THE TWO GREAT SOLITARY WAVES OF RUSSELL. 337 



But the pressure in the quiescent part varies with the depth only ; and depends not at all on 

 the square of the depth ; consequently there is a discontinuity of the law of pressure in passing 

 from the wave to the quiescent fluid. This is of course an impossibility; and tlierefore our 

 equations, though they may represent the properties of the wave with as much accuracy as the 

 experimental observations, cannot be regarded as the exact representatives of a possible wave 

 motion. But as they are rigidly deduced from the two hypotheses which Mr. Russell considered 

 to be experimentally justified, it follows as a necessary and indisputable consequence that it 

 is impossible for the particles of a permanent wave to move in the manner here assumed, viz., 

 so that those which are in a vertical plane at right angles to the axis of the canal should always 

 continue in a vertical plane during the transit of the whole wave. This hypothesis, as we have 

 seen, leads us to an impossible result; and it is of importance to notice that this impossibility could 

 not have been affirmed to be a necessary consequence of our hypotheses had methods of approxi- 

 mation been followed in our investigations, because it obviously depends on quantities which are 

 small. 



It appears then that the pressure at the junction of the moving fluid with the quiescent 

 fluid cannot practically be such as our two hypotheses require it should be, yet as the hypothesis 

 respecting the continuance of particles in the same vertical plane is certainly known to be verv 

 nearly true, as nearly true indeed as observation has been able to discriminate, we may expect 

 that it is the other hypothesis which deviates more sensibly from experiment. To the want 

 of permanency of the wave therefore we must look for the experimental confirmation of the 

 impossibility we have just discovered. We will therefore now turn to Mr. Russell's experiments 

 for evidence upon this point. 



At page 327 of The Report on Waves, we find what the author has designated the History of 

 a Solitary Wave of the First Order, from observation. A .vave such as we have been 

 investigating was generated in a canal such as we have supposed. The depth of the level fluid 

 was 5.1 inches; and k — h or the altitude of the crest of the wave above the general level was at 

 first 1.31 inches. An inspection of the table shews that the crest of the wave gradually fell, with 

 so rapid a degree of degradation, that in five minutes it was reduced to -08 inches, the wave 

 having in that time described ll6o feet. The velocity of the wave in the same time fell from 4.21 

 feet per second to 3.61 feet per second ; the diff'erence being -6 or one-seventh part of the whole 

 original velocity. It is evident from this statement that the degradation of the wave was a rapid 

 process, and that the consequent effect upon the velocity was considerable. 



These effects, which are much greater than could have been caused by imperfect fluidity or 

 friction against the sides and bottom of the canal, I consider are fully accounted for by the 

 circumstance above-mentioned, viz. the impossibility there is that the pressure should be continuous 

 and the wave at the same time permanent if the motions of the particles are such as we supposed 

 them to be, and which experiment shews they very nearly are. We have certainly proved the 

 trutii of these two alternatives ; — if particles continue in a vertical plane while a wave passes them, 

 then the wave cannot be permanent; — and, if the wave be permanent then the motions of particles 

 once in a vertical plane cannot preserve them in a vertical plane while the wave passes them. 



In proportion as one of our two hypotheses is more nearly true the other is farther from being 

 accurately true. Degradation of the wave is therefore the natural consequence of the law which 

 we have assumed for the motions of the fluid particles; and if that law be an experimental truth, 

 as we believe it is to a close degree of approximation, then the gradual destruction of the wave 

 is a necessary conse(|Ucnce, resulting not from friction alone, nor from im])erfect fluidity, but 

 chiefly from the manner in vvhich motion is initially communicated to the fluid particles. 



Strictly speaking, our investigations have been conducted on two hypotheses which arc 

 incompatible with each other; but experiment shews that, though they may not be accurately 

 true, they are ap|)roximately correct and comj)atible : and we claim for the results of our 

 Vol.. VIII. I'aiit III. X X 



