354 Sir William Thomson on Stationary 



on the east side of Trumpington Street, Cambridge, or in 

 the race of Portland or Islay overfalls. The train of 

 diminishing waves which we see in the wake of each little 

 irregularity of the bottom would, of course, extend to infinity 

 if the stream were infinitely long, and the water absolutely 

 inviscid (frictionless); and a single inequality, or group of 

 inequalities, in any part AB of the stream would give rise 

 to corrugation in the whole of the flow after passing the 

 inequalities, more and more nearly uniform, and with ridges 

 and hollows more and more nearly perpendicular to the 

 sides of the canal, the farther we are from the last of the 

 inequalities. Observation, with a little common sense of the 

 mathematical kind, shows that at a distance of two or three 

 wave-lengths from the last of the irregularities if the breadth 

 of the canal is small in comparison with the wave-length, or 

 at a distance of nine or ten breadths of the canal if the 

 breadth is large in comparison with the wave-length, the 

 condition of uniform corrugations with straight ridges per- 

 pendicular to the sides of the canal, would be fairly well 

 approximated to ; even though the irregularity were a single 

 projection or hollow in the middle of the stream. But the 

 subject of the present communication is simpler, as it is 

 limited to two-dimensional motion; and our inequalities are 

 bars, or ridges, or hollows, perpendicular to the sides of 

 the canal. Thus, in our present case, we see that the con- 

 dition of ultimate uniformity of the standing waves in the 

 wake of the irregularities is closely approximated to at a 

 distance of two or three wave-lengths from the last of the 

 inequalities. 



Let SA, SB denote two fixed vertical sections of the canal 

 at infinitely great distances beyond A and beyond B. It will 

 simplify considerations and formulas if we take SB at a node 

 (or place where the depth is equal to b, the mean depth), and 

 we therefore take it so; although this is not necessary for the 

 following kinematic and dynamical statements: — 



I. The volumes of fluid crossing SA and SB in the same 

 or equal times are equal; or, in symbols, 



au = bv = M (2), 



where M denotes the volume of water passing per unit of time. 



II. The excess (positive or negative) of the work done by 

 p on any volume of the water entering across SA, above the 

 work done by q on an equal volume of the water passing away 

 across SB is equal to the excess of the energy, potential and 

 kinetic, of the water passing away above that of the water 

 entering. Hence, and by (1), taking the volume of water 



