KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 47. N:o 4. 10 



2a i j , 2 ... a * 2 



V/rJ <'■ 



for x = o, to the double amount for x oo ; 



II. a wave converging towards x = o from the right, of amplitude diminishing from: 



o 



2a /' ,,,, ... 



e _ " ' • cosxAdA 



I '.' 



— ex 



for .r = o, to zero for x oc ; 



III. a standing oscillation, simple-harmonie in t i me and of a phase differing by - 



from that of the velocity of the tidal current, and falling off rapidly, as a function of x, 

 with increasing vallies of that variable. 



Since for x = o the amplitudes of the diverging and converging waves will be equal, 

 the motion on the whole may be briefly described as being for small values of x a standing 

 oscillation, having a node in the origiri, and gradually developing with increasing values of x 

 into a 'progressive wave. 



It is to be remembered, if actually construeting the wave-profile, that from the 

 expressions of the velocity-potentials it follows that a positive tidal current is directed 

 towards increasing values of x, and that this positive current attains its maximum at 

 the epoch zero. 



The configiiration of the ridge here introduced is specially adapted for numerical 

 calculation, because all coefficients may be reduced to tabulated functions. The first 

 step will be to find the real root /, say by successive approximation. If the ridge is sup- 



2ir 

 posed to be sensible only for a certain, not too great, portion of the wave-length -7-, 



we may put: 



■T X 



I g- " 2 >? cos x Xd X = I er " 2 ; - 2 d X . 

 6 ö 



Since /, anyhow, is no very great number, the term 



2« , C !» • , ,- 



— — sin a t . sin /. x \ er " ; " sin / Kd /, 



v« J 



X 



will be found to be rather insensible, compared to the standing wave contributed by 

 the imaginary roots; for that wave again we use the identities: 



I e~ " 3; - 2 er l vU -*)dX = - év x e^" 2 1 e~ " 2 du 



x l v 



ax ~r a 



01-, 



v 2a 



j e-"- 1 ' 1 é^'- x Hl= é~ l v*e ia2 J e-^du 



