RECENT PROGfRESS IN HYDRODYNAMICS. 85 



pagatcd so that this condition is satisfied. It is curious that sudden waves 

 of condensation are permanent, whilst those of rarefaction are not so. 



When the length of the wave is very great compared with the depth 

 of the fluid, it is clear that the vertical motion of any particle is very 

 small compared with the horizontal. A theory based on the neglect of 

 the vertical motion is called a theory of long waves, and had been very 

 fully treated by Lagrange, Airy, and others. If in addition to small ratio 

 of depth to length, the ratio of height of wave to depth is also small, 

 then to the first order of these quantities the wave is of a permanent 

 type. But if the height of the wave has a finite ratio to depth of fluid 

 Rayleigh ' has shown that it is impossible for the waves to rnaintain their 

 form. In order that it should do so the force of gravitation oiight to 

 vary as the inverse cube of the height above the bottom of the fluid. In 

 the'same paper he points out that in a canal of slowly varying section, if 

 the velocity of the stream be less than that of a free wave, a contraction 

 of the channel produces a decrease in wave-height, and vice versa. The 

 opposite happens if the velocity of the stream is greater than that of a 

 free wave. 



The theory of irrotational waves of pei-manent type, considered _ by 

 Stokes, proceeds on the assumption that an infinite series of similar 

 waves follow one another, or that the wave-length is finite ; and we have 

 seen how, in this case, the solution is unique for given velocity of 

 propagation. But this theory does not take account of the question 

 whether a solitary wave can be thus propagated ; in fact, it is clear that 

 the functions determining such a wave must not be expressible as a series 

 of circular functions, as the motion is essentially non-periodic. That 

 such waves exist has been known for a long period, since the experiments 

 of Scott-Russell brought to light the wave called by him the solitary 

 wave. In the last report, Stokes has mentioned Earnshaw's attempt at 

 giving a mathematical theory of this kind of wave, and has pointed out 

 the objection to it, that it requires a finite change of velocity and pressure 

 at the beginning and end of the wave. Rayleigh has noticed that the 

 cause of this is that Earnshaw's wave contains molecular rotation, whilst 

 the fluid beyond the wave is at rest. A satisfactory theory^ has been 

 given by Boussinesq,^ when the curvature of the wave-profile is of such 

 a magnitude that d*i//cM may be neglected. He deduces that the 

 velocity of the wave is given by s/ gh where h is the primitive depth of 

 the fluid, which is Russell's result, as found by experiment. The form 

 of the profile is given hy y = a sech^ a!A/(3a/4 h^), a being the height 

 of the wave above the original surface. The centre of gravity of this 

 wave is at one-third of its height above the undisturbed level. 

 Boussinesq introduces a quantity, connected with any swelling of fluid 

 propagated along the surface, which he calls the moment of instability. 

 For all swellings of equal energy this moment! s least when the svrelling 

 is the solitary wave, which keeps its form ; when the moment is not 

 quite a minimum, the form of the swelling will oscillate about the 

 permanent form given above. It follows, from the analysis, that a 

 negative permanent wave is impossible, a result also of Russell s 



' ' On W.aves,' Phil. Mag. (5), i., p. 257 (1876). 



2 < Theorie des ondes ct des remous qui se propagent le long d'nn canal rec- 

 tangulaire horizontal, en communiquant au liquide contenu dans ce canal des vite.sges 

 senpiblement pareiUes de la surface au fond,' LiowiUe (2), xvii. (1872), p, 55, 



