250 Mr. J. McCowan on the 



which is almost an axiom with some glacialists*. In short, I 

 have not been able to find any thoroughly satisfactory solu- 

 tion of the difficulty, — a difficulty which is not without 

 precedent elsewhere f. For the present I must be content to 

 state it, and let it remain among the riddles in which the 

 Glacial Epoch is so fruitful. 



XXY. On the Theory of Long Waves and its Application to 

 the Tidal Phenomena of Rivers and Estuaries. By J. 

 Mc Cowan, M.A., B.Sc, Assistant Lecturer on Natural 

 Philosophy, University College, Dundee J. 



THE theory of the long wave has received considerable 

 attention since the time of Lagrange, who obtained the 

 well-known first approximation for its velocity in a channel 

 of uniform rectangular cross section §. In 1839 Green || 

 obtained the corresponding result for a channel of triangular 

 section, and KellandU that for a channel of any uniform 

 section. So far, however, only the first approximation had 

 been discussed. Sir George B. Airy, in his treatise on 

 " Tides and Waves " in the Encyclopaedia Metropolitana 

 (1845), gave for the first time, so far as I am aware, the 

 exact equation on which the theory of the long wave in a 

 channel of uniform rectangular section depends : he gave no 

 general solutions, but discussed certain problems very fully 

 by methods of approximation**. I do not know that any 

 advance on Airy's treatment has since been published. The 

 subject is briefly discussed in Prof. G. H. Darwin's article 

 on " Tides " in the Encyclopaedia Britannica (ninth edition), 

 but only an abstract of Airy's results is given. 



In the following paper I have sought to give a fairly 

 complete discussion of the motion of long waves in a channel 

 of any uniform section, but I have not sought to consider 

 channels of varying section. For the case of waves propa- 



* It must be remembered that here we are among- the mountains in a 

 locality where the excavating powers of the glaciers should be at a 

 maximum ; so the difficulty cannot be eluded by the analogy of a stream 

 which will excavate where the descent is rapid and deposit where this is 

 gentle. 



f Quart. Journ. Geol. Soc. 1885, p. 514. 



X Communicated by the Author. 



§ Berlin Memoirs, 1786. 



|| Trans. Oamb. Phil. Soc. vol. vii. 



51 Trans. Roy. Soc. Edinb. vol. xiv. 



** An interesting account of Airy's results is given by Stokes in his 

 "Report on Recent Researches in Hydrodynamics," B. A. Reports, 

 1846. 



