SECT. 5] 



783 



computation which can be undertaken. To get a uniquely determined solution 

 of tlie problem in all grid points on the boundary, the elevation of sea-level ^ 

 is prescribed. Now in each grid point the differential equation is replaced by an 

 equation of finite differences : 



{x,y)-^-^f{x,y) 



Yyfi^^y) 



f{x + l,y)-f{x-l,y) 

 21 



f{x,y + l)-f{x,y-l) 

 21 



(10) 



^2 ^2 



fix + 1, y) +f{x - 1, y) +f{x, y + l) +f{x, y-l) 



-m^,y)- 



On introducing these terms into the differential equation (7) or (9) a system of 

 linear equations arises. The number of equations is equal to the number of 

 unknown values and in principle there are no mathematical difficulties in 

 solving them. This method is applicable to examples in either one or two 



Fig. 15. Red Sea — range of M2 constituent. (After Defant, 1919.) 

 .... observed xxx calculated 



dimensions. Moreover the computation of tides in a one-dimensional channel 

 can be arranged in such a way as to make it possible to express the values of ^ 

 in the grid points by the boundary values in a direct manner. This has been 

 done by Defant. After this method he investigated a large number of elongated 

 channels and seas, and especially he obtained remarkable agreement between 

 observed and computed tides in areas of very small width. Fig. 15 gives the 

 result for the Red Sea (Defant, 1919). In two-dimensional cases the amount 

 of computing work increases with the number of grid points, and, in addition, 

 it becomes complicated in cases of small values of the determinant of the 

 above-mentioned system of linear equations. As already noted, the choice of the 

 grid-size is determined by the demand that all essential features of the tides in 

 the sea under consideration are taken into account. On the other hand there is 

 an upper limit for the number of grid points caused by the amount of computa- 

 tion work. Normally, iteration processes have been used for solving the equa- 

 tions. This method has been applied to the tides in the North Sea, the Enghsh 



