450 SURFACE WAVES. [CHAP. IX 



stance, of wavelets of about two-thirds of an inch in length, which will 

 continually increase in amplitude until they transcend the limits implied 

 in our approximation. 



248. We resume the investigation of the effect of a steady 

 pressure-disturbance on the surface of a running stream, by the 

 method of Arts. 226, 227, including now the effect of capillary 

 forces. This will give, in addition to the former results, the ex- 

 planation (in principle) of the fringe of ripples which is seen in 

 advance of a solid moving at a moderate speed through still 

 water, or on the up-stream side of any disturbance in a uniform 

 current. 



Beginning with a simple-harmonic distribution of pressure, we 

 assume 



<f)/c = x + fte ky sin kx, | , 



^/c=-y + @e k ycoskx j * '' 



the upper surface coinciding with the stream-line -^ = 0, whose 

 equation is 



y = /3 cos kx ........................... (2), 



approximately. At a point just beneath this surface we find, as 

 in Art. 226 (8), for the variable part of the pressure, 



= /3 {(kc* g) cos kx + /JLC sin kx} ............ (3), 



where fj, is the frictional coefficient. At an adjacent point just 

 above the surface we must have 



= 

 p ' p dx* 



^^{(ktf-g-k^coskz + fjLCsmkx} ......... (4), 



where T is now written for TJp. This is equal to the real part of 



-g- k*T - ific) e ikx . 

 Writing P for the coefficient, we find that to the imposed pressure 



/W 



-^- = Pcoskx ............................ (5) 



will correspond the surface-form 



~ 9 ~ k*T) cos kx IJLC sin kx . 



