452 On Stationary Waves in Flowing Water. 



mediate between D and the smaller depth, which we shall 

 call D', in the undisturbed stream above. But however gra- 

 dually the initiating irregularity may have been instituted, 

 this travelling of an elevation up-stream must develop a bore; 

 because the velocity of propagation is, as it were, different in 

 different parts of the slope, being v c/D' at the commence- 

 ment of the slope, and ranging from this, through v^D / , to 

 \/gD as the depth rises from D r to D ; so that, as it were, 

 the brow of the plateau in its advance up-stream overtakes the 

 talus, till the slope becomes too steep for our approximation. 

 The inevitable bore and " broken " water (inevitable without 

 viscidity of the water, or some surface-action preventing the 

 excessive steepness) would modify affairs down-stream in a 

 manner which it is difficult to imagine. It becomes, there- 

 fore, interesting to see how it may be avoided, whether by 

 surface-action, or by giving some viscosity to the water. It 

 is more interesting to do this by surface-action, and to allow 

 the water to be perfectly inviscid, so that our standing waves 

 down-stream may be perfectly unimpaired. And we may do 

 it very simply by covering the free surface all over (up-stream 

 and down-stream) with an infinitely thin viscously elastic 

 flexible membrane, stiffened transversely (after the manner of 

 the sail of a Chinese junk) by rigid massless bars with ends 

 travelling up and down in vertical guides on the sides of the canal . 

 If we suppose the motion of these ends to be resisted by forces 

 proportional to their velocities, and the membrane to exercise 

 (positive or negative) contractile tensional force in simple 

 proportion to the velocity of the change of its length in each 

 infinitely small part ; we have a mechanical arrangement by 

 which is realized the mathematical condition of a surface 

 normal pressure varying according to normal component 

 velocity of the otherwise free surface, and in simple propor- 

 tion to this normal velocity when the slope is infinitesimal. 

 By making the viscous forces sufficiently great, we may make 

 the progress of the rise of level up-stream as gradual as we 

 please, and perfectly avoid the bore. We may also make the 

 progress of the procession of stationary waves down-stream as 

 slow as we please. The form of the water-surface over the 

 inequality or inequalities, and to any distance from them, 

 both up-stream and down-stream, is not ultimately affected 

 at all by the viscous covering ; and it becomes, as time 

 advances, more and more nearly that of the mathematical 

 solution for steady motion, which I hope to give, with graphic 

 illustrations drawn according to calculation from the solution, 

 in Fart III. 



