154 



DYNAMICAL THEORY OF SOUND 



and '842a, and so on, the numbers varying slightly however 

 with the value adopted for or. According to Poisson, the 

 values of m for the above modes are given by 



ra 2 a 2 = 8-8897, 38-36, (2) 



on the hypothesis that a- = J. 



The complete theory of the free circular plate was worked 

 out by Kirchhoff in a celebrated memoir (1850). It appears 

 that the gravest of all the normal modes has two nodal 

 diameters, and no nodal circle. Its frequency is 



5234. /(- 



according as we adopt the value cr \ or a = % of Poisson's 

 ratio. The figure shews the 

 configuration of the nodal lines 

 in the simplest cases of one and 

 two nodal diameters. 



The theory of a circular 

 plate clamped at the edge has 

 been treated by Poisson and 

 others. In the first two symmetrical modes it is found that 



ra 2 a 2 = 10-2156, 39'59, (3) 



respectively. In the second of these modes there is a nodal 

 circle of radius '38 la. The theory has been applied by Lord 

 Rayleigh to calculate the natural frequencies of a telephone 

 plate. 



Fig. 56. 



Fig. 57. 



In the case of a square plate we have to depend almost 

 wholly on observation, there being at present no exact theory. 

 As in the case of the square membrane ( 53), the nodal lines 



