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 



&amp;lt;f&amp;gt;/c = - x + ftefr sin lex, \ . 



^/c-y+tf**coefcr J 



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

 equation is 



y = ft 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, 



= ft {(kc* - g) cos kx + yuc sin kx} ............ (3), 



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

 above the surface we must have 



p p x* 



= ft {(kc z -g- &T) cos kx + pc sin kx] ......... (4), 



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



ft (kc* -g- k*T - ific) e ikx . 

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



will correspond the surface-form 



_ (k^ ~ 9 ~ k*T f ) cos kx nc sin kx 



(5) 



