82 



Sir W. Thomson. On the 



[Feb. 3 y 



Example (1). — As a first example take deep-sea waves; we have 



/w = v / »'- ••••••• w 



which reduces (4), (3), and (10) to 



»-*v^ (12) 



and x = \y/ g -A, ...... (13> 



1 ±( cos g +sin |f) = _L cos (g_^ . (14> 

 gV2x%\ 4>x 4x / gx \4<x 4/ 



u — 



which is Cauchy and Poisson's result for places where x is very great 

 in comparison with the wave-length 2ir/fi f that is to say, for place and 

 time snch that gt 2 /4<x is very large. 



Example (2). — "Waves in water of depth D : 



*^v4iiTS£} (15 > 



Example (3). — Light in a dispersive medium. 

 Example (4). — Capillary gravitational waves : 



=V(i +T -) (16> 



Example (5). — Capillary waves : 



f(m) = V(Tm) (17) 



Example (6). — Waves of flexure running along a uniform elastic 

 rod : 



/(m) =,»./?, (18) 



V w 



where B denotes the flexural rigidity, and w the mass per unit of 

 length. 



These last three examples have been taken by Lord Rayleigh as 

 applications of his generalisation of the theory of group -velocity ; and 

 lie has pointed out in his " Standing Waves in Running Water " 

 {London Mathematical Society, December 13, 1883) the important 

 peculiarity of Example (4) in respect to the critical wave-length which 

 gives minimum wave-velocity, and therefore group -velocity equal to 

 wave- velocity. The working out of our present problem for this case, 

 or any case in which there are either minimums or maximums, or both 

 maximums and minimums, of wave- velocity, is particularly interest- 



