150 DYNAMICAL THEORY OF SOUND 



When a force Z per unit area acts on a circular membrane, 

 the equation (1) is replaced by 



it being supposed, for simplicity, that there is symmetry as 

 regards the distribution of Z and the consequent displacements 

 If, further, Z vary as cos (pt + a), we have 



Z 



(23) 



If Z be independent of r, so that the impressed force is 

 uniform over the membrane, the solution of (22) is obviously 



t=-lp + CJ t (kr), ............... (24) 



and determining the constant C so that f =0 for r = a, we find 



(25) 



(ka) 



The amplitude becomes very great whenever fca approximates 

 to a root of (5), i.e. whenever the imposed frequency approaches 

 that of one of the symmetrical free modes. When, on the 

 other hand, the imposed vibration is relatively slow, ka is 

 small, and we have by (4) 



(26) 



approximately. This is the statical deflection corresponding to 

 the instantaneous value of the disturbing force. 



55. Uniform Flexure of a Plate. 



The theory of the transverse vibrations of plates stands in 

 the same relation to that of bars as the theory of membranes 

 to that of strings. The reader will understand from this com- 

 parison that the mathematical difficulties are considerable, arid 

 will not be surprised to learn that some of the most interesting 

 and, at first sight, simple problems remain unsolved. On the 

 other hand the subject readily admits of experimental illustra- 

 tion. If the plate be horizontal, and fixed at one point, the 



