the plate during the effective time of action of the wave is negligibly small. 

 The solution of Equation [10] that represents the plate as starting from rest 

 at I = and ( = Is then easily verified to be 



dt pc — am ^ ' i. -^ i 



see TMB Report 480, page 25. 



The corresponding total pressure on the plate above hydrostatic is, 

 from Equations [9] and [15], 



p + p = m — -o- = ^ [pee "" - ame 14 



^ '^ dt pc — am \ ^ I ^ ^ 



