22 ZEILON, ON TIDAL BOUNDARY-WAVES. 



giving a wave-length of some 12 cm. The amplitude of the progressive wave was by 

 calculation found to be: 



2a = 0,030 c 



with a maximum elevation of the ridge equal to 



JJ 



I n ' 



In the figures, the vertical scale is so chosen as to make this amplitude about 3 /io 

 of the height of the ridge, thus the curves would correspond to 



0,3 2 



C ' ==Z ' ' 



0,0 3 1 ' n 



giving a maximum current veloeity of 11,3 cm. This of course, apart from the chosen 

 dimensions of the ridge, is considerably more than could be tolerated by the present theory. 



E. Boundary-waves caused by a moving Ridge- 



The preceding theory has two drawbacks, which may prove serious. One refers 

 to the supposition of a very flat ridge, and is quantitatively not to be mastered by analysis 

 at present, whereas the assumption of a feeble tidal current is at any råte formally to be 

 dispensed with. I prefer, here, not to discuss this problem directly but to attach it to 

 the following investigation, which also may be useful for other purposes. 



Imagine the same channel as before, the circumstances and notations being just 

 the same but for there beiyig now no tide in it. The water being initially held at rest, find 

 the motion resulting from a disturbance of the bottom level, such as may be realised by 

 making the bottom consist of an elastic material, and pushing it up and down from below 

 at certain points, so as to form alternately small elevations and depressions of it. 



The first step will naturally be to suppose the equation of the bottom to be: 



y = — h + y cos kx . e inl ; 



then, to the first order of magnitude, this surfaoe will be nioving with the normal veloeity: 



i)y , . , 



^r = iov cos kx . e lot . 

 dt 



Departing from the veloeity-potentials: 



tp' = (a'coshky + ftsinhky) cos kx .e int 

 (/< = (a cosh ky + (i sinhky) cos kx . e inl , 



there is then, for y = — h, the condition to be satisfied: 



dy dij 



Ft dy' 



