116 DYNAMICAL THEORY OF SOUND 



is exactly the same as in the case of the transverse vibrations of 

 a string. The frequencies of the various modes are given by 



N = sc/2l, 00 



where s = l, 2, 3, .... The result is unaffected by permanent 

 tension in the wire. 



When the rod is free, the condition of zero stress at the 

 ends gives f ' = for x = and x = I. Introducing this condition 

 in (5) we find 



F' ( ct ) = f (ct), F'(ct + l)=f(ct-l), (8) 



for all values of t. The former of these gives on integration 



F(ct)=f(ct), (9) 



no explicit additive constant being necessary since it may be 

 supposed included in the value of f(ct). The second relation 

 then gives 



f(ct + l)=f(ct-l)+C. (10) 



The constant G is connected with the total momentum of the 

 bar in the direction of its length. We have, from (9) and (10), 



f f dx=c \\f\ct - x) +f(ct + x)} dx = cC. . . .(11) 



Jo Jo 



Since nothing essential is altered if we superpose any uniform 

 velocity in the direction of the length, we may assume the 

 mass-centre to be at rest, in which case (7=0. The formula 

 (10) then shews that the residual motion is periodic, since 

 everything recurs when t increases by 2l/c. 



In the analytical process for ascertaining the normal modes 

 we assume that f varies as cos (nt -f e), whence 



and 



(WIT W 'T'N 

 4 cos +sin ) cos (nt + e) (13) 

 c c / 



The conditions that 9j-/3a? = for x and x = I require B 0, 

 sin (nl/c) = 0, whence 



nJ/c = 6-7r, (14) 



where 5 = 0, 1, 2, 3, ..., the scale of periods being harmonic. 

 The nodes (f = 0) are given by cos (sirx/l) = 0, and the loops, 



