24 ZEJLON, ON TlDAL BOUNDARY-WAVES. 



and there will be the following relations: 



. -B. K -6.x 



A = : , a 



1(7 IG 



Thus: 



The disturbance in the boundary, caused by the tidal wave when checked by a 

 bottom-ridge at x = o, is, to the first power of its maximum current-velocity c, equi va- 

 lent to a certain periodical simple-harmonic impulse exercised from the bottom in the 

 neighbourhood of x = o upon the adjacent particles of the fluid. 



Our second, and more important, application will be to assume for the bottom: 



y = — h + f(x smot), 



f being still a function of the usual kind. This represents a ridge of constant configuration, 

 oscillating about the point x = o, its maximum distance from there being on each side - 



Its velocity of movement is equal to: 



c cos at. 



According to Fourier's theorem 1 :. 



o 



00 00 



fix — sinff£) = i dk f f(l) coskil — x + -sinat) di 



\ o ! 



-oo 



Now, from the known formula 1 : 



e t*cosea = 22i m J m (x) cosma), 



where J m are the successive Bessel functions of the first kind, you have: 



[kc\ 



liv 



= 



oo / 7 v 



2 sin k{Ä — x) 2 ^«.- + i —I sin (2 v + \)at 



M — 



lkc\ 



)x / 7 v 



= 2 cos k (A — a*) 2 J-2v cos ~ vai 



= 2 cos k (A - x) ^ J, „ (^ \ e 2 " ' "' 



+ 2 i sin k{l — x) 2 J> v + i (y ) e (a ' + > >»'<" 



In the first of these two sums is included the term independent of /: 



tkc\ 



2coak(X — x).J \^\ 



which of course will cause no wave-motion. 



1 Por convenience 2 ■/.. is written instead of i li«- </„ of the usual notation 



