26 ZEILON, ON TIDAL BOUNDARY-WAVBS. 



ti = — C COS at 



u = — ccoso^ — 



<)if \x + sinat\ 



dx 



(Q 

 x + sintra 



Ox 



Forming now the general pressure expression: 



V , d<p 1 , 



(to denoting the resultant of the fluid velocity) for both sides of the boundary, we need, 

 on equating p to p, only retain terms depending on fp or <p, since the periodic velocity 

 impressed upon the system is assumed to satisfy separately all hydrodynamical condi- 

 tions. We thus obtain: 



d(p[x + - sinat) d fp' ix + -sinat) 



1 .. , (>< l 1 r ^ , 0( P' 



— - g . 2 c cos at . ^ + - a . 2c cos at. -J- 



2 S ox 2 > dx 



-2%) +2 Ml,)' 



It might be remarked that the last quantities are everywhere small; consequently, 

 since 



dtp [x + -sin an a 



\ o dip dtp 



, A — - = -77 + ccosa^.^-. 

 dt dt dx 



we have once more the pressure-condition of the preceding chapter already used to de- 

 termine r io (x). 



As far as concerns points not far away from the ridge, we then by this operation 

 obtain the true solution, provided that the influence of the vertical motion of the boun- 

 dary may be neglected. This will be possible, if the vertical amplitude of the surface-tide 

 is small compared to the total depth of the channel, and from known laws of long-wave 

 motion this is the case, if the current velocity is small compared to the velocity of propaga- 

 tion C of the tide. 



At considerable distances from the ridge, the propagation of the boundary waves 

 is necessarily influenced by the wave-character of the tide. Tf, for simplicity, the tide 

 be a standing wave, we shall have the current velocity 



— c cos K x . cos at. 



ft is then easily proved that, when executing the transformation of 



