41 



55 



in which p^ represents the actual pressure p on the surface. At the same 

 time, elements in contact with the plate will be moving according to some 

 other equation such as Equation [l6]. The symbol [d^z/dt^] t -± in any equa- 

 tion may be taken to refer always to the acceleration of an element of the 

 liquid surface, whether free or in contact with the plate. 



The Impulse 



It is noteworthy that the total impulse on any point of the plate 

 should not be affected by the occurrence of cavitation. For the pressure on 

 the plate is always the_ same as that on the liquid surface, according to the 

 assumptions that have been made. Hence, from Equation [2U], the total im- 

 pulse per unit area on the plate due to the waves, up to a time at which the 

 plate has come to rest and all effects of diffraction have ceased, is 



/ = fip-Po)dt = 2/p. 



dt 



[61] 



where p^ is the pressure in the Incident wave. The integral of the left-hand 

 member of Equation [24] ,with respect to the time vanishes in the end, since 

 dz/dt begins and ends at zero. The intervention of cavitation has no effect 

 upon Equation [6l ] . 



A Proportionally Constrained Plate 



The problem becomes much simplified and can be treated completely 

 if the very arbitrary mathematical assumption is made that both plate and 

 liquid surface move proportionally and in the same manner, so that their dis- 

 placements are both represented by equations of the type of Equation [28] but 

 with different values of Zc{t) during the cavitation phase. Cavitation then 

 appears and disappears simultaneously at all points of the plate. Successive 

 phases of such motion are illustrated in Figure 23. 



Figure 23 - Illustration of Cavitation 



According to the Assimption of 



Proportional Constraint 



The left-hand figure shows the initially flat dia- 

 phragm; in the middle, cavitation has occurred, but 

 both diaphragm and liquid surface are assumed to be 

 deformed in the same proportional manner; right, the 

 cavitation has disappeared simultaneously over the 

 entire diaphragm. 



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