29 43 



revealed by analysis will find their counterpart in the behavior of actual 

 structures. 



The Non-Compresslve Case 



For a proportionally constrained plate, in a rigid plane baffle, a 

 definite treatment can be given of the non-compressive case that was dis- 

 cussed previously in general terms. In the Appendix, Equations [126] and 

 [127], the following statement is proved: 



At any time when the acceleration has been sensibly uniform, at 

 least during the immediately preceding interval of length D/c, where D is the 

 maximum diameter of the plate. Equation [29] reduces temporarily to the ordi- 

 nary differential equation, 



(M + M,)|^ = 2Fi + * [35] 



where 



M, = ^ff(x,y)dsjf{x;y')^ [36] 



Here Af, may be regarded as the effective mass of the liquid that 

 is following the plate; it represents the same loading of the plate by the 

 liquid that would occur if the liquid were incompressible. The kinetic en- 

 ergy of the liquid that follows the plate is M,(dZj/dt)^/2 ; see the Appendix. 

 Thus, when the acceleration varies sufficiently slowly, the release effect 

 produces the loading by the liquid as calculated from non-compressive theory. 



An analogous result for an unconstrained plate is difficult to ob- 

 tain, but it may be inferred that even in this case there will be some degree 

 of approach to the motion as calculated for incompressible liquid whenever 

 the acceleration of the plate satisfies the condition Just stated. A rough 

 estimate of the accelerations to be expected in such cases can probably be 

 made by assuming some plausible type of proportional constraint and using 

 Equations [35] and [36]. 



Some Simple Types of Proportional Constraint 



Several forms of proportionally constrained motion were, in effect, 

 treated by Butterworth (1 ). His formulas do not contain the factor 2 that 

 arises from the reflection of the wave, and the retardation in time is omitted 

 after a brief mention of it; hence his results are in reality those that would 

 be produced in incompressible water by a pulse of pressure having the same 

 form as the incident wave. 



