KUNOL. SV. VET. AKADEMIENS HANDLINGAR. BAND 47. V<> 4. 6 



static pressure below the free surface. The channel nexi should be filled wifch non-homo- 

 geneous water; \ve assumc as the simplesl possible case thal it contains a bottom-layer 

 of salt-water and an upper layer of fresh-water, separated ironi each o1 her by a horizonl .1! 

 surface of discontinuity. Now, with that arrangement, there is still a long-wave-molion 

 possible, with a pressure at any point equal to the statical one, the whole motion, for a 

 small difference of density between the two fluids, differing very little from the case of 

 homogeneous water. The vertical variation of velocity will still be infinitesimal of the 

 same order as before, and there will be no pressures, however small, tending to deform 

 the common bonndary in any abnormal way. In nature, as a rule, the boundary surface 

 is inclined to the hori zontal by some small angle, but the same conclusion will still be seen 

 to hold, viz. that surface will continue to move with the tide in very much the same way 

 as if being an imaginary surface in homogeneous water. 



In order to obtain any considerable disturbance of the boundary, there must be in- 

 troduced some hindrance to the free progression of the tide. Now, as a *tidal boundary - 

 wave» is to be reckoned such a Avave, of the tidal period of course, which at the boundary 

 has a much greater amplitude than at the free surface. Such a wave, according to 

 Stokes' formulas, has a small wave-length, compared to that of the ordinary tide. 

 From general principles it follows that the hindrance introduced, if developed in a Fourier 

 series, must contain terms of wave-lengths very much smaller than the wave-length of 

 the impinging tide. Such a hindrance will then effect a vertical variation of velocity 

 which, even though very small in itself, yet is very great, compared to the infinitesimal one 

 proper to the primary tide. As possible mechanical arrangement s are to be mentioned: 

 1. the imposing of a local pressure upon the free surface; 2. the disturbing of the even level 

 of the bottom by ridges stretching across the channel. 



The latter arrangement is certainly the most important, and the one to be studied 

 here. But it is also very probable that, forex ample, an area of unusually high barometric 

 pressure may act in the same way. Both problems, however, allow nearly quite the same 

 treatment; accordingly that of the disturbing bottom ridge may be treated as the typical 

 one. 



We thus arrive at the arrangement deviced b}^ Nansen & Helland-Hansen as a 

 possible instance of tidal boundary-waves. 



The case of a local surface pressure perhaps possesses its chief interest in the extreme 

 case that the long-wave motion has an infinite period, that is, degrades into a constant 

 stream passing the area of depression. Then are obtained, if the pressure is caused by a 

 current of air impinging upon the surface, the waves found by J. W. Sandström. 1 If 

 again the forcive consists in a floating vessel, the resulting wave-motion will be equiva- 

 lent to that causing the phenomenon of »dead- water ». 2 



The problem of the disturbing bottom ridge is defined as follows: Given a 

 rectangular channel, filled to a certain height with water of given density, and containing above 

 that a layer of water of smaller specific gravity (the difference of density being assumedto be 



1 Dynamische Versuche mit Meerwasser, Annalen fur Hydrographie und Maritime Meteorologie, 1908, Hcft I. 



2 The theory of which is given and experimentally illustrated by V. W. Ekman: On dead-water, Scientific 

 results of the Norwegian North Polar Expedition 1893 — 1896, XV. 



