80 



Sir W. Thomson. On the 



[Feb. 3, 



February 3, 1887. . 



Professor STOKES, D.C.L., President, in the Chair. 



The Presents received were laid on the table, and thanks ordered 

 for them. 



The following Papers were read : — 



I. " On the Waves produced by a Single Impulse in Water of 

 any Depth, or in a Dispersive Medium." By Sir W. 

 Thomson, Knt, LL.D., F.R.S. Received January 26, 1887. 



For brevity and simplicity consider only the case of two-dimensional 

 motion. 



All that it is necessary to know of the medium is the relation 

 between the wave- velocity and the wave-length of an endless proces- 

 sion of periodic waves. The result of our work will show us that the 

 velocity of progress of a zero, or maximum, or minimum, in any 

 part of a varying group of waves, is equal to the velocity of progress 

 of periodic waves of wave-length equal to a certain length, which may 

 be defined as the wave-length in the neighbourhood of the particular 

 point looked to in the group (a length which will generally be inter- 

 mediate between the distances from the point considered to its 

 next-neighbour corresponding points on its two sides). 



Let f(m) denote the velocity of propagation corresponding to wave- 

 length 27r/\. The Fourier- Cauchy-Poisson synthesis gives 



u = \ dm cos m — £/(m)] (1) 



Jo 



for the effect at place and time (x, t) of an infinitely intense disturb- 

 ance at place and time (0, 0). The principle of interference as set 

 forth by Prof. Stokes and Lord Rayleigh in their theory of group- 

 velocity and wave- velocity suggests the following treatment for this 

 integral :— 



"When x—tf(m) is very large, the parts of the integral (1) which lie 

 on the two sides of a small range, fi— a. to + vanish by annulling 

 interference ; /t being a value, or the value, of m, which makes 



s {m[x-«/(»)]} = 0; 



(2) 



