Waves in Flowing Water. 451 



term are neglected in the approximation (24) to the value of 

 the second ; and we have, as our final approximate result, 



l?=i 9 v(l- e2 J'^ mD ) (25). 



There is no difficulty in understanding the permanent 

 steadiness of the motion which we have now been considering: 

 to any finite distance, however great, on either the up-stream 

 or down -stream side of the inequalities, if the water in the 

 finite space considered is given in this state of motion, and 

 if water is admitted on the one side and carried away on the 

 the other side conformably. But it is very interesting and 

 instructive to consider the initiation of such a state of things 

 from an antecedent condition of uniform flow over a plane 

 bottom. Suppose, as the primary condition, an inequality, 

 whether elevation or depression, to exist in the bottom, but 

 to be carried along with the water, so that the flow of the 

 water is everywhere uniform and in parallel lines. If the 

 inequality is an elevation above the bottom, our supposition 

 is that the whole projecting piece, moving with the water, 

 slips along the bottom. If the inequality be a depression in 

 the bottom, the more awkward supposition must be made of a 

 plasticity of the bottom, and the form of the inequality 

 carried along, while the bottom is kept rigidly plane before 

 and after this depression. 



Suppose, now, the inequality is gradually or suddenly 

 brought to rest, what will be the resulting motion of the 

 water ? The question is identical with that of finding the 

 motion of water in a canal, when by an external force, such 

 as that of a towing-rope, a boat is gradually or suddenly set 

 in motion through it ; or, rather, it would be identical if the 

 boat were a beam filling the whole breadth across the canal, 

 so that the motion of the water shall be purely two-dimen- 

 sional. I hope in a later article (Part III. or Part IV. of the 

 present series) to investigate the formation of the proces- 

 sion of standing waves in the wake of the obstacle, and its 

 gradual extension farther and farther down-stream from the 

 obstacle, the motion having become sensibly steady in the 

 its neighbourhood, and becoming so to greater and greater 

 distances down-stream by the completion of the growth of fresh 

 waves. The disturbance sent up-stream from the initiating 

 irregularity must also be considered. Equation (15) shows 

 that whether the irregularity be an elevation, as in our first 

 diagram (fig. 1), or a depression, as in fig. 2, a rising of 

 level must travel up-stream, at a velocity relatively to the 

 water which we know must be s/gT)J, where D ' is inter- 



