584 Prof. E. H. Barton on the Lateral 



Since each end of the bar may be of any one of these three 

 types, we have six cases in all. These may be classed as 

 follows : — 



I. Both ends fixed. 

 II. Both ends free. 



III. Both ends supported. 



IV. One end fixed and one free. 



V. One end free and one supported. 

 VI. One end supported and one fixed. 



Thus for any specified bar we have to satisfy (5) for every 

 point and one of the pairs (6) to (8) for one end t T = 0, and 

 the same or another pair for the end x = l. 



Solution of Equations. — We now assume, with Lord Rayleigh, 

 that the motion is harmonic. Thus y may be written 



y — uo,Q>^y-~m't\ (9) 



where I is the length of the bar and m is an abstract number 

 depending on the order of the tone. The substitution of (9) 

 in (5) gives 



dSi 



m 



u (10) 



dx* I" 



Suppose u = e pmxil is a solution of this. Then, on differ- 

 entiating, we find jo 4 =l, i. e., p~ + \/— 1 or +1. Thus u 

 can be represented as the sum of four terms each involving- 

 one of the values of p and an arbitrary coefficient. Hence, 

 after a little transformation, we may write 



u = A cos mx-ll + B sin mtc/l + Ce mx ' 1 + DjT mx / 1 . . (11) 



This, put in (9), is a solution of (5), and deserves a little 

 notice previous to fitting it to (6)-(8). The simplest case is 

 when the motion is strictly periodic with respect to x when 

 C and D vanish. 



Then, if wave-length, period, and frequency are denoted 

 by \, t, and N respectively, we see that 



2tt/X = t7?//, or \=27rl/m, (12) 



27r/T=Kbm°/l 2 , ..or 2irl :2 lfcbm 2 ^X : /27rfcO, .. (13) 



and ^ T /elm 2 ■ „ k / —.- ,^. s 



N= _ =m ._ pV ^. (14) 



We thus have the known relations that : — 



(1) Wave-length varies directly as length of the bar and 

 inversely as m. ■ 



(2) The frequency o£ any given tone in the series of 



