4U SURFACE AND OTHER EFFECTS 



at a fixed surface and the term poCo dzc/dt the rarefaction resulting from 

 its motion. As time increases this pressure is modified by the arrival of 

 diffracted waves from adjoining points on the plate and baffle, these 

 being represented by the last two terms of Eq. (10.17), which act to 

 reduce the pressure gradient. 



,A second limiting case is that in which the acceleration of the plate 

 changes slowly enough that the propagation time can be neglected. 

 This corresponds to evaluating d%/dt^ in the integral preceding Eq. 

 (10.17) at the same time r for all points, as if the motion and pressure 

 at all points were the same as in noncompressive flow. In this limit, 

 Eq. (10.17) becomes 



(10.19) Pit) =2P,(0-^S 



3 dt^ 



The second term on the right has the form of an inertial reaction of the 

 accelerated flow near the plate, which in the equation of motion for 

 plate and water becomes equivalent to an added mass (2poa/3) per unit 

 area (at the center of the plate) . The load on the plate thus approaches 

 a doubling of pressure less the inertia of noncompressive flow. This 

 transition is a simple example of what Kennard has called the reduction 

 principle, by which "the action of a wave tends continually to change 

 into or reduce to the type of action that is characteristic of incompres- 

 sible liquid." Considerations of this kind are often useful in simpli- 

 fied analysis of the pressure distribution, but it is important to observe 

 that this limiting behavior is brought about by compressive equalization 

 of pressure differences and the limiting case results only if the time for 

 this to occur can be neglected. The noncompressive approximation 

 has been treated by Butterworth (15), and by Kennard (57) in a com- 

 prehensive discussion of the effects of pressure waves on diaphragms. 



10.6. Motion of a Circular Plate 



A. The equation of motion for a plate under plastic tension. If a de- 

 forming plate is assumed to have geometrically similar profiles at all 

 times, only the equation of motion for an element of the plate need be 

 solved, as this solution determines the rest of the profile. As an illus- 

 tration, consider the assumed parabolic deformation of a circular plate, 

 subject to a pressure P{t) at its center, negligible restraint by the 

 medium behind it, and a stress (Jn{t) normal to the surface as a result of 

 its deformation. If m is the mass of unit area of the plate, the equation 

 of motion for lyiit area at the center of the plate is 



