MEMBRANES AND PLATES 151 



configuration of the nodal lines can be exhibited by means 



of a little sand previously strewn on the surface. When any 



particular normal mode is excited, the sand is shaken away 



from the places of vigorous motion, and accumulates in the 



neighbourhood of the nodal lines. Usually the plate is set 



into vibration by bowing at right angles to the edge, and the 



desired mode is favoured by touching the edge with the fingers 



at one or more nodal points. If, as in the case of a rectangular 



plate fixed at the centre, the point of support is a nodal point 



>f several normal modes, a great variety of beautiful figures 



nay be obtained. An extensive series of diagrams of results 



obtained in this way were given by Chladni; many of these are 



eproduced in the current manuals of experimental acoustics. 



In the theoretical treatment it is assumed that one of the 

 principal axes of strain and stress is normal to the faces of the 

 plate, and that the corresponding stress vanishes. Putting, 

 then, p s = in the formulae (9) of 42, we find, for the 

 remaining principal stresses, 



)> p> = E'( > + a ei ) t ......... (1) 



where E f = E/(l - <r 2 ) ...................... (2) 



If R l , R 2 be the principal radii of curvature at any point of the 

 plate, when bent, we have, by an investigation similar to that 

 of 45, 



C^Z/R!, e 2 = z/R 2 , .................. (3) 



where z denotes distance from the medial plane. If we consider 

 a rectangular element of the plate bounded by lines of curvature, 

 and denote by h the half-thickness, this leads to bending 

 moments 



per unit length of the respective edges, in the planes of the two 

 principal curvatures. 



The potential energy per unit volume is 



ef) .......... (5) 



