BAES 127 



Combining the results for the two classes it is found that 

 wZ/7T = 1-50562, 2*49975, 3'50001, ..., ...(19) 

 where the values for the symmetric and asymmetric types 

 alternate. The subsequent numbers are adequately represented 

 by s + |. The fact that the frequencies are approximately 

 proportional to 3 2 , 5 a , 7 2 , ... was ascertained, from observation 

 alone, by Chladni*. 



To examine the form assumed by the bar in any normal 

 mode, we require the ratio of the arbitrary constants, as deter- 

 mined by (6) or (13). Thus in the case of symmetry we have 

 rj = C (cos | ml cosh mx + cosh ml cos mx) cos (nt + e), (20) 

 where m is a root of (7). The curve may be traced with the 

 help of a table of hyperbolic functions, and the positions of the 

 nodes found by interpolation. The form assumed in the 

 gravest mode is shewn in Fig. 45. The nodes here are at a 

 distance of '224 of the length from the ends. 



Fig. 45. 



The corresponding formula for the asymmetric modes is 

 77 = G (sin J ml sinh mx + sinh ml sin mx) cos (nt 4- e), (21) 

 where m is determined by (14). 



47. Clamped-free Bar. 



The next most interesting case is that of a bar clamped at 

 one end and free at the other. Here also there is an advantage 

 in taking the origin at the middle point of the length )-. The 

 terminal conditions then are 



^=0, v = o [* = -jq (i) 



* E. F. F. Chladni, born at Wittenberg 1756, died at Breslau 1827. 

 Distinguished by his experimental researches in acoustics. These are recorded 

 in his book Die Akustik, Leipzig, 1802. 



t Greenhill, 1. c. 



