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 



r e~ ilcx 7 



2 7r ie iKlX + I TT * ( 1U )&amp;gt; 



Jo &-*-&amp;lt;! 



and the latter to 



dk 



o 



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



~ dk... ....(v), 



and the latter to 



-ikx 



r p-ikx 



+j.jSf (vi) - 



We have here simplified the formulae by putting v = 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, 

 ^.y = - *?sanc& + F(x) (10), 



V #2 ~ K l 



and, for x negative, 



2?T . -r., *. /i-i\ 



^) ............... (11), 



where 



1 ([&quot;&quot;coskxjj [ x coskx,j} . 



=~ \\ -j --- dk- ,- dk&amp;gt; ....... (12). 



K 2 - K! (Jo k + K! Jo k + K 2 J 



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

 tions Ci !#, Si #!#, Ci K&, Si K.&, 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 (^TT/KJ). 



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

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

 of length 27r/K lt and on the up-stream side by a train of the 

 shorter wave-length 27r/; 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. 



