Tides in Estuaries 467 



given a mathematical solution for the change in profile of a breaking flood 

 wave, which reproduces exactly all the stages a wave goes through. 



The changes in shape which the waves undergo in a river are also influenced 

 by the fact that the river water flowing downstream forces the sea water 

 back, which increases the steepness of the wave. 



Poincare (1910, p. 409), following computations by Saint -Venant, has 

 given a mathematical most extensive theory of the river tides, taking into 

 account all important factors. Furthermore, Krey (1926) has given a survey 

 of the tide waves in estuaries with applications of a hydraulic nature. The 

 tide in channel-like bays and estuaries are characterized by the same basic 

 equations, as apply to the co-oscillating tides in adjacent seas (p. 143). As for 

 small channels and estuaries the Coriolis force is of no importance, the general 

 equations reduce to the one-dimensional case. Taking account of the friction 

 at the bottom and designating by w, v\ and r the mean values of velocity, 

 vertical motion and bottom stress, respectively, over the entire cross-section S 

 of the width b, we obtain the equations of motion and the continuity equation 

 in the following form: 



cu _du , dr\ b 

 dt b dx 



(XIII. la) 



The bottom stress (friction at the bottom) t can be considered proportional 

 to the square of the mean current velocity: 



t = ku\u\ , 



where k = 2-6 < 10 3 (see p. 346, XI. 32). Hansen (1956) has used these equat- 

 ions to compute for the estuary of the Ems River the total tide distribution up 

 to about 100 km upstream (Herbrum) from the boundary values at the mouth 

 into the North Sea (near the Island of Borkum; heights of the water level on 

 (25 June 1949) and from the freshwater transport of the Ems near Herbrum. 

 The method of boundary values was applied, which was also developed by 

 Hansen (pp. 372 and 362) to compute the tides of the North Sea. The section 

 of the Ems between Borkum and Herbrum, measuring 102 km in length, was 

 subdivided into intervals of 2 km each; the variations of the water levels were 

 numerically determined for points 4, 8, 12, ..., 100 km from the mouth and 

 the current velocity for points 2, 6, 10, ..., 98 km. In Fig. \91a, the estuary 

 of the Ems is shown on the left-hand side, while to the right there are given 

 the observed and computed water levels and current velocities in this river 

 (25 June 1949) for some points of the estuary as shown to the left. The agree- 

 ment between the observed and computed water levels and between the cur- 

 rent velocities is quite remarkable and even shows details of the curve shape. 

 Much earlier, hydraulic specialists have developed several methods to compute 



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