239] WAVES IN UNIFORM CANAL: GENERAL THEORY. 431 



mined as above, gives the velocity-potential of a possible system of 

 standing waves, of arbitrary wave-length 2-7T/&, in an unlimited 

 canal of the given form of section. Now, as explained in Art. 218, 

 by superposition of two properly adjusted systems of standing 

 waves of this type we can build up a system of progressive waves 



&amp;lt;f) = P l cos (kas + at) (9). 



Hence, contrary to what has been sometimes asserted, progressive 

 waves of simple-harmonic profile, of any assigned wave-length, are 

 possible in an infinitely long canal of any uniform section. 



We might go further, and assert the possibility of an infinite 

 number of types, of any given wave-length, with wave-velocities 

 ranging from a certain lowest value to infinity. The types, how 

 ever, in which there are longitudinal nodes at a distance from the 

 sides are from the present point of view of subordinate interest. 



Two extreme cases call for special notice, viz. where the wave 

 length is very great or very small compared with the dimensions 

 of the transverse section. 



The most interesting types of the former class have no longi 

 tudinal nodes, and are covered by the general theory of long 

 waves given in Arts. 166, 167. The only additional information 

 we can look for is as to the shapes of the wave-ridges in the 

 direction transverse to the canal. 



In the case of relatively short waves, the most important type 

 is one in which the ridges extend across the canal with gradually 

 varying height, and the wave-velocity is that of free waves on 

 deep-water as given by Art. 218 (6). 



There is another type of short waves which may present itself 

 when the banks are inclined, and which we may distinguish by 

 the name of edge-waves, since the amplitude diminishes expo 

 nentially as the distance from the bank increases. In fact, if the 

 amplitude at the edges be within the limits imposed by our 

 approximations, it will become altogether insensible at a distance 

 whose projection on the slope exceeds a wave-length. The wave- 

 velocity is less than that of waves of the same length on deep 

 water. It does not appear that the type of motion here referred 

 to is of any physical importance. 



