152 DYNAMICAL THEOBY OF SOUND 



If we substitute from (3), and integrate over the thickness, we 

 find for the potential energy per unit area of the plate 



The formulae (4) may be applied to the case of a flat bar of 

 rectangular section, uniformly bent by two opposing couples 

 MJ), where b denotes the breadth. Along the free edges we 

 have M 2 = 0, and therefore 



JZr' = -rJZr' (7) 



The bending moment is accordingly 



M 1 b=%Ebh 3 /R l) (8) 



by (4). This agrees, as it must, with 45 (5), since o> = 2bh, 

 K * == *h?. The formula (7) shews that when a bar of rectangular 

 section is bent in a plane parallel to one pair of faces, an opposite 

 or " anticlastic " cur- 

 vature is produced in 

 the plane of the cross- 

 section, the ratio of 

 the curvatures being 

 identical with Pois- 

 son's ratio &. This 

 circumstance has 

 been made the basis 



of practical methods Fig. 54. 



of determining <r, by 



Cornu* (1869) and Mallock (1879), the curvatures being 

 measured by optical or other means. 



It follows from the above that a perfectly free rectangular 

 plate cannot vibrate after the manner of a bar, with nodal lines 

 parallel to one pair of opposite edges, since couples would be 

 required, about the remaining edges, to counteract the tendency 

 to anticlastic curvature. 



56. Vibrations of a Plate. General Results. 

 In a vibrating plate the directions and amounts of the 

 principal curvatures will in general vary from point to point. 



* A. M. Cornu (1841 1902), professor of physics at the Ecole Polytechnique 

 1871 1902. Famous for his experimental determination of the velocity of light, 

 and for other important contributions to optics. 



