782 



HANSEN 



[chap. 23 



and, for two dimensions, equation (7). These differential equations are of the 

 second order. This means that | is uniquely determined in an area G (Fig. 14) 

 if the value of ^ is prescribed on the boundary C or Ci and C2. From this the 

 possibility arises of computing the tides in a channel, closed at both sides, or 

 in a landlocked ocean or sea without any aid from observations. The boundary 

 condition requires the normal component of velocity at the boundary to be 

 zero ; or, expressed in another way, the direction of current must be parallel to 

 the direction of the coastline. In this sense, only the ocean, lakes and a few 

 adjacent seas, as for instance the Baltic, are landlocked. In all other cases a 

 sea has some communication with neighbouring seas, so that it is necessary to 

 set up artificial boundary lines C2 as shown in Fig. 5. Now the tides are uniquely 

 determined within the area O by prescribing the sea-level | on C2, while the 

 normal component of velocity must be zero on Ci. But in this connection 

 difficulties arise, as there is a lack of observations in the open sea, and, there- 

 fore, it is highly desirable to record the elevation of sea-level in such areas. 



(a) 



(b) 



Fig. 14. Scheme of (a) landlocked ocean: C boundary, O open ocean; and (b) ocean partly 

 bounded by coastal boundary C\ and partly limited by an artificial boundary C^- 



Tackling the tidal problem in such cases, it becomes necessary to estimate 

 these values on the artificial boundary C2. Sometimes it is possible to get in- 

 formation to some extent by evaluating observations of tidal currents, as, for 

 instance, in the northern entrance to the North Sea. 



A number of mathematical theories exists for solving the above mentioned 

 systems of differential equations (Courant and Hilbert, 1924). A numerical 

 method is briefiy described below ; this is a generalization of Defant's method. 

 Problems of this type are sometimes named boundary- value problems. 



As the computation in elongated channels gives rise to no difficulty, it seems 

 to be sufficient to discuss only the two-dimensional problem. The sea G under 

 consideration (Fig. 5) is covered by a grid with mesh-size I. The boundary of 

 this grid should approximate the boundary Ci of the sea as accurately as 

 possible. On the other hand the number of grid points increases with decreasing 

 mesh-size I. The final choice of mesh-size I should consequently be made 

 taking into account the desired degree of approximation and the amount of 



