590 Lateral Vibration of Bars treated simply. 



values are taken from Lord Rayleigh's calculations, but are 

 here given to three places of decimals only. The first column 

 shows the odd values of tt/2 for comparison sake. The other 

 columns have been calculated as for the free-free bar in 

 Table II. 



Nodes for Fixed- Free Bar. — The positions of the nodes for 

 a fixed-free bar may be found analytically as for the case 

 already cited, and have been calculated by Seebeck and by 

 Donkin. For the early overtones these are as follows : — 



Second tone, 0*2261 of length from free end. 



Third tone, 0*1321 and 0*4999 do- do. 



Fourth tone, O0944, 0'3558, and 0*6439 do. do. 



A near approach to a fixed-free bar is made in the behaviour 

 of one prong of a tuning-fork. On fixing the fork with its 

 prongs horizontally one over the other, touching in the neigh- 

 bourhood of the nodes and bowing the end of the prong, 

 the higher tones may be elicited and the nodes observed by 

 chalk or other powder. Operating in this way on a large 

 fork of 128 per second, whose length to the inside of the 

 hollow was 10} J inches, the following observations were 

 made. The second tone (or first overtone) showed a node 

 at 2| inches ; or, taking the length of the prong to be 

 11 inches to the fixed part, 0*216 of its length. This com- 

 pares well with the theoretical value 0*2261 . On placing two 

 fingers in the right positions as found by trial and bo win o- 

 suitably, the third tone was elicited. This showed nodes at 

 1*5 and 5*5 inches from the free end. Taking the fractions of 

 the length 11 inches, these give respectively 0*136 and 0*5, 

 which again are close to the theoretical values of 0*1321 

 and 0*4999. 



Remaining Cases of Vibrating Bars. — A bar vibrating 

 with one end free and the other supported is like half of a 

 free-free bar when vibrating with a node in the middle. 

 Again, the vibrations of a bar fixed at one end and supported 

 at the other are like those of one half of a rod with both 

 ends fixed and vibrating with a central node. 



There is now but one case left, namely, that of a bar with 

 both ends supported. Referring to equation (15) and the 

 conditions (8) , we find that for x = 0, P = and Q = 0. Again, 

 for the end# = Z, i. e., as r =m } we have R — S = and sin ra = 0, 

 i. e., m — nir, where n is any integer. Thus equation (15) 

 becomes reduced to a single term involving sin a/. We may 

 therefore write, for the bar supported at each end. by eq. (9), 



?/ = Ksin-y— cos/ — j 2 — t + e). . . . (24) 



