BAES 129 



The frequencies of the whole series of normal modes, after the 

 first, are approximately proportional to 3 2 , 5 2 , 7 2 , . . . , as found 

 experimentally by Chladni. The accurate solution gives, to 



five places, 



ml/7r = -59686, V49418, 2-50025, (17) 



In the modes which follow the first we have respectively one, 

 two, three, ... internal nodes. The annexed figure shews the 

 gravest mode. 



Fig. 46. 



Other problems, which are however of less interest, may be 

 obtained by varying the terminal conditions. We will only 

 notice the case where both ends are " supported," i.e. fixed in 

 position but free from terminal couples. The conditions then 

 are, by 45 (14), 



77 = 0, 7/"=o |>=iq (is) 



In the symmetrical class we have 



77 = G cos mx . cos (nt -f e), (19) 



with cos ^ml = 0, whence 



mll7r = l, 3, 5, (20) 



In the asymmetric class 



77 = G sin mx . cos (nt + e), (21) 



with raZ/7r = 2, 4, 6, (22) 



The frequencies are, by 46 (2), proportional to the values of 

 m 2 , and so to the squares of the natural numbers. 



The foundations of the theory of the transverse vibrations 



were laid by D. Bernoulli (1735) and Euler (1740). The 



latter also gave the numerical solution of the period equation 



in a few of the more important cases. In more recent times 



L. 9 



