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XLV. On the Propagation of Waves along connected Systems 

 of Similar Bodies. By Lord Rayleigh, F.B.SM 



l/'OR simplicity of conception the bodies are imagined to be 

 J- similarly disposed at equal intervals (a) along a straight 

 line. The position of each body, as displaced from equilibrium, 

 is supposed to be given by one coordinate, which for the rth 

 body is denoted by i/r r . A wave propagated in one direction 

 is represented by taking -^r r proportional to e i( - nt+r P\ If we 

 take an instantaneous view of the system, the disturbance is 

 periodic when r/3 increases by 27r, or when ra increases by 

 27r«//3. This is the wave-length, commonly denoted by X; so 

 that, if k = 2ir/X, k = /3/a. The velocity of propagation (V) is 

 given by Y = n/k ; and the principal object of the investiga- 

 tion is to find the relation between n or V and \. 



The forces acting upon each body, which determine the 

 vibration of the system about its configuration of equilibrium, 

 are assumed to be due solely to the neighbours situated within 

 a limited distance. The simplest case of all is that in which 

 there is no mutual reaction between the bodies, the kinetic 

 and potential energies of the system being then given by 



T=iA £$, P=iC S^, . . . . (1) 



similarity requiring that the coefficients A , C be the same 

 for all values of r. In this system each body vibrates inde- 

 pendently, according to the equation 



A f,+ C f,.=0, (2) 



and 



™ 2 = C /A (3) 



The frequency is of course independent of the wave-length in 

 which the phases may be arranged to repeat themselves, so 

 that n is independent of k, while V equal to n/k varies 

 inversely as k, or directly as X. The propagation of waves 

 along a system of this kind has been considered by Rejmolds. 

 In the general problem the expression for P will include 

 also products of yjr r with the neighbouring coordinates . . . 

 i/r r _ 2 , tyr-i, tyr+u ^V+2 ■ • •> and a similar statement holds 

 good for T. Exhibiting only the terms which involve r, we 

 may write 



T= . . . +±A f 2 r - A^rfr-l-Alfrfr+l 



— A^f r ^ r -2 — A 2 l/r r ^ r+2 — . . ., .... (4) 



* Communicated by the Author, 



