Lord Kelvin on Groups of Deep-Sea Waves. 11 



but we see distinctly the outward zeros 2, 3, 4, 5, 6, 7, 8. 9, 

 and indications of the inward zeros 9, 8. The whole train of 

 zeros for time 4v/7r, shown and ideally continued to the 

 middle by numbers, is 1. 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3 : 

 sixteen in all. 



Zero 3 has passed out o£ the range of diagram 6, but we 

 see in it distinctly the outward zeros 4, 5, ... . 12, and 

 an indication of the pair 33, 33, which has come into ex- 

 istence before the time 8 s/ir. The whole train of zeros for 

 time 8.^/77-, indicated by numbers, is 1, 2, ... . 32, 33, 33, 

 32, .... 4, 3 ; sixty-four in all. 



(2) Illustration* of the Indefinite Extension and Multiplication 

 of a Group of Two-Dimensional, Deep-Sea Waves 

 Initially Finite in Number. §§ 114-117. 



§ 111. The water is left at rest and free, after being 

 initially displaced to a configuration of a finite number of 

 sinusoidal mountains and valleys — five mountains and four 

 valleys ; in the diagrams placed before the Society. The 

 initial group of waves, shown in diagram 1, of tig. 35. is 

 formed by placing side by side, at distances equal to z (taken 

 as unity), nine of the curves of diagram 1, fig. 34, alternately 

 positive and negative. Diagrams 2 and 3, of fig. 35, are 

 made by corresponding superpositions of the curves of 

 diagrams 5 and 6, of fig. 34, Thus what, according to the 

 known law of deep-sea periodic waves (§ 19 above), would 

 be definitely and precisely the wave-length,, if the number.^ 

 of crests and hollows were infinitely great, would be 2 : and 

 as we are taking <7 = 4, the period would be \/ /r rr^ and the 

 propagational velocity would be 2/ yV. 



§ 115. Immediately after the water is left free, the dis- 

 turbance begins analysing itself into two groups of waves, 

 seen travelling in contrary directions from the middle line of 

 the diagram. The perceptible fronts of these two groups 

 extend rightwards and leftwards from the end of the initial 

 single static group, far beyond the '' hypothetical fronts, ' 

 supposed to travel at half the wave- velocity, which (according 

 to the dynamics of Osborne Reynolds and Rayleigh, in their 

 important and interesting consideration of the work required 

 to feed a uniform procession of water- waves) would be the 

 actual fronts if the free groups remained uniform. How far 

 this if is from being realised is illustrated by the diagrams of 

 fig. 35, which show a great extension outwards in each 

 direction far beyond distances travelled at half the " wave- 

 velocity/' While there is this great extension of the front> 



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