472 



FOURTH REPORT — 1834. 



is the hydrostatic amplitude ; and as p is very small in propor- 

 tion to jy, the one amplitude will be double the other. 



Funk, after having demonstrated that Dubuat's method gave 

 too great an excess, admitted that the surface was a concave 

 arc of a parabola, at a distance from the dam of twice the hydro- 

 static amplitude, of which the perimeter would be ^ H'/> ; con- 

 sequently at any distance from the dam, the height of the swell- 

 ing below the surface of the current is 2 H' —p a— V H' ^ — ^H' 



3 H' . 

 p a. When a=— — this height is zero, from whence the point or 



the surface of the swelling joins the current. The amplitude 



3 H' . 



will then be — — , ox\\ time the hydrostatic amplitude. 



The following are the results : 



This question has also been examined by Messrs. Belanger 

 and D'Aubuisson*. 



In like manner the contractions of rivers, by natural and ar- 

 tificial causes, occasion a I'ise in the surface of the water equal 

 to the difference between the heights of the water before and 

 after the contraction : the same applies to the piers of bridges 

 and to jetties ; both cases have been examined by Dubuat, Eytel- 

 wein, and Funk. Funk in particular made several experiments 

 on the swell occasioned in the river Weser by the bridge of 

 Minden. The mean breadth of the river was 180"7l metres, the 

 mean depth 5*37 metres; the produce of the water was 1318 

 cubic metres ; the height of the swelling was found to be 0*383 ; 

 the sum of the openings of the bridge was 96'03 metres ; the 

 velocity of the river before the swelling was 1*358 metres 



(1318 \ 

 ~ TftrvT v¥7 ) ' ^^^ ^^^ velocity of the upper surface of the 



I 



• Page 162 of tlie Traite d'Hi/draulique a l' usage des Ingenieurs ; par J. T. 

 D'Aubuisson de Volsins : Paris 1834. See also VenturoU di Meccanka e d'ldrau^ 

 Ilea. Milan 1818. 



