252 Sir W. Thomson on the Waves produced by 



whereas my own experiments show that both the torsional 

 and longitudinal elasticities of iron and steel are decreased by 

 about 2J per cent, when the temperature is raised from 

 0° C. to 100° 0. 



I have dwelt longer than I should otherwise have done on 

 this part of my paper because I find that even our best text- 

 books relating to elasticity and sound still retain what I am 

 convinced is an error. 



XXVII. On the Waves produced by a Single Impulse in Water 

 of any Depth, or in a Dispersive Medium. By Sir William 

 Thomson, LL.D., F.R.S.* 



EOR brevity and simplicity consider only the case of two- 

 dimensional motion. 



All that it is necessary to know of the medium is the rela- 

 tion between the wave-velocity and the wave-length of an 

 endless procession of periodic waves. The result of our work 

 will show us that the velocity of progress of a zero, or maxi- 

 mum, 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 the preceding and 

 following waves). 



Let/(w) denote the velocity of propagation corresponding 

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

 gives 



u= I dm cos m\_x — tf(m)~] .... (1) 



Jo 



for the effect at place and time (#, t) of an infinitely intense 

 disturbance at place and time (0, 0). The principle of inter- 

 ference, as set forth by Prof. Stokes and Lord Kayleigh in 

 their theory of group -velocity and wave- velocity, suggests the 

 following treatment for this integral : — 



When x — tf(in) is very large, the parts of the integral (1) 

 which lie on the two sides of a small range, fi — u to p + ot, 

 vanish by annulling interference ; /jl being a value, or the 

 value, of m which makes 



d/dm{m[x-tf(m)]}=0 .... (2); 



* Communicated by the Author, having been read at the Meeting of 

 the Koyal Society, 3rd February, 1887. 



