25 39 



The resulting equation can be written in the form 



^SiTP^l.f '''.-'- -r t.^1 



In this form the eq.uation holds, indeed, quite generally, for any liquid sur- 

 face that is nearly plane and effectively unlimited in lateral extent, even 

 when the surface is partly or wholly in contact with a solid body. See Ap- 

 pendix, Equation [103]. The equation fixes the acceleration of the surface 

 at each point in terms of previous accelerations at all points and the vari- 

 ous pressures. 



Furthermore, with similar changes. Equation [l8] can be applied to 

 the motion of the liquid surface as exposed in a hole in a movable plane baf- 

 fle lying on the surface. 



It may be noted that the liquid surface does not exhibit the same 

 kind of resilience that is characteristic of ordinary elastic bodies. Thus a 

 rubber ball dropped onto the floor bounces back. If the surface of a liquid 

 similarly impinges upon a rigid obstacle, however, there is no rebound. Dur- 

 ing the impact the surface undergoes momentary negative accelerations of 

 large magnitude, and Equation [24] shows that these accelerations must be ac- 

 companied by a positive pressure acting on the surface, and also, therefore, 

 on the obstacle. However, on the assumption that only a limited part of the 

 surface was in motion, the integral in Equation [24] ultimately fades out 

 without changing sign, and the corresponding part of p must, therefore, do 

 the same. Since negative values of p do not occur, there is no tendency for 

 the liquid surface to leave the obstacle. 



Elastic rebound such as that of the rubber -ball is exhibited only 

 by bodies, solid or liquid, whose dimension perpendicular to the surface of 

 contact is effectively finite. 



IMPULSE PER UNIT AREA DUE TO THE WAVES 



Before considering solutions of the equations of motion, the fol- 

 lowing interesting conclusion concerning the impulse may be noted. 



Suppose that the plate, after having been at rest until a certain 

 instant, moves in any manner and then comes permanently to rest again. If 

 it is surrounded by a baffle that also moves, let the baffle likewise come 

 to rest. Let / denote the total impulse per unit area caused by the Incident 

 waves or jp^dt, where p„ is the excess of pressure above hydrostatic pressure 

 and the Integral extends over all time. Then, for a plate in a wide plane 

 baffle, it turns out that 



/ 



=2fp.dt [25] 



