KUNGL. SV. VET. AKADEMIENS HANDLING AK. BAND 47. N:o 4. 27 



x into x + cos /Cx . sin o / , 

 a 



the result will be approximately true but for quantities of the relativt order of smallness 

 • so, denoting by [r /0 | a quantity of the same order of magnitude as »? , you have for 

 the true amplitude: 



c „ . .\ . , cK 



t , c ,, \ ,. , cK 



= ij \x + - cos A x . sin ut\ + | .»,„ ] ■ 



Now [»;„] is a small quantity of the first order, and . another, so their product is to be 



neglected. 



The transformation will of course serve to annihilate (exactly) the oscillating 

 motion of the ridge. Thus we may state: 



When a tidal current of maximum intensity c, small compared lo the velocity of propa- 

 gation C, is running past a fixed ridge, the resulting displacement of the boundary will be 

 found with sufficient approximation by replacing in the solution of the »oscillaiing-ridge 

 problem »: 



x by x + - cos Kx . sinat 



J a 



(or more generally 



x by x + ä , 



3 denoting the horizontal displacement in the tidal wave), and then adding the vertical 

 displacement of that surface, due to the tide when not disturbed by the ridge. 

 Now returning to the formula I of p. 25, we write it: 



00 + 1 



*- - k iJ ^LZZ* m *"■'■'■■ (t) «/«« " ka -*■ 



Assuming, a,s will be verified presently, that this series has a rapid convergency 

 only a number of terms need to be taken into account, small enough to make all the 

 smaller roots K v of the equations 



D v = O 



very small and approximately equal to 



ra 

 r K = 



Vg(h + h') 



These roots will not then, properly speaking, contribute to the boundary-wave. The 

 larger roots again may be denoted by /„, and approximately we have 



X v = VA l = VY., 



