(i ZEILON, ON TIDAL BOUNDARY WAVES. 



small); suppbse the whole bulle of water to be moving to and fro in a simple-harmonic long- 

 wave-motion of given period and amplitude. Find the disturbing effect produced upon this 

 original sta te of things by the raising up, from the hitherto even bottom, of a small ridge stretch- 

 ing uniformly över the whole breadth of the channel. As usual in this kind of problems, 

 the height of that ridge must be supposed small, compared to the depths of the flm ds; in 

 the nature of the problem is also implied that its linear dimensions should be small, com- 

 pared to the wave-length of the tide. 



B. Nolution of the Hydrodynamical Problem in a simple Case. 



Taking the axis of x along the channel, let y - o be the mean position of the surface 

 of separation; y=K and y = -h will be taken to represent the mean position of the frec 

 surface and the bottom plane respectively. The density of the upper fluid may be q 

 and that of the lower one q. 



Denoting by — the tidal period, the corresponding wave-length of the original 



2 T 



tide is " K and determined by the equation: 



£=C=Vg{h + h'), 



vvhere C is the velocity of propagation. This wave-length, being very great compared to all 

 other linear quantities used, we will assume to be practically infinite, i. e. we confine oursel- 

 ves to points x lying not too far away from the disturbing ridge, or, more precisely, we 

 are to consider the resulting motion only at distances from the ridge which are small com- 



pared to "W. Doing so, we may simply represent the tidal motion by a periodic velocity 



along the axis of x: 



u== u' = ce int , 



being by tlie nature of the wave the same for all values of y between —h and h', and its 

 variation with x now also being neglected. This also implies that the vertical motion up 

 and down of the boundary with the tide is left out of account when calculating the dis- 

 turbing effect upon that surface of the presence of the ridge, a supposition which will hold 

 true as long as the amplitude c is not too great. It is thus supposed that, in order to 

 obtain the total motion of the boundary, we have to superpose upon the original tidal 

 motion of it the disturbance caused by the ridge in the idealized case of an infinite wave- 

 length. 



The periodical velocity now introduced is in the usual way derived from the velo- 

 city potentials, identical for the two fluids: 



(p = (fi' == — cxe inl , 



vvhere, as in all our notations, the accentuated letter relätes to the upper fluid. In the 

 first instance the problem is to be discussed under the supposition that c is small enongh 



