232 Long Waves in Canals and Standing Waves in Closed Basins 



increase as the range of tide increases, and the effect of friction is to make 

 the surge decrease as the range of tide increases. 



The same problem was tackled by Doodson in a similar manner. He used 

 for a model a long uniform gulf about 100 miles long and about 21 fathoms 

 deep. He had also taken account of the non-linear terms in the equations 

 of motion, which introduces some mathematical difficulties. The numerical 

 methods have been applied to cases where storm surges have been super- 

 posed upon the tides, with the maximum of the surge occurring near high 

 water, or near low water, or near one of the two half-tide levels. All four 

 cases demonstrate that the character and magnitude of the interaction are 

 not clear, but it is shown that the apparent surge is dependent upon the coef- 

 ficient of friction and that the magnitude of this is greatly affected by tides. 

 The actual apparent surge at the mouth of the gulf is affected by the reflexion 

 which takes place at the head of the gulf and thus the correlations of apparent 

 surge with meteorological data are complicated by the reflected oscillation. 



In recent times, numerical methods have been used for the solution of 

 the system comprising the equations of motion and the equation of continuity, 

 taking full account of lateral and vertical friction as well as of external forces 

 at the water surface (atmospheric pressure and wind), in order to investigate 

 surges in marginal and adjacent seas and to find their causes. Hansen 

 (1956) has given a procedure that is based upon boundary and initial values. 

 The differential quotients in the direction of the x- and v- co-ordinates ap- 

 pearing in the equations are approximated by quotients of differences de- 

 termined according to a grid with a width of mesh /. In this way, a system 

 of common differential equations of time differential quotients is formed, 

 which are all linear. This means: the time variation of the velocity compo- 

 nents and the water levels in the grid-points are known if the function 

 values are given. Substituting the time derivation by finite values, it is 

 possible to compute numerically the function values for a subsequent time 

 from the given function values at the initial time. 



In the application of this method to the water level in the North Sea at 

 the time of the catastrophic surge on 31 January/ 1 February 1953 (Holland 

 Storm) the North Sea grid, given in Fig. 98a, was used. The meteorological 

 observations were taken from weather maps prepared at 3-hourly intervals. 

 The following boundary conditions were chosen: The water level remains 

 undisturbed at the northern limit of the area, running from Scotland to 

 Norway. The normal component of velocity vanishes along the coasts. This 

 holds also for the entrance to the Skagerrak, which is not treated here. The 

 flow through the Dover Straits is assumed to be proportional to the water level. 

 We wish to point out that no other observations of water level or of water trans- 

 port were used in the computation. This computation of water levels and mean 

 velocities for the entire duration of the surge was done by the electronic com- 

 puter BESK. Figure 98 a gives the isolines of water level on 31 January 1953 



