KUNGL. SV. VKT. AKADEMISKS HANDLINGAR. BAND 47. NO 4. 25 



The boundary-disturbance corresponding to that oscillating ridge, may now be 

 immediately written: 



i=. it/ t/ 



o — * 



CO ' 



( ■*-' / sinn fc« . D»,, + 1 \ a ' J 



2; 



-i 



witb : 



A, - D(fio) — (/ 4 o 4 (ocoth fcAcoth kh' + o') — ira-gkoicoth kh + coth kh') + g- k- (o — o'). 



F. Case of a strong Tidal Current passing across the Bottom-ridge. 



Before discussing this formula, its importance for our original problem is to be noted. 



2iT 



Letting — be the tidal period, impress upon the channel with the whole bulk of water 

 contained in it the velocity along the axis of x: 



— c cos a t. 



This amounts to the replacing, in all the formula?, of x by: 



c . 

 x + sm o t. 

 o 



The result must be shown to satisfy the hydrodynamical conditions. Now there are 

 conditions of three types. The first says that, if 



<p(x),cp'(x) 



are the original velocity potentials, the functions 



C C 



ii (x + sin at), w'(x + sin at) 

 g o 



must still be solutions of 



<)x s "*" ihf ' 



and this, evidently, is true. The second condition, relating to the continuity of normal 

 motion at the three surfaces: bottom, common boundary, and free surface, is also imme- 

 diately seen to be satisfied. Finally, we have the pressure conditions. Taking for exam- 

 ple the common boundary, we denote by u and u the velocities along the axis of x ou 

 both sides of it; we then have: 



K. Sv. Vet. Akad. Handl. Band 47. X:o i. i 



