784 HANSEN [chap. 23 



Channel, the Irish Sea and the North Atlantic Ocean. Holsters (1959) used this 

 method to obtain information in Belgian coastal areas. Rossiter (1958) con- 

 tinued the work of Proudman and Doodson in connection with oceans bounded 

 by parallels of latitude and longitude using methods of finite differences. In 

 this special case he pointed out : "Thus the distribution of a tidal constituent 

 is obtained without the use of observations from a knowledge of the tide 

 generating forces only. The method may be extended to oceans with boundaries 

 of irregular shape." In all other cases, apart from this more theoretical task 

 treated by Rossiter, the magnitude of the M2 constituent is prescribed on the 

 boundary, partly based on coastal observations and partly on estimated values 

 on the artificial boundary. To obtain some idea of the accuracy of these methods 

 applied to actual seas, it is necessary to compare theoretical values and ob- 

 served data. This is possible to a certain degree in the North Sea (Hansen, 

 1952) and also in the English Channel and the Irish Sea (Doodson et al., 1954). In 

 the North Atlantic Ocean there is only a small number of observations avail- 

 able from oceanic islands. This comparison shows that there is some con- 

 formity, and it seems to be useful to tackle certain tidal problems by means 

 of this method. Fig. 16 gives as an example observed and computed |i and I2 

 values for the southern part of the North Sea. Boundaries are shown by open 

 circles with crosses and interior points by full circles. The |i and ^2 values are 

 drawn up as vector components. This method may be used to get information 

 on the tides in the open sea, if these are known on the boundary. As already 

 mentioned, this method is usually applied to the principal M2 constituent. In a 

 similar way all other constituents may be ascertained, and with this the tide 

 can be represented by a series of terms like equation (1), but it should be kept 

 in mind that this is feasible only in the case of linear equations. This is true in 

 the deep sea, but in shallow waters the hydrodynamical equations are essentially 

 non-linear. This is the reason why the method discussed above is not applicable 

 to tides in tidal estuaries or in shallow bights. Model trials can solve these 

 problems of shallow- water tides and have been undertaken for both scientists 

 and coastal engineers. 



Besides hydraulic and electric models often used by engineers, several 

 methods have been developed to compute the tides in such areas as, for ex- 

 ample, the Netherlands. Dronkers (1935) extended the values of tide and tidal 

 currents into power series of the space variable x, and in this way he arrived at 

 solutions of the non-linear equations. Fig. 17 shows the difference between 

 observed and computed sea-level on the Nieuwe Waterweg. Schonfeld applies 

 the characteristic method to tidal problems and compares his results with 

 observations in a hydraulic model. 



In 1960 Pekeris gave a lecture concerning his computation of the principal 

 constituents of the tides for the ocean as a whole. Starting with the hydro- 

 dynamical equations in spherical co-ordinates, but without friction terms, he 

 introduced a time factor and, after eliminating the components of velocity u 

 and V, he arrived at an equation similar to (7) taking into account the boundary 

 condition — normal flux equals zero — expressed in terms of derivatives of ^. He 



