410 SURFACE AND OTHER EFFECTS 



For spherical deformation as in Fig. 10.5b the increase A^ for a small 

 deflection Zc is given by A A = wzj^ and the plastic work is 



Wp = iraohzc^ 



The energy absorbed by the plate thus varies as the square of its central 

 displacement. 



An analysis based on the idealization of a uniform plastic tension is 

 clearly very approximate, and neglects such effects as anisotropic 

 stresses, resistance of the plate to local bending, and tangential propa- 

 gation of plastic effects under dynamic loading.^ The stress and energy 

 relations do, however, provide a rough description of the plastic resist- 

 ance to deformation, and together with a formulation of the pressure in 

 the adjacent water permit at least a crude analysis of the resulting 

 motion. 



B. Diffraction effects at a target. The pressure in the water at a 

 deformable surface must differ from that in the incident wave in such a 

 way that the normal component of particle velocity is the same as the 

 corresponding velocity of the plate. If the incident pressure is a plane 

 wave and the target is an infinite one, all parts of which undergo the 

 same motion, this requirement is satisfied in the linear (acoustic) ap- 

 proximation by superimposing reflected plane waves. If, however, the 

 target is of finite extent, the pressure and velocity must vary over its 

 face in such a way as to be compatible with the velocity and stress in 

 the plate. The latter must be different, for example, at the center and 

 at the supported edge of a plate, and some system of waves other than 

 plane waves of infinite extent must be necessary. The mathematical 

 analysis of such cases, originally due to Kirchhoff, is described by the 

 physical concept of diffracted spherical waves originating at points on 

 the boundary, which combined with the incident wave satisfy the 

 boundary conditions and hydrodynamical equations. A linear super- 

 position of this kind is only possible if the hydrodynamical equations are 

 linear, and the analysis to be described therefore applies exactly only to 

 waves of acoustic intensity. The presentation given here is adapted 

 from Kirkwood's discussion.^ 



In the linear approximation, wave propagation in the water is char- 

 acterized by the velocity potential <^, from which the excess pressure 

 and particle velocity are obtained by 



u = - grad (p, P - Po = Po-~ 



ot 



^ A discussion of such effects in circular diaphragms is given by Hudson (48). 

 ^ Reference (61), see also the more general discussion by Lamb (65), pp. 49Sff. 



