Kundt's law of selective dispersion. 23 



vibratory systems, not affected by viscosity, is on the average half 

 potential and half kinetic. The ultimate foundation of this im- 

 portant general principle seems to be adequately contained in the 

 following remark, and more abstruse considerations are hardly 

 necessary. The characteristic of all kinds of steady standing 

 vibrations or undulations in a non-dissipative system, however 

 complicated in its structure, is that each particle of the system 

 describes an orbit in simple harmonic motion. For each particle 

 therefore the two kinds of energy, kinetic and potential, are equal 

 on the average ; and this includes their equality in the aggregate. 

 For uniform trains of progressive waves similar considerations 

 remain applicable. The argument extends also to the most com- 

 plex types of dispersive media, in cases where absorption is 

 negligible : it is to be noticed that the energy that is ' propagated ' 

 with the group-velocity is there the energy in the ether, together 

 with that of the vibratory disturbance of the stationary molecules, 

 — which latter is on the theory of cyclic systems of Kelvin and 

 Routh a perfectly definite quantity added on to the intrinsic 

 kinetic energy of each molecule. 



3. The principle involved in propagation by groups may be 

 somewhat generalised, thus affording insight into its essential 

 character. An equation 



y = Af(nw — pt) 



represents a disturbance whose profile is of the form y = Af(n%), 

 propagated onward with uniform velocity pjn. When the form is 

 a periodically undulating one the equation represents a progressive 

 wave-train of definite wave-length. The problem is to specify the 

 general features, if any, presented by an aggregate of related 

 progressive forms, all differing but slightly from a central type 

 y x f (n oc — p t) but otherwise undetermined. We may represent 

 one of these component forms by By = BA ./{(n Q + Bn) x — (p Q + Bp) t), 

 where Bn and Bp are treated as infinitesimal. The definiteness of 

 type of the forms under consideration implies that n is given as 

 some function of p, so that Bn/Bp is here a function of p and so 

 constant, say U . The application of Taylor's theorem allows us 

 in general to express the component form above mentioned as 



Sy = 8Af(n sc—p t) + (8n.cc — Bp.t) BAf (n^x — p t). 



This expansion holds so long as Bn.x — Bp.t is small: that is, it is 

 valid for the neighbourhood of a point travelling with the definite 

 velocity 8p/8n, say U . Under this restriction, the aggregate of a 

 crowd of such closely related progressive forms is therefore repre- 

 sented by an equation 



y = <$>(x- U t) F(n oc-p t). 



