67 ■ 81 



THE MOTION OF A PLATE, DIAPHRAGM OR LIQUID SURFACE 



Suppose that, In the case Just considered, the target consists of a 

 plate or diaphragm, initially plane. Then its equation of motion will be 



mi = p + 4> — Pf, [ T 01 ] 



where m is its mass per unit area, and stands for the difference between 

 the hydrostatic pressure Pp on the front face and the pressure p^ on the back 

 face, plus the net force per unit area due to stresses, if any. Or, by [TOO], 



TOz = 2p, + </. - -^ f- z, A dS [102] 



The displacement is assumed here to remain small enough so that its component 

 parallel to the Initial plane can be ignored. Equation [102] is an integro- 

 dlfferential equation for z, which is a function both of time and of position 

 on the plate. 



The "target" may actually consist wholly or in part of the free 

 surface of the liquid, for nothing in the calculation of the pressure rests 

 upon the assumption of a solid target. At any point where the surface is 

 free, or, for that matter, at any other point as well, z will represent the 

 normal displacement of the liquid surface. 



At a point on the free surface, [100] may conveniently be written 



_p 

 2 



'-f^z,_^dS = 2p, + po-p [103: 



where p is the external pressure on the surface. The integral extends as 

 usual over the entire plane. This equation, when needed, can be formed from 

 [102] by setting m = and 4> = p^ - p . Here p^ Includes atmospheric pressure 

 and may differ from p because of an accelerational pressure gradient in the 

 liquid. At any point where the surface of the liquid is in contact with a 

 plate or diaphragm, [102] will continue to hold. 



THE CASE OF PLANE WAVES 



If the plate remains accurately plane, and if p, is uniform over it, 

 then z is also uniform and hence is a function of ( only. Thus in the inte- 

 gral in [102] the quantity z , j. is a function of t and s only. Hence in 



c 



this integral dS may be replaced by Znsds, representing a ring-shaped element 

 of area on the plane, and 



f— z, , dS = 27rrz, , di 



Now a dot over z ,_j.is equivalent to differentiation with respect to the 

 argument ( - s/c, so that 



