1839.] 



THE CIVIL ENGINEER AND AUCMITECT.S JOURNAL. 



.335 



tion through its whole depth, though the velocity near the bottom is 

 but very small. 



We find the area of the cross section equal to 

 (/J+K) B 

 and if u represents the average velocity of the current in the pool, we 

 have 



Q_ 



" - (h+k) B 

 Now, let us represent the height which corresponds to this velocity 

 by the letter H, then we have, according to formula I, 



H = 0.0155 V- 

 and by substituting for v its value, we get 



For finding the true height of the surface of the pool above the top 

 of the dam, or the height A B = A, we have therefore to deduct the 

 value of H from the value of /i', and we arrive at the formula 



3Q \S_nn,../_Q_ 



AB. 



= {- 



,0155 



■r 



And 



V 



2« I I I (i+A-)B 



if we put the co-efficient a = 7.5 and B = l,we have 



This formula contains in the substractive member the value of h it- 

 self. As this term of the equation, however, is comparatively small, 

 it will be sufficiently correct in practice, to find the value of /i by ap- 

 proximation, without making the formula more intricate by further 

 reduction. 



Example I. 



Suppose a dam of 500 feet long and 1 1 feet high, has been con- 

 structed across a river of the same width, the average depth of which 

 in time of a high freshet is ten feet, and its discharge at the same 

 time 25,000 cubic feet, per second. How much will the water rise 

 above the top of the dam, if all circumstances are favourable to the 

 discharge, and the co-efhcieut d is put =: 7.5 ? 



The above formula for //, is here 



| 3x25000 ,a_o,^s5/ 25000 

 \ 15x500 / 



;500 

 Now, let us assume h = 4.5 



then is h — yiOO 



, I 



((A-f-lI)X500i 



0.0155 

 — O.lGl 



/ 25000 ■» ' 

 \15.5x50o) 



25000 

 i500 

 or, !i =: 4.G41 



therefore, h = 4.48 feet. 



This result is near enough to the assumed value, and therefore suffi- 

 ciently correct. 



3. 

 When the top of tlie dam is lower than the surface of the lower 

 pool, and the water in the upper pool arrives at the head of the fall 

 with a certain velocity. 



The annexed diagram may represent the case in question, 

 and we will represent the depth of the river below the 



dam, or E D, by the letter A 



The height of the fall from the upper level to the lower 



level, or A E, by - - . - H 



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



The length of the dam, or width of the river, by - - I 



The quantity of water discharged per second, by - - Q 



The line C F may represent the bed o'f the river corresponding to 

 the average height h of the water. 



To simplify the demonstration of this case, let us suppose the water 

 in the upper pool form a perfect level without current, and not con- 

 sider the effect which the whirl below the dam, caused by the fall of 

 the water, has upon the discharge. 



The quantity of water discharged through the height A E, will then 

 be found by formula III. 



= §a mi/H 



The body of this water above the level L E presses upon the body 



of water below, included between the dotted lines L E and AI B, which, 

 therefore, will be forced to pass off through the height E B. 



Let us now imagine a pipe E H (i I B, of the width of the river, and 

 the height E B resting on top of the dam, with one vertical opening 

 E B at the dam, and another horizontal opening H G at the surface of 

 the lower level, below the f dl. The body of water included between 

 the lines L E and M B, w'ould then pass through this pipe, and be dis- 

 charged at the surface of the lower level with a velocity corresponding 

 to the pressure of the water above, or due to the height A E. The 

 velocity of the water flowing through the height E B is therefore 

 found, according to formula I 



= av/H 



and the discharge 



=:EB./. av/H= al(h—k) i/H 

 The discharge through the height A B is equal to the sum of dis- 

 charges througii A E and E B, and therefore 



Q = !S a /Hv/H+ a /(//— A) t/H or 



VI q,-CXl(^l\ + h—k) ^n 



and from this we find 



Q 



VII 



H=- 



The value of H must be found here by approximation, as in for- 

 mula V. 



With respect to the velocity of the current in the upper pool, Mr. 

 Eytelwein offers a formula for the value of H, the application of which 

 is very difficult on account of its perplexitj'. The following demon- 

 stration, however, will biing us near enough to truth, and furnish a 

 formula which will he found sufficient to all practical purposes. 



When H has been found by formula VII, we have then an approxi- 

 mate value for the average depth of the upper pool, or 

 A D = H -f // 



The area of the profile of the upper pool is therefore 

 = / (H -t- h) 



From this we find the average velocity of the current in the pool 



^ Q 



I (H + h) 



which velocity is owing to the current of the river above, independent 

 of the fall of the water over the dam. 



According to formula II, we find the height, corresponding to this 

 velocity 



1, " 



; 0.0155 . 



VIII 



H: 



AE = 



I (H + h) 



which ought to be deducted from the value H in formula VII, as we 

 have done in case No. 2, in order to arrive at the true height of the 

 fall. 



We therefore arrive at the formula. 



a ■'/■' (i H-f /i— K)' —0.0155 \ Z(H-t-A) ( 



The objection can be made against this formula, that the current of 

 the upper pool may be reduced by the resistance of the water below, 

 and that then the value of H is found too small. 



To examine this question, we must distinguish several cases. The 

 first case is, when a dam forms a breast-dam, with no lower slope. The 

 falling water will here produce a whirl, the effect of which will not 

 extend far below the dam, and will have little influence on the current 

 of the tail-water. The second case, when the dam has a long slope 

 forming an inclined plane, or better, an inverted parabola, on which 

 the water glides down. The lower body of water, after having moved 

 down the slope, shoots off in a more horizontal direction, not affecting 

 the bed of the river immediately below the dam, but pushing ahead 

 the tail-water, the current of which consequently will be increased. 

 Without reference to the form of dams, other considerations present 

 themselves with respect to the depth of the water. When the river 

 is not deep, and the lower level but little above the top of the dam, 

 the escape of the tail-water will be increased by the mechanical mo- 

 mentum, produced by the luight of tk/ull of the water, rolling downi 

 the slope, and the resistance offered to the current of the upper level, 

 will be therefore decreased. On the other hand, when the dam is 

 very low and the water very high, the momentum of the falling water 

 will be increased proportionably by the gemral iiicnasi of the velucily 

 of the river, and will therefore also increase the velocity of the tail- 

 water below the fall, so as not to resist the current above. 



It appears, therefore, that we may apply the above formula, without 

 any deduction, in all nises fiivoarahk to the escape of the tail-water. 

 When the construction of the dam, and the features of the river, how- 

 ever, are uti/avutirabte to the discharge of the tail-water, then we must 

 reduce the value of the substractive member of the formula. 



2 C 



