Propagation of Disturbance in a Dispersive Medium. 161 



as in water of finite depth — the wave velocity V has a finite 

 limiting value ; a non-dispersive medium (V constant) 

 appears as a particular case in which the effect may begin 

 abruptly at each point after suitable delay. Lord Rayleigh 

 remarks that in the ordinary theory of waves on water the 

 velocity of propagation of waves of expansion is regarded as 

 infinite, and doubtless some such dynamical assumption can 

 be found in most cases ; but the mathematical expressions 

 may be modified and another point of view is suggested. 

 The previous expression (3) gives equal values for negative 

 as for positive values of t ; in fact the whole disturbance is 

 built up of standing waves and may be regarded as a 

 vibration in which at the instant £ = the displacement is 

 confined to the region near .r = and the velocity is every- 

 where zero. The appearance of an effect at every point 

 immediately after this instant appears connected with the 

 occurrence of a disturbance at each point before the instant 

 in question, apart from considering velocity of propagation. 



If we attempt to form an expression which shall be zero 

 for negative values of t and shall give suitable values for t 

 positive, we find again that the effect begins without delay 

 at all points ; but this now appears connected with the fact 

 that for certain periods no wave motion is possible. In 

 other words, a Fourier analysis with respect to period divides 

 the disturbance into two types ; to each period there corre- 

 sponds in the one case a wave motion and in the other a 

 disturbance- established in equal phase throughout. Each 

 part of the expression implies an effect beginning without 

 delay at all points, but it is suggested that the existence of 

 both types is essential for this result — the character of the 

 dispersion of the medium being dependent upon the range of 

 period and the manner in which wave motion is impossible. 



In order to avoid some complication of formula? in localizing 

 the initial disturbance, illustrations are taken in propagation 

 along connected systems of similar bodies. 



2. Consider the case of a stretched string itself without 

 mass, but carrying loads (//,) at equal intervals (a) ; let 1\ be 

 the tension of the string and let y r be the transverse dis- 

 placement of a particle. The potential and kinetic energy 

 functions are * 



v = +£<*-*-#+£(►«-*)»+ 1 (4) 



* Lord Rayleigh, Scientific Papers, iv. p. 342. 

 Phil Mag. S. 6. Vol. 19. No. 109. Jan. 1910. M 



