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XLII. On Stationary Waves in Flowing Water. — Part I. 

 By Sir William Thomson *. 



THIS subject includes the beautiful wave-group produced 

 by a ship propelled uniformly through previously still 

 water, but the present communication f is limited to two- 

 dimensional motion. 



Imagine frictionless water flowing in uniform regime 

 through an infinitely long canal with vertical sides ; and 

 bottom horizontal except where modified by transverse ridges 

 or hollows, or slopes between portions of horizontal bottom 

 at different levels. Included among such inequalities we may 

 suppose bars above the bottom, fixed perpendicularly between 

 the sides. Let these inequalities be all within a finite por- 

 tion,, AB, of the length, and let / denote the difference of 

 levels of the bottom on the two sides of this position, positive 

 if the bottom beyond A is higher than the bottom beyond B. 



Now, let the water be given at an infinite, or very great, 

 distance beyond A, perpetually flowing towards A with any 

 prescribed constant velocity u, and filling up the canal to a 

 prescribed constant depth a. It is required to find the 

 motion of the water towards A, through AB, and beyond B 

 as disturbed by the inequalities between A and B. This 

 problem is essentially determinate; and it has only one solu- 

 tion if we confine it to cases in which the vertical component 

 of the water's velocity is everywhere small in comparison 

 with the velocity acquired by a falling body falling from a 

 height equal to half the depth. Let b be the mean depth, 

 and v the mean horizontal velocity at very great distances 

 beyond B; and (to have w to denote wave -energy) let w be 

 such that 



(iv* + igb)b + w (1) 



is the whole energy, kinetic and potential, per unit of the 

 canal's breadth and per unit of its length. In cases in which 

 the water flows away unruffled at great distances from B, w 

 is zero. But, in general, the surface is ruffled, and the water 

 flows " steadily " between the plane bottom and a corrugated 

 free surface, as in the well-known appearance of water flow- 

 ing in a mill-lead, or Highland burn, or in the clear rivulet 



* Communicated by the Author, having been read before Section A 

 of the British Association, Birmingham, Sept. 7, 1886. 



t I have since found, in a sufficiently practical form, the solution for 

 the wave- group produced by the ship, which I hope to communicate to 

 the Philosophical Magazine for publication in the November number. — 

 W. T., September 13, 1886. 



