8 REPORT—1846. 
I have had occasion to refer to what Mr. Airy calls a standing oscillation 
or standing wave. To prevent the possibility of confusion, it may be well 
to observe that Mr. Airy uses the term in a totally different sense from Mr, 
Russell. The standing wave of Mr. Airy is the oscillation which would re- 
sult from the coexistence of two series of progressive waves, which are equal 
in every respect, but are propagated in opposite directions. With respect to 
the standing wave of Mr. Russell, it cannot be supposed that the elevations 
observed in mountain streams can well be made the subject of mathematical 
calculation. Nevertheless in so far as the: motion can be calculated, by 
‘taking a simple case, the theory does not differ from that of waves of other 
classes. For if we only suppose a velocity equal and opposite to that of the 
stream impressed both on the fluid and on the stone at the bottom which 
produces the disturbance, we pass to the case of a forced wave produced in 
still water by a solid dragged through it. There is indeed one respect in 
which the theory of these standing waves offers a peculiarity, which is, that 
the velocity of a current is different at different depths. But the theory of 
such motions is one of great complexity and very little interest. 
Theory of Solitary Waves.—It has been already remarked that the length 
of the solitary wave of Mr. Russell is considerable compared with the depth 
of the fluid. Consequently we might expect that the theory of long waves 
would explain the main phenomena of solitary waves. Accordingly it is 
found by experiment that the velocity of propagation of a solitary wave in a 
rectangular canal is that given by the formula of Lagrange, the height of the 
wave being very small, or that given by Prof. Kelland’s formula when the 
canal is not rectangular. Moreover, the laws of the motion of a solitary 
wave, deduced by Mr. Green from the theory of long waves, agree with the 
observations of Mr. Russell. Thus Mr.Green found, supposing the canal 
rectangular, that the particles in a vertical plane perpendicular to the length 
of the canal remain in a vertical plane; that the particles begin to move 
when the wave reaches them, remain in motion while the wave is passing 
over them, and are finally deposited in new positions; that they move in 
the direction of propagation of the wave, or in the contrary direction, ac- 
cording as the wave consists of an elevation or a depression*. But when we 
attempt to introduce into our calculations the finite length of the wave, the 
problem becomes one of great difficulty. Attempts have indeed been made 
to solve it by the introduction of discontinuous functions. But whenever 
such functions are introduced, there are certain conditions of continuity to 
be satisfied at the common surface of two portions of fluid to which different 
analytical expressions apply; and should these conditions be violated, the 
solution will be as much in fault as it would be if the fluid were made to 
penetrate the bottom of the canal. No doubt, the theory is contained, to a 
first approximation, in the formule of MM. Poisson and Cauchy; but as it 
happens the obtaining of these formule is comparatively easy, their discus- 
sion forms the principal difficulty. When the height of the wave is not very 
small, so that it is necessary to proceed to a second approximation, the theory 
of long waves no longer gives a velocity of propagation agreeing with expe- 
riment. It follows, in fact, from the investigations of Mr. Airy, that the velo- 
city of propagation of a long wave is, to a second approximation, WV g(h+38h), 
where # is the depth of the fluid when it is in equilibrium, and 4+£ the 
height of the crest of the wave above the bottom of the canalf. 
* Transactions of the Cambridge Philosophical Society, vol. vii. p. 87. 
+ Tides and Waves, art. 208. In applying this formula to a solitary wave, it is necessary 
to take for h the depth of the undisturbed portion of the fluid. In the treatise of Mr. Airy 
the formula is obtained for a particular law of disturbance, but the same formula would have 
