CHAPTER V 



MEMBRANES AND PLATES 



52. Equation of Motion of a Membrane. Energy. 



The vibrations of membranes are not very important in 

 themselves, and the conditions assumed for the sake of mathe- 

 matical simplicity are, moreover, not easily realized experi- 

 mentally. The theory is however, for a two-dimensional system, 

 comparatively simple, and the results help us to understand in 

 a general way the character of the normal modes in other cases 

 where the difficulties of calculation are much greater, and 

 indeed often insuperable. 



The ideal membrane of theory is a material surface such that 

 the stress across any line-element drawn on it is always in the 

 tangent plane. We shall consider only cases where the surface 

 in its undisturbed state is plane, and is in a state of uniform, or 

 "homogeneous," stress; i.e. it is assumed that the stresses across 

 any two parallel and equal lines are the same in direction and 

 magnitude. We further suppose, for simplicity, that the stress 

 across any line-element is perpendicular to that element. It 

 follows, exactly as in hydrostatics, from a consideration of the 

 forces acting on the contour of a triangular area, that the stress 

 (per unit length) is the same for all directions of a line-element. 

 This uniform stress is called the " tension " of the membrane ; 

 we denote it by P. Its dimensions are those of a force divided 

 by a line, or [MT~*]. 



We take rectangular axes of x, y in the plane of the 

 undisturbed membrane, and denote by f the displacement 

 normal to this plane. The surface-density (i.e. the mass per 



