16 ZE1LON, ON TIDAL BOUNDARY-WAVES. 



/. The tidal wave, travelling in a channel containing two layers of ivater of differ- 

 ent densities, when passing across an elevation of ihe bottom of the channel, will cause 

 waves to be generated in the common boundary of the fluids; 



II. These waves will diverge from above ihe ridge in the form of two infinite trains 

 of simple-harmonic waves (provided, of course, the tide is itself si)n ple-harmonic) travell- 

 ing with a constant velocity, 1 one to the right, the othcr to the left of the ridge; 



III. The »boundai-y-tide» will have the tidal period and the wave-length (and velo- 

 city of propagation) proper to free boundary-waves of that period; 



IV. At the same time there will only be a relatively insensible wave of that wave- 

 length in the upper surface, but the »surface tide» will suffer a diminntion of amplitvde 

 corresponding to the energy consumed to maintain the boundary -ti de. 



D. (Jhai-jicter of the Wave-inotioii in the Neighbourhood of the Ridge. 



The statements above, though containing the chief results of this paper, say nothing 

 about the manner in which the waves develop themselves from above the ridge. That 

 study will be greatly facilitated by exténding the disregard of the upper-surface distur- 

 bance to be possible, not only as far as concerns the diverging wave, but also for contri- 

 butions to the first integral in I bis p. 10 from imaginary roots of the determinant. This 

 will amoimt to the neglect, throughout the integration, of 



O" 



k' 



and will give us: 



co()e iat C kdk 



+00 



with 



■ge 1 " 1 ! fca/c 1 .... .... , 7 . 



t J S inhkh.j Jf {X)elW - X)dX 



J = o- 2 o(coth kh + coth kti) — gk(o — o'). 



It is immediately seen that the correctness of the omission made will be assured, if 

 there are no roots, real or imaginary, of: 



.7 = 

 having so small an absolute value that for these roots 



O" 2 



k 

 could not be assumed to have a very small absolute value. This, however, will be 

 seen not to be the case. 



Although some of the conclusions derived in this chapter would hold for any 

 symmetrical ridge, I prefer to introduce a special configuration of it. I take: 



1 The constancy of the volocity of propagation essentially depends <>n bhe asenmption <>!' a small '•. 



