Lord Rayleigh : Hydrodynamical Notes, 181 



of the dispersive medium — expressing that the initial elevation 

 (£ = 0) is concentrated at the origin of x. When t is great, 

 the angles whose cosines are to be integrated will in general 

 vary rapidly with k, and the corresponding parts of the 

 integral contribute little to the total result. The most 

 important part of the range of integration is the neigh- 

 bourhood of places where kx + at is stationary with respect 

 to k, i. e. where 



•±'3-° ^ 



In the vast majority of practical applications da/dk is 

 positive, so that if x and t are also positive the second 

 integral in (1) makes no sensible contribution. The result 

 then depends upon the first integral, and only upon such parts 

 of that as lie in the neighbourhood of the value, or values, of 

 k which satisfy (2) taken with the lower sign. If k 1 be such 

 a value, Kelvin shows that the corresponding term in t] has 

 an expression equivalent to 



_ cos (cr-jt — k x x — \if) ..,. 



V ~ </{-2irtd**/dk 1 *Y W 



cr l being the value of a corresponding to k Y . 



In the case of deep-water waves where <T = ^/(gk), there is 

 only one predominant value of k for given values of x and t, 

 and (2) gives 



h = gt*l4*», *i=gt/2x, .... (4) 

 making 



and finally 



(Tit — k X X— l7T = gt 2 /4x-~ ^7T, .... (5) 



?7= -\- cos 



2tt^x% 



{£-0. ••■'•«» 



the well-known formula of Cauchy and Poisson. 



In the numerator of (3) &i and k A are functions of x 

 and t. If we inquire what change (A) in x with t constant 

 alters the angle by 27r, we find 



*{•*(-£)£}-* 



so that by (2) A = 2ir/k 1 , i. e. the effective wave-length .V 

 coincides with that of the predominant component in the 

 original integral (1), and a like result holds for the periodic 



