42 



t hat is 

 dtp 



ZEILON, ON TIDAL BOUNDARY-WAVES. 



dtp 



dt 

 or 



coskx + -,,- . sinÄ;x , = cfccos at(cp — (i) sin kx — ek cos at . tp cos kx — cktp . sinkx + ek tp cos kx 



= — ek cos at . ip + cktp 



dep 

 Ht 



-rr = ck cos at(cp — j3)~cktp. 



Hence, resolving and differentiating, we obtain: 



ld<[ dtl> 



cU> = c cos a t . tp + t -3T , c ~ 



k dt dt 



, dtp ld 2 (p 

 eosmot.tp + e cos ut. d - t + ^ 



Q 



± , a . Idtp dep ,, ,. J a> 1 d 2 tp 



C<p — C COS (Ti (f/> — /?) jT-TT , C-iy = — Cff Sl« (/£((/> j3) + C COS (J^. -TT T-jT^" - 



Introducing this in the equations above, we finally have: 



Di(cp,tp) = '/^(^(o — g') — kc 2 a cos 2 a t) + ' ca a sin ot .tp + 2cacosot . tp' = fcc 2 cos 2 a i . 6 

 -^2 (9>> t / ; ) = V ig (q — ?') — kc 2 a cos 2 at) + —7; — \- caa sin ai . (p — 2ca cos ai . tp' = — ca sin at . &, 



where the accents denote differentiation with respect to time, as a system of differential 

 equations to be satisfied by <p and tp, and determining the wave-profile as a f unction of time. 



It is easily verified that the solution of chapter B, part I, is included here, if c is 

 assumed to be small. An interesting special case, easily studied, is also that of a = 0. 



The general treatment of the equations, however, will obviously prove rather labo- 

 rious, and if developed in a series, the solution would obtain a very complicated aspect. 

 It may suffice to verify directly our previous solution, exposed in the text. 



For the oscillating ridge we had : 



00 00 



-1 v Cv'-a 2 p{qk — v 2 a 2 coth kh') . . , lkc\ 7 , i ,,,. .,,. .,, 

 ' '' 'T J sinn kh . D v \a I J 



Replacing here 



x by x + sin at, 

 a 



and neglecting the upper surface disturbance, it follows that the solution for the fixed 

 ridge may be written: 



00 qo 00 00 



»; = ' [ - I '' (,) dk i /(/.) cos k(l — x)dX + - ( ^dk ff(k) sin k (Å - x)dl. 



