Tides in Estuaries 465 



estuary. According to Franzius (1901, p. 229), this happens on the Elbe 

 and the Weser, where there are at the same time two tide waves; on the 

 Amazon River, whose tide zone is nearly 1000 km long, there are sometimes 

 even seven or eight tide waves present at the same time. Assuming that the 

 tidal curve at the mouth is represented by a simple sinus curve, we have in 



, , . 2n « a—h , b . 2n ,wttt ,\ 



h + r] = a + bsm — t or t = ~r ^h^Y 1, (XIII. 1) 



which h is the height of the undisturbed water-level of the river, y\ the eleva- 

 tion above this level and a can only be slightly different from /?. The water 

 surface behaves like a "bore" and according to (VI. 13) and (V. 12) respect- 

 ively its water volume travels upstream with velocity: c = j (gh)(l + f r)/h). 



T 



During a full period T, the amount of water \ cr\dt penetrates into the 



6 

 river. If we neglect the small term (a—h) 2 , the integral gives 



ha = l{& I a). (XIII. 3) 



This means that the mean water level does not correspond to the sea level 

 at the mouth of the estuary, but is lower by %(b 2 /a). This difference is notice- 

 able ; for a = 6 m below the mean sea level and an elevation of 3 m above it, 

 this difference in level is 0-25 m. This is, in fact, what the observations show 

 (see Levy, 1898). 



h+; 



—r 



Fig. 196. Relationship between velocity of progress V, velocity of current c and height ?/ 



of a bore. 



The retarding influence of the friction appears particularly, according to 

 M. Moller (1896, p. 479), in the correlation between thevelocity of propaga- 

 tion V of the flood wave, the velocity of the current c and the tidal range r\. 

 If there is a "bore" out at sea with a velocity of propagation V which travels 

 upstream (see Fig. 196), and the velocity of the current is c x out at sea and c 2 

 in the river, the relative velocities compared to the bore are V— c x and F— c 2 . 

 Moller now imagines the bore to be fixed and applies Bernoulli theorem to 

 the current; then 



( V- c 2 f - ( V- o) 2 = 2gi\ , (XIII. 4) 



and the equation of continuity gives 



h ( V- c 2 ) = Qi + ri) ( V- c x ) , (XIII. 5) 



30 



