KUNGL. sv. VET. AKADEMIENS HANDLINGAR. BAND 47. \:n 4. 7 



for higher powers of it to be neglected. It might seem that this would be a necessary conse- 

 quence of the neglect of the vertical tidal motion; however, as shows the extreme 

 case ff = 0, when the tidal period is very great, the latter supposi tion m;i\ be satisfieH 

 when the former is not. We will return to this point further on. 



For the disturbed motion we assume the velocity-potentials: 1 



7 ' = cé at ( — x + sin kx(a' cosh ky + (i' sinh ky)) 

 (p = ce iot ( — x + sin k x (a cosh k y I (i sinh ky)). 



We shall have to follow the usual method of breaking np the funetion of x representinj: 

 the ridge according to Fourier's theorem, and the assumption above is made to corre- 

 spond to the corrugated bottom: 



y = — h + y cos kx. 



There should then be no motion normal to this surface, so we ought to have: 

 I. fory = — h+*/coskx: 



d<p < ,( i , . fp , 



— -7T- = — -t;- cos (n, x) — t~- cos (n,y) = 0. 

 on Ox (>ii 



Along that corrugated surface: 



cos(n,x) dy , . , 



—7 = =z- = ky sin kx. 



cos (n,y) ax 



Higher powers of the quantities a, [I, a, (i' being always neglected, we thus have: 



ky sin kx + k sin kx ( — « sinh kh + ;! cosh kh) = , 

 or: 



1) y = — a sinh kh ^- ,icosh kh . 



The conditions to be satisfied with respect to the common boundary are: 

 1 1. That of continuous normal motion, or: 



dy dy 

 this simply gives: 



2) § = /. 



II T. The pressure equation: 



p = ]>', for y = 0. 



Assume as the equation of the disturbed boundary: 



1 For the following analysis reference ough< fco be made fco Arts. 2'M) — 24."> of H. Lambs Hydrodynamics, 

 3rd ed. 1906). 



