KUNGL. SV. VET. AKADKMIKNS HANDLINGAR. BAND 47. V<> 4. 31 



G. Genera] Gonclnsions for Gases nof covered by the Theory. 



Supposing that analysis fails to give a quantitative account, it will be indispensable 

 to-settle, as far as possible, to what extent the conditions of the theory are essential to the 

 qualitative appearance of the phenomena. Fortunatcly this may be done with a good 

 deal of safety. The physical reason of the boundary being disturbed, of course, is that 

 the bottom water is forcedly heaved upwards when running up the hill, and this will 

 occur whenever the tide is crossing the obstaele. Assume then a couple of imaginary 

 cross-sectional walls, enclosing the whole of the barrier; you will have to the right of the 

 right wall a rectangular channel of throughout uniform depth, and the same to the left of 

 the left wall. Regarding now, for example, the boundary surface at the left end of the 

 channel to the right, that surface is there, to a certain length, because of the periodic 

 upheavals of bottom-water, forced to move up and down. Suppose now the ridge to be 

 too high for the preceding theory to be applicable; it will still be assumedlow enough for 

 a considerable portion at least of fresh-water to pass freely across it. Then we shall 

 still, to the left of the right channel, have such a forced motion of the boundary; only that 

 with a very high ridge it may be far mo reintense than before. For all points at a suffi- 

 cient distance from that left end, the boundary, however, may move with its proper- free 

 motion. It will thus resemble, say a string, one end of which is subject to a forced periodic 

 motion. The natural consequence is then known to be the propagation outwards from 

 the point of disturbance of waves of that period, and the wave-length determined by the 

 physical properties of the string. This is not a mere illustration. In fact, for all values 

 o small enough to make the corresponding wave-lengths of boundary waves very long 

 compared to the depths of the two fluids, there is no sensible dispersion, and the immediate 

 consequence is that for a small o, i. e. a long period of disturbing force, the equation 

 of motion of the boundary will be: 



c denoting the constant velocity of propagation of long boundary-waves, and cp the dis- 

 turbing force. If <( is everywhere zero except at the left end of the channel, the problem 

 will be to solve: 



dt- dx 9 - 



with 



and gives the solution 



\ = «/■' ( — /) for, say % = a , 



, x — a 



C 



Since the function ip will certainly, at any råte, be a periodic, w r hen not simple- 

 harmonic, function of time, there will be, for the right channel, an infinite train of boun- 



