ON WAVE MOTION. 409 



and this is the group velocity, when the wave length, X, and A..,, and 

 velocity, V, and Vj, is so nearly equal, that we take 



and is the number of waves per unit of time, is the number of 



2tt Air 



waves per unit of length. 



The effect is to make the amplitude of rise and fall at one spot to 

 diminish, and then increase again, in the long period T. 



This is observed when we follow with the eye the crest of a particular 

 wave of the sea. After a maximum height, it is seen to die down 

 gradually, and meanwhile another wave has been growing up behind it. 



Group velocity of sea waves was known to the sailor of all time, in 

 his proverbial formula of every tenth or third wave a big one, SeKaKVfjbta 

 or TpLKVfjLia, according to the fetch of the sea, great or small, across the 

 Ionian or inside the ^Egean. 



Ovid records it on his voyage into exile across the Ionian Sea. 



" Qui venit hie fiuctus, fluctus supereminet omnes. 



Posterior nono est, undecimoque prior." 

 " Vastius insurgens decinife ruit impetus undae." 



Thus in sea rollers, where A varies as V^, if every nth wave is a big one, 

 between the n — 1 and n + lth (Ovid's nine and eleven, with n=10), 



(8) G = 



half the harmonic means of V„_] and V„+i. 



When V is assumed to vary as some power, q, of A, 



(») l^=s I- «=('-«' ^- 



Thus in deep water waves, rollers over the sea, when the velocity 

 varies as the square root of the length, q=\, G=^V. 



In short capillary waves, Scott Russell's tertiary waves, produced by 

 drawing the point of a stick through water, q= — ^, as also in the waves 

 of lateral vibration of a bar, and then G=|T, and the group velocity is 

 not observed, except when a stone is thrown into water, in the dark 

 streaks of reflexion seen overtaking the capillary waves. 



In his Report on Waves to the British Association, 1844, Scott 

 Russell quotes Homer's simile, of the waves on a cornfield as well as the 

 sea (Iliad ii 144, xx 227), imitated by Shelley, but incorrectly, in 



* the ripe corn under the undulating air undulates like an ocean.' 

 But here each cornstalk vibrates or undulates independently in the 



