with a horizontal water particle velocity, uniform over the depth, of 



I ^ 1/2 1 



2 -/gd L (1+ 7?/d)'^^ -1 J 



(D-2) 



These results assume a horizontal bed and negligible vertical accelerations. 

 To the first order of tj / d the results are the same as derived by Airy in 

 his finite annplitude, long wave theory. Evidently, since -n can be both 

 positive or negative according to position in the wave, the crest velocity 

 n^ust exceed that of the trough and the wave must eventually break or form 

 a bore. Eq. (D-1) is the same as Eq. (38), used in the text in the discussion 

 of tidal ingress up the Columbia River. 



It is of interest that if 17/d is small in Eq. (D-1), then 



^ [ 1 + I ( 7y/d) ] 



(D-3) 



and the velocity of progression exceeds that of a solitary wave. This 

 result is given also by Milne -Thomson (I960, p. 418) by a different method; 

 as also by Bouasse (1924, p. 301). 



We may transform Eq. (D-2) to the form 



2Vg^ [(l+d/7? )^/2 - (d/7^ )^^^ ] (D-4) 



and hence conclude that if the wave were to run on to dry ground (for which 

 d tends to zero), then the water velocity would approximate 



- 2ygi7 



(D-5) 



This equation has no very special meaning because of its neglect of 



D-2 



