398 SURFACE WAVES. [CHAP. IX 



The interpretation of these results is simple. The first term of 

 (19) represents a train of simple-harmonic waves, on the down 

 stream side of the origin, of wave-length 2?rc 2 /^, with amplitudes 

 gradually diminishing according to the law e~^ x . The remaining 

 part of the deformation of the free-surface, expressed by the 

 definite integals in (19) and (20), though very great for small 

 values of #, diminishes very rapidly as x increases, however small 

 the value of the frictional coefficient /j^. 



When /*! is infinitesimal, our results take the simpler forms 

 7TC 2 ~ . f 00 cos kx 



f rfliQmx 



2vr sin KX + I 2 c?m (21), 



Jo w + # 



for a? positive, and 



7TC 2 pcos&tf 77 f 00 rae ma; , /rtox 



T- y = l. ^T7*=J ^+7^ m (22) 



for a? negative. The part of the disturbance of level which is 

 represented by the definite integrals in these expressions is now 

 symmetrical with respect to the origin, and diminishes constantly 

 as the distance from the origin increases. It is easily found, by 

 usual methods, that when KX is moderately large 



me~ mx d 1 _3J_ 5! 



Q /it \ fC /C JU tC JU rC Ju 



the series being of the kind known as semi-convergent. It 

 appears that at a distance of about half a wave-length from the 

 origin, on the down-stream side, the simple-harmonic wave-system 

 is fully established. 



The definite integrals in (21) and (22) can be reduced to known functions 

 as follows. If we put (&+K) x=u, we have, for x positive, 



r 



cos kx 77 /&quot;* cos (KX - u) _ 

 -; -- dk= I - 2 - Lfo 



k +* f u 



= - Ci KX cos KX + (far Si KX} sin KX ...... (ix), 



where, in the usual notation, 



n . f 00 cosu , . f u siuu , 



(J1U-I - du. Siu= I --- du ............... (x). 



ju U Jo U 



