44 



ZEILON, ON TIDAL BOUNDARY-WAVES. 



Now, to pass to the case of a fiiiite, though very great, wave-length will not pro ve 

 very difficult. The functions <p and ip will now also be functions of x, which may at any 

 råte be developed in certain Fourier series of the argument Kx. 



The same will also be the case for the functions cp and ip entering in the expression 

 of the potentials. It will be convenient for us to introduce two functions which we 

 denote by 



(pcosKx and tp cos K x, 



and which are defined by certain trigonometric series: 



cpcos Kx = a + a { cos Kx + a\ sin Kx + a 2 cos 2Kx + a\ sin 2Kx + • • • 

 tp cos Kx = b + 6, cos Kx + b\ sin Kx + b 2 cos 2Kx + b' 2 sin 2Kx + ■■■ 



Now, departing from the undisturbed tidal velocity potential 



O 0' t smKx . „ 



- = — = — cos o t . — =: — . cosn A y , 

 c c K J ' 



(where of course cosh Ky may be put equal to 1), we construct for the upper fluid: 



m< l sinKx , „ (3 cosh (k + K) {y — ti) . 

 u A=cosot.[- —g-.coshKy — ^- „ „ : „u j.j.r 



2 sinh kti~ 



r— 

 v— 



cosh(k — vK){y — ti) 



x + 



sinh kti 



cos (k — v K) x\ + etc. 



Now suppose that the Fourier developements of the functions cp cos Kx and 

 ip cos Kx are converging rapidly enough, so that only terms of orders v, so small that 



v K 



are still very small numbers, need be retained. Then we may write: 



O' 



— =COStf£ 



c 



sinÄ^a; (i cosh k (y — ti) 



K 



sinh kti 



sin kx.cosKx) 



cosh k (y — ti) 

 sinh kti 



. cos Kx [tp sin kx — ip cos kx) 



and of course for the lower fluid the velocity potential will be similarly transformed. If, 

 however, the trigonometrical series obtained for tp and xp should prove badly convergent, 

 the abbreviated expressions for O and O' will anyhow be valid for small values of y, thus 

 especially also for y = rj. 



The expression for 0' and the analogous one for O, whether everywhere to be sim- 



