L 160 ] 



XIII. On the Instantaneous Propagation of Disturbance in a 

 Dispersive Medium. By T. H. Havelock, M.A., D.Sc, 

 Armstrong College, JYew castle-on- Tyne*. 



1, TN a recent paper in this Magazine (July 1909) Lord 

 X Rayleigh discussed some cases of instantaneous pro- 

 pagation o£ a limited initial disturbance, and pointed out 

 that, although the physical assumption which permits an 

 infinite velocity may be obvious, there is an apparent 

 paradox when the disturbance can be analysed by the 

 Fourier method into simple waves of which the maximum 

 wave-velocity is finite. In the following note two further 

 cases which admit of exact solution are examined ; the 

 mathematical expressions are modified by a further Fourier 

 analysis into periods so as to express more suitably a dis- 

 turbance which has a definite beginning in time, and the 

 result appears to emphasize the connexion of the phenomenon 

 with the dispersive character of the medium. 



Let y denote the effect at a position x and time t of an 

 initial disturbance in a dispersive medium ; for an initial 

 displacement cos ko\ with no initial velocity ; the disturbance 

 is given by 



y = cos («Vi) cos tea, (1) 



where V is supposed a known function of k. 



Generalizing by Fourier's method we Lave for an initial 

 displacement f(x) , 



1 r* i™ 



y = — I d/c cos fcYt I /(<») cos k(co — x)d(o. 



^Jo Jo 



(?) 



If the initial disturbance is limited practically to a line 

 through the origin it is usual to write 



y— \ COS (fcYt) COS (fCic)dK, ... (3) 

 7Tj 



where for convergence a factor e~ K ^ may be introduced 

 under the integral sign, and the limit taken for y zero. 



The cbief regular features of the disturbance may be 

 obtained by interpreting (3) in terms of groups of waves 

 travelling out from the origin. The same expression (3) 

 was used by Lord Rayleigh in discussing the initial motion 

 due to a limited disturbance. He showed that in general 

 the effect begins at all points without delay, even when — 



* Communicated by the Author. Presented to the British Association 

 at Winnipeg. 



