SECT. 6J TIDES 767 



Within the present Hmitations tidal research received a remarkable stimula- 

 tion from the work that has been done in connection with seiches. This special 

 kind of oscillation in Lake Geneva had been observed and discussed by Forrel. 

 Chrystal (1904) gave a hydrodynamical explanation of this kind of wave, and 

 his theory was in good agreement with observations. Sterneck (1920) and Defant 

 (1919) recognized that a transition and extension of this theory to tidal prob- 

 lems should be possible and successful. To avoid mathematical difficulties, they 

 discussed only elongated channels of variable depth, width and cross-sections. 

 The essential idea of these scientists was not to attempt to find an exact 

 mathematical solution of the problem but instead to concentrate all efforts 

 toward an approximate solution of numerical type. This was done in the 

 following manner. In the equation of motion the derivative with respect to 

 the space variable x was replaced by finite differences : the continuity equation 

 was integrated with respect to the same variable. In effect, therefore, the 

 channel was divided into a number of intervals each with constant depth and 

 width. The numerical computation off and u was possible without any difficul- 

 ties. Defant applied his method to a large number of elongated seas, and the 

 results he obtained are in good agreement with observations. The application 

 of these numerical methods, based on finite differences, appear in principle to 

 be a renunciation of an exact mathematical solution. But in this connection it 

 should be remembered that the hydrodynamical differential equations are 

 derived by similar processes from difference equations (Lamb, 1932). So far 

 the idea of replacing the differential quotients by finite differences seems to be 

 useful and correct. But it should always be kept in mind that, before applying 

 this difference method, it is necessary to prove the degree of approximation and 

 its numerical stability. In this respect there are still uncertainties. It therefore 

 seems desirable to start from the hydrodynamical equations. Collatz (1955) 

 shows by means of simple examples that the transformation of differential 

 equations into difference equations should be done cautiously. 



The difference methods can be applied to general problems ; for example, to 

 those concerned with shallow or stratified waters acted upon by tides, wind 

 stress, or other forces. It can be expected that in the future these methods will 

 be used to a greater extent in problems of practical dynamical oceanography, 

 since modern electronic computers now exist to overcome the large amount of 

 calculation. 



Beside the exact mathematical and the difference methods, there are a 

 number of other proposals to solve the problems of tides in oceans and seas. 

 These problems are usually made easier by linearizing the hydrodynamical 

 equations, but in all cases this is still not sufficient to obtain analytical solutions. 

 Sometimes it is convenient to start with the exact differential equations and 

 at a later stage to continue with numerical methods. Another procedure consists 

 of determining the exact solution for small areas of constant depth and com- 

 bining these into a more general solution for an extended area. Altogether, in 

 the treatment of the tidal problem, there exists a large number of possibilities 

 combining exact and numerical procedures. For example, the characteristic 



