766 HANSEN [chap. 23 



make it possible to eliminate the time variable. Three partial differential 

 equations with only space derivatives result: 



{-iG + r)u-fv + g— (l-l) = ( 



^_ia + r)v+fu + gy (i-l) = \ (6) 



The next step is to eliminate the components of velocity, u, v. The following is 

 an equation of sea-level 



-[/2 + (r-io-)2]| = 

 



(7) 



A corresponding equation is found for spherical co-ordinates. This partial 

 differential equation is of the second order and of elliptic type. That means, ^ 

 is uniquely determined in the interior of an area if on the boundary | or the 

 derivative of f or a combination of these functions is known. There are 

 numerous methods for solving this type of equation. The solutions depend on 

 the shape and depth of the ocean and on the parameters of the equation. Only 

 in very simple cases is it possible to give the solution in analytical terms of well- 

 known mathematical functions. In principle these solutions have to be ascert- 

 ained for each ocean or sea separately. The amount of work increases with the 

 complication of the area under consideration. For this reason, in the classical 

 theory of tides, the investigations are restricted to geometrically simple basins. 

 These are : an ocean covering the whole earth, oceans bounded by two parallels 

 of latitude and oceans bounded by meridians. In order to get some mathematical 

 simplifications, it was assumed that the depth was a function of latitude and/or 

 longitude. 



Considerable complications in these equations arise from the Coriolis force. 

 (This force also complicates the calculations of the tides in a rectangular basin 

 with constant depth.) Elevations of tides and components of tidal currents, u,v, 

 are only representable by infinite series, as has been shown by Taylor (1920). 

 An analytical solution is only known for a rotating basin bounded by a circle 

 (Lamb, 1932). The elevation $ is represented by Bessel functions. The results 

 of this classical theory do not give information about the tidal oscillation in an 

 actual ocean or sea. It may be possible to approach the solution of this problem 

 by using modern electronic computers, although this method has not yet been 

 applied. There are now a number of methods allowing one to tackle this problem 

 in a more direct manner ; these are not restricted to the simplifications of the 

 classical theory which demand the linearity of the hydrodynamic equations. 



