MEMBEANES AND PLATES 153 



Shearing forces will also be called into play normal to the plane 

 of the plate. The circumstances are somewhat complicated, 

 but the deduction of the equation of motion for the body of the 

 plate is a straightforward matter, and presents no real difficulty. 

 A more serious question arises when we come to the conditions 

 to be satisfied at a free edge. It appears that the simple 

 condition of strain which has been postulated as the basis of 

 the formulae (4) of 55 cannot be assumed to hold, even 

 approximately, right up to the edge. In the immediate neigh- 

 bourhood of the edge, i.e. to a distance inwards comparable 

 with the thickness, a peculiar state of strain in general exists, 

 one remarkable result of which is a shearing force on sections 

 perpendicular to the edge, of quite abnormal amount. 



For the further development of the subject reference must 

 be made to other works*. We merely quote a few of the more 

 important results which have been obtained, relating chiefly to 

 plates whose edges are free. 



It is found that for a plate of given lateral dimensions the 

 frequency (n/2?r) of any particular normal mode is given by 



*-'.#.*, (1) 



where, as in 46, m is a constant, of the nature of the reciprocal 

 of a line, given by a certain transcendental equation, and p 

 denotes the volume-density. For plates with geometrically 

 similar boundaries the frequency accordingly varies as the 

 thickness, and inversely as the square of the lateral dimensions. 

 In the case of a perfectly free circular disk the nodal lines 

 are circles and equidistant 

 diameters. In the symmetrical 

 modes, which were investigated 

 to some extent by Poisson 

 (1829), we have nodal circles 

 alone. Thus in the gravest 

 mode of this type we have a 



nodal circle of radius '678a, where a is the radius of the disk ; 

 in the next mode there are two nodal circles of radii '39 2a 



* See Lord Kayleigh. Theory of Sound, chap. 10 ; Love, Theory of Elasticity, 

 Cambridge, 1906, chap. 22. 



