334 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[Skptkmber, 



of water support at a certain ileptli, is prnportionate to the velocity 

 with which they tend to escape. Tliis velocity is hypothetically 

 equal to that acquired by bodies falling through the same space. The 

 velocity of a body, acquired at the end of (ho first second of its fall is 

 = 2 X It'.l = 32'2 feet, and if we denote the diHerent velocities by 

 (' and V, and the respective heights liy k and H, then according to the 

 laws of gravity 



is « : V : : v/' : VH 



and V — 



Vv' H 

 IT 



IG-l, we have 

 /32-2= H 



If we take v = 32*2, and h =z lG-1, we have 

 I. 



and 



II. 



H = 



V- 



; 0-0155 V2 



8-0242" 



The quantity S.024 is called the hypothetical co-efficient for falling 

 bodies, and this co-efficient will be here generally denoted by the letter 

 ex. In applving the above rule lo the motion of water, the case is 

 somewhat different under different circumstances. Du Buat and 

 Eytelwein have made a number of satisfactory experiments to fix co- 

 efficients for the velocity of water in different circumstances. 



According to these experiments, for instance, the value of the co- 

 efficient for the discharge of water over a waste-weir, of common con- 

 struction, is found to be- - - - - - ::= 5.7 



For large and well constructed dams, where all circum- 

 stances are favourable to the discharge, - - := 7.5 



Before we can proceed to demonstrate the discharge of water over 

 dams, we have to examine the laws under which water generally will 

 be discharged, when under a certain head. 



• >l 



The annexed diagr;^i represents a vessel, Q R, filled with water 

 up to A. Suppose that sufficient water is flowing in to keep the sur- 

 face at the same level, and that there are several small openings, P, E, 

 B, above each otlier in the vertical line A B, in one side of the vessel. 



The jets of water streaming througli the opening F, E, B, are repre- 

 sented by the horizontal dotted lines, P M, E H, B G. 



Let us put A P = .r ; the velocity with which the water rushes 

 through the opening P, be = t/ ; and tlie co-efficient of tliis velocity 

 be = c( . 



So is, by fornnda 1. 



y = a \/ X. 



The same is applicable to every other opening B, with a head of 

 pressure = A B ; and if we denote A B by /(, and the corresponding 

 velocity by v, we have 



BG^ 



The same is true for every other absciss and ordinate, as A P, and 

 P M, and from this it follows, that the curved line A M H G, which is 

 formed by the extreme points M, G, &c. of the dotted lines, repre- 

 senting the velocities of the water-jets, forms a parabola. If we now 

 imagine the vertical line A B consists of a great number of such small 

 openings, than the amount of water, or the sum of all the water-jets, 

 may be represented by the area of the parabola. The superficial 

 content of the parabola A B G is 



= 3 A B. B G = i » A 

 If we denote the width of the perpendicular narrow opening or slit 

 A B, by I', the awount of water discharged through this slit will be 



Now, suppose the great rectangular opening, A B C D, consists of 

 a large number of sutdi vertical openings, and let be 



AC = BD =/ 

 and the discharge through that rectangle = Q, then we have 



and by substituting for v, its value = j. ^Ji, we have the discharge 

 per second, or 



Q =§a Ih^h 



III. 



and 



IV 





In investigating the state of water, when obstructed by dams, three 

 ditl'erent cases present themselves. 



1. 



When a dam serves only on a waste-weir, and the pool above it 

 forms an extensive sheet of water, the surface of which is kept at the 

 same level, without any perceptible current. 



In the annexed diagram, B D represents the dam or weir; the line 

 K A, the level of the upper pool ; and C F, the bed of the river or 

 reservoir, corresponding to the average depth of the water. 



The body of water, discharging over a dam, will sink considerably 

 below the level of the surface of the pool, before it reaches the breast 

 of the dam, forming a curve tangential to the surface of the jiool. 



The formulsE 111 and IV apply to this case exactly. The height h, 

 or the head of the fall, is in the diagram represented by the lines 

 K L := A B, the elevation of the surface above the top of the dam. 



If we, therefore, know the quantity of discharge per second, we 

 find by the formula I V the height corresponding to it ; and if the 

 height is known, we find the discharge by formula III. 



The height of the water above the edge of the dam, or B E, and 

 the contraction of it below, is here not taken in consideration, as it is 

 of no practical use. 



2. 



When, as in the first case, the comb, or top of the dam is above the 

 surface of the lower pool, and the water in the upper pool arrives at 

 the head of the fall with a certain velocity. 



With reference to the above diagram, let us term the point K in 

 the surface of the upper pool, where the water is horizontal, or nearly 

 so, or has yet about the same inclination as the pool farther up, the 

 head of tliefall. 



The elevation of this point B above the top of the dam, or 

 A B, may be denoted by the letter - - - - h 



The height of the dam, or B D, by K 



The average width of the pool, by - - - - - B 



The length of the dam, by ...... I 



The quantity of discharge over the dam per second, in cubic 

 feet, by ...... ...Q 



The line C F represents the bed of the river, (corresponding to the 

 average depth) as well as the base of the dam, and all the heights are 

 calculated from it. 



If we now supjiose the upper pool forms a still water w itliout any 

 current, then we have the former case, and if we represent the fall, or 

 A B, by the letter Ti', we find according to formula IV 



._3Q_): 

 \2ai I ) 



But in the present case the water arrives at the head of the jiool, 

 with a certain velocity due to the current in the river above the pool, 

 and this velocity comes to the aid of the velocity of discharge, caused 

 by the height of the fall. 



The velocity of the discharge is therefore equal to the velocity, due 

 to the height of the fall, plus the velocity, due to the current of the 

 pool. But the quantity of discharge remaining the same, and the 

 velocity being increased, the height of a discharging body of water 

 will be reduced in a proportion corresponding to the increased velo- 

 city. The water in the pool| is in consequence of the current in mo- 



