280 



Theory of the Tides 



% +2,. JK cos9 = - »1(q + ^ , 

 dt RsmftdX \ qJ 



-r + Zvjucosv = — — £ + - 

 <7/ 3z \ £ 



The external force X 7, Z has a potential £? if it can be represented by 



]_8D _J 0£ _a£? 



~R8§> ~ Rsind 8X ' 8z ' 



In these equations the frictional terms have been neglected. If we neglect the 

 vertical acceleration which is small compared to the acceleration of gravity 

 and neglect also the vertical component of the Coriolis force, which is for 

 all practical purposes without importance, the third equation of motion is 

 reduced to the hydrostatic equation. If we put g = 1 and substitute R for 

 the depth R + z assuming z is small in comparison with the earth's radius 

 and, further, if we consider that p ~ gij and Vj = —Q/g the two first equa- 

 tions of motion can be written: 



W 



2covcosd 



i r 



R8& 



8 



J r 2coucos& = — - 

 dt R sm&BX 



{rj-rj) 



(IX. 4) 



To this we have to add the equation of continuity for variable water 

 depth /?, 



By] 1 \ 8hus'm& . 8hv\ 

 dt + Rsmd \ 8& h 81 ] 







If the tide-producing potential is of a periodical nature, u, v and r\ will 

 be also, and we can assume that all these quantities contain the time factor 

 exp \i(at+sX+e)}, where s is an integer. These equations become, if the 

 depth h is only a function of #, and if the boundaries to the sea coincide with 

 parallels of latitude: 



iau — lwv cos& = 



Z 8 , 



v), 



IPS 



iav + 2o)ucos& = — ^ — „ (?? — vj) , 



1 {8hus'm& . . , \ 

 Rsmvy 8& ) 



(IX. 6) 



u and v can be computed as functions of r t from the two first equations and 

 if these values are introduced into the third equation, we obtain a differential 

 equation for r\ alone, whose solution gives the tide waves generated and 



