SURFACE AND OTHER EFFECTS U3 



maintained, as a more exact solution obviously determines what the 

 profile must be. This question has been examined by Kirkwood, who 

 finds that the initial phases of deformation are more nearly conical in 

 form, and a change in shape occurs with time of deformation and time 

 variation of incident pressure, as is also found experimentally. The 

 paraboloid form is, however, fairly typical, and we assume for sim- 

 plicity that the deflection z{r, t) is given by 



z{r,t) = Zc{t) (1 - rVa2) ^ > 

 = ^ < 



where the central deflection zjt) depends only on time and the edge is 

 fixed {z{a, t) = 0). Substituting this expression for z in Eq. (10.15) 

 gives for the pressure at the center of the plate 



P(0=2P.(0-W2.)j"(f)^(l-5).. 



o 



since Un{r, t) = dz/dt. Changing the variable of integration to 

 T = t — r/co and integration by parts gives finally Kirkwood's result 

 that 



(10.17) 



where the diffraction time dd = a/co is the time for a wave to be propa- 

 gated from the edge of the plate to its center. The pressure for other 

 points on the plate can of course be similarly evaluated, but involves 

 greater mathematical complications which is unnecessary for the present 

 purpose. 



The significance of Eq. (10.17) for the pressure at the center of a 

 plate is more clearly brought out by considering the limiting cases to 

 which it reduces under special conditions. In the initial phases of the 

 motion for which t < dd, we have zdt — dd) = and the integral is 

 small because Zc{t) = for r < 0. The pressure is then very nearly 



(10.18) Pit) = 2P,{t) - PoCo^ 



at , 



This, however, is simply the pressure in front of an infinite plate moving 

 with velocity dzc/dt, the term 2Pi(t) representing doubling of pressure 



