Theory of Long Waves. 251 



gated in one direction only the complete solution is given, 

 and the general solution for propagation in both directions is 

 obtained in finite terms for a system of different forms of 

 channel. One of these forms of channel is peculiarly 

 interesting in respect of the fact that long waves of any kind 

 are propagated along it without change of form : in all other 

 forms of channel long waves change in form as they advance. 

 For such a channel, which may be called a channel of uniform 

 propagation, the general solution takes a specially simple 

 form, and I have devoted a section to its discussion. 



Perhaps the most important application of the theory is to 

 the explanation of the tidal phenomena of rivers and estuaries. 

 In the treatise on " Tides and Waves " the application was 

 made by Airy with great success. As, however, the results 

 were obtained from approximations only, I have thought it 

 desirable to re-discuss some of them with the aid of the exact 

 solutions. As was to have been expected, they are, so far as 

 they go, confirmed in the main ; but one of them, to which 

 some importance has been attached, is found to be erroneous. 

 I refer to the explanation of what are called double tides, 

 sometimes observed in rivers at stations sufficiently far from 

 the mouth. The splitting up of the wave, however, indicated 

 by Airy's solution, and on which the explanation depended, 

 is only apparent and depends on the fact that the approxi- 

 mation used is not sufficiently exact for stations far up the 

 river : in the exact solution there is no trace of such a 

 division. 



1. The Equations of Motion, 



Consider a canal or horizontal channel of any uniform 

 cross section filled initially to a certain depth with water or 

 other liquid. Suppose that this liquid is now disturbed in 

 such a way that all particles which are initially in any hori- 

 zontal or vertical straight line perpendicular to the length of 

 the channel receive a common horizontal displacement and 

 are then left free with a common velocity parallel to the 

 length of the channel; the displacement and velocity, however, 

 being such that the inclination of the path of any particle to 

 the length of the channel is everywhere very small and the 

 motion continuous. Such a disturbance will in general give 

 rise to a wave-motion in which "long waves" with parallel 

 crests are propagated in both directions along the channel. 

 The nature of this motion we proceed to investigate. 



Taking the axis of x parallel to the length of the channel, 

 along its bottom say, let £ be the abscissa at time t of a plane 

 perpendicular to this axis containing a set of particles which, 



