KUNGL. SV. VKT. AKADKMIKNS HANDLINGAR. BAND 47. Vo 4. I :i 



o-(y(coth kh t coth kh') — gk (q — q') = 0. 



This might seem a little dubious when o is very small, since then also x will be a \<i\ 

 small number; it will be seen further 011 that, with thc small a proper to (lic t ides, the ap- 

 proximation will still be fairly good. It is however necessary that 



Q- Q' 



should be small. 



These remarks will allow us an important conelnsion. The ratio: 



a 

 K 



gives us exactly that wave-velocity of the original long-wave motion assumed ironi the 

 beginning to be a practically infinite number. Again, for the second root, from the pro- 

 perties of the function &, it follows that 



0(x) 



is a very small quantity, so that, corresponding to /., there ivill be no sensible disturbance 

 of the upper surface. In accordance with this, the equation determining /. is the same as 

 should have been obtained, if from the beginning we had supposed that surface not to be 

 sensibly disturbed by the ridge. Indeed, for a small difference of density, that equation is 

 the known one giving the wave-length of free waves in the boundary of two fluids enclosed be- 



liveen rigid watts, for the period — 



The physical reason of the occurrence of the waves in formulae II: 1 — 4, corresponding 

 to the root K, must be a twofold one. They partly represent the influence that the 

 presence of the ridge would have upon the long wave, if the water were homogeneous, 

 and partly they indicate the decay of the upper-surface-elevation, due to the energy subtracted 

 for the ?naintenance of the boundary-waves corresponding to the root -/. Since the assump- 

 tion of an infinite wave-length involves an infinite energy, the effective calculation of 

 that consumption of energy would perhaps seem somewhat illusory; however, we need 

 only note the fact that the boundary-waves represent energy, and that this energy must 

 be supplied by the original long wave. 



C. Application to the Tides. 



The preceding investigation, so far, is the purely qualitative solution of a hydrody- 

 namical problem, but, before claiming a real physical value for it, it must be shown that 

 the effects predicted will be of a sensible magnitude. The drawings of Fig. 2, to be 

 discussed presently, will, as based upon effective calculation, show this as far as concerns 

 conditions of laboratory. No doubt, from the fact that the theory would give boim- 



2 71 



dary-waves of an infinite amphtude for a bottom-corrugation of the wave-length — , 

 it might be inferred that by suitable bottom arrangements (making use, if necessary, of 



