452 SURFACE WAVES. [CHAP. IX 



These integrals are of the forms discussed in Art. 227. It appears that 

 when x is positive the former integral is equal to 



. *-ikx 



W" 



and the latter to 



r x,-tfcc 



dk 



/*-* 



Jo " ^2 



On the other hand, when x is negative, the former reduces to 



~*^dk... 



and the latter to 



o-Vcx 



/ e -ikx 



^+/.FH5* () 



We have here simplified the formulae by putting / = after the transfor- 

 mations. 



If we now discard the imaginary parts of our expressions, we obtain the 

 results which immediately follow. 



When fjf is infinitesimal, the equation (9) gives, for x positive, 

 7rT l _ 2?r . -.-,, , (~\c\\ 



and, for x negative, 



where 



' 1 f f cos h ji f 00 cos h 77 ) /n\ 



=- -\\ -= -- dk-l f - dk}- ....... (12). 



K Z - K! (Jo & + *! Jo k + tc 2 j 



This function F(x) can be expressed in terms of the known func- 

 tions Ci KI#, Si !#, Ci #2#, Si /Cgd?, by Art. 227 (ix). The disturb- 

 ance of level represented by it is very small for values of x, 

 whether positive or negative, which exceed, say, half the greater 

 wave-length ( 



Hence, beyond some such distance, the surface is covered on 

 the down-stream side by a regular train of simple-harmonic waves 

 of length 2 < 7r//e 1 , and on the up-stream side by a train of the 

 shorter wave-length 27r//c 2 . It appears from the numerical results 

 of Art. 246 that when the velocity c of the stream much exceeds 

 the minimum wave-velocity (c m ) the former system of waves is 

 governed mainly by gravity, and the latter by cohesion. 



