120 Free Procession of Waves in Deep Water. 



where m is a large positive numeric. Thus, remarking that 

 cay [ — 6)-= —cay (6), and say ( — 6)=. —say (6), we have, by 

 (43) and (41) in (32), , 



Q = \J - { [cay (m) — cay{s/ 2t™)] 2 + [say (m) -say{V 2irn) ] 2 \i (44) ; 



" if 



and therefore, when m and n are each very large, Q=0. 

 Because n is large we still, as in (36), have R=0 ; and 

 therefore the motion is approximately zero, at any consi- 

 siderable number, n, of wave-lengths from the origin, so long 

 as m in (43) remains large. As time advances, m decreases 

 to 0, and on to — oo : and, watching at the place x = n\ we 

 see wave-motion gradually increasing from nothing, till it 

 becomes the regular procession of waves represented by (39); 

 and continues so unchanged for ever after. When m=0 } 

 that is to say, at the time 



t = 2cox/g (45), 



Q has attained half its final value. The point x where this 

 condition is fulfilled at time t may be called the mid-front of 

 the procession. It travels at the velocity ^g/co, or half the 

 wave-velocity ; which agrees with the result of Stokes. 



We may arbitrarily define " the front " as the succession of 

 augmenting waves which pass between the times corresponding 

 to m= + 10 and m= —10 (or any other considerable number 

 instead of 10). Thus the time taken by the front, in passing 

 the place x=rik 1 is 4:0o)~' l ^27rn. The space travelled by the 

 mid-front in this time is 20gco~ 1 V2irn, which may, arbitrarily, 

 be defined as the length of the front. It increases in pro- 

 portion to s/n ; and therefore in proportion to s/t, as said 

 above. The effect upon phase of the changing waves in the 

 front ; due to the fluctuations of e, and to the law of augmen- 

 tation of Q from zero to its final value ; is to be illustrated by 

 calculations and graphic representations, which I hope will be 

 given on a future occasion. 



The rear of a wholly free procession of waves may be quite 

 readily studied after the constitution of the front has been 

 fully investigated, by superimposing an annulling surface- 

 pressure upon the originating pressure represented by (12) 

 above, after the originating pressure has been continued so 

 long as to produce a procession of any desired number of 

 waves. 



