247-249] SURFACE DISTURBANCE OF A STREAM. 451 



Let us first suppose that the velocity c of the stream exceeds 

 the minimum wave-velocity (c m ) investigated in Art. 246. We may 

 then write 



K,)(K 2 -k) .................. (7), 



where K^ tc 2 are the two values of k corresponding to the wave- 

 velocity c on still water; in other words, ZTT/KH 27r//c 2 are the lengths 

 of the two systems of free waves which could maintain a stationary 

 position in space, on the surface of the flowing stream. We will 

 suppose that K&amp;lt;&amp;gt;&amp;gt; K^. 



In terms of these quantities, the formula (6) may be written 



_ P (k K!) (K 2 k) cos kx /// sin kx , . 

 y = T&quot; (k-Kjfa-ky + p * 



where /// = pc/T . This shews that if p! be small the pressure is 

 least over the crests, and greatest over the troughs of the waves 

 when k is greater than K. 2 or less than K I} whilst the reverse is 

 the case when k is intermediate to te lt K 2 . In the case of a pro 

 gressive disturbance advancing over still water, these results are 

 seen to be in accordance with Art. 165 (12). 



249. From (8) we can infer as in Art. 227 the effect of 

 a pressure of integral amount Q concentrated on a line of the 

 surface at the origin, viz. we find 



Q r (k- ^(Kz-k) cos kx - fjf sin kx ( . 



y - *TI - Jo -^tf&i-ky+jfi- 



This definite integral is the real part of 



e**dk 



r 



J o 



The dissipation-coefficient // has been introduced solely for the purpose of 

 making the problem determinate; we may therefore avail ourselves of the 

 slight gain in simplicity obtained by supposing pf to be infinitesimal. In this 

 case the two roots of the denominator in (i) are 



k = 



where v=p. /(ic. 2 -K l ). 



Since * 2 &amp;gt; K i&amp;gt; &quot; is positive. The integral (i) is therefore equivalent to 



i f r &amp;lt;p*dk r _j tta tifc__i 



K 2 - Kl -2i v \J A-fa + fr) Jo k-( K2 -iv)f~ 



292 



