408 REPORTS ON THE STATE OF SCIENCE. — 1919. 



Group Velocity. 



.S. Waves over the sea are not pure waves, but a blend of different 

 wave lengths ; just as white light is a blend of the monochromatic parts 

 of the spectrum. 



A simple pure wave of length A, and period T, advancing with 

 velocity V, is given by 



(1) y = b sin — - {x—Yt) = b sin [mx — nt), 



A. 



o 2V Stt 



writing vi for - , and n for — - = — , for economy in space of printing. 

 A A 1. 



This pure wave is replaced, in Stokes' explanation, by a blend of 



two such waves, nearly equal, 



(2) 2/i = ^1 s'^ {m^x—n^t-'rf.i), Vi = ^2 sin {m^x — n^t-^-f..^. 

 Their resultant, at the same time, /!, and same place, x, is given by 



K^) y\ +y-2 = 2 (^1—^2) [sin (m,a;— 7iii + C|) ± sin {vLrX—n.^t 4- £2)] 



= (&i±^2) gijjM("^i-'"2)a;-(ni-?i.) i + e.-e.] 



CO^ U('"l+'"2)a?~(nj+7l2)i + e, +€2] 



of which the second factor represents a wave of length 



277 _ 2 



A, A2 

 the harmonic means of the wave lengths A; and A.,, moving with velocity 



(4) 



A I Ao 



and when the two waves are nearly equal, this is equivalent to the pure 

 wave motion of (1), with Vj = ¥3 = V, A, = A2 = A. 



The first factor represents a wave motion of long period T and great 

 length L, the equivalent of Scott Eussell's primary solitary wave, in 

 which 



'Ao 



