786 



[chap. 23 



had to solve a system of more than 300 hnear equations. This was done by 

 inverting the matrix of this system. The result seems to be in good agreement 

 with tidal observations. 



Another computation was made by Gohin in 1960 for the North Atlantic. He 

 found remarkable agreement between observations and computation. 



Fig. 17. Height of sea-level in the Nieuwe Waterweg at Kruiteland. (After Dronkers, 

 1935, Fig. 2.) Full curve after observations; broken curve after computations. 



7. The Application of DifFerence Methods to Initial -Boundary Problems 



Recently finite difference methods have also been applied to non-linear 

 hydrodynamic differential equations in one and two dimensions. By generaliza- 

 tion of Defant's idea, all derivatives of space and time variables in the system (3) 

 are replaced by finite differences. 



If for a certain time t — to the elevation ^, the components of current velocity 

 u, V and the external forces are known, the equations (3) allow the time deriva- 

 tives of I, u, V to be determined. This means it is possible to have at least 

 approximate information about these values at a following time t + r, where t 

 is sufficiently small. In this way the variations off, u, v are determined step by 

 step for all time. It must be realized that this process is only a formal one, and 

 it must be ascertained that this method becomes convergent ; in other words, 

 the numerical computation must be stable. 



This problem differs from the task discussed above (the boundary-value 

 problem) : 



1. The system of differential equations is now of the hyperbolic type. In the 

 former case, ^ was the solution of a differential equation of elliptic type. 



2. No time factor is introduced; functions are not restricted to simple 

 harmonic terms but an arbitrary time dependence is allowed : the same is valid 

 for the boundary values. 



