OF THE FALL UNDER BRIDGES.
The piers of a bridge form an obstacle in the way of the waters, and will cause them to rise above the general level. The same body of water which flows in the open channel must be conveyed through the openings of the bridge. The narrower that passage is, the swifter must be the current. And this additional swiftness is only to be produced by a descent from a greater height. Con n n sequently, the water will accumulate above the obstruc tion, until it runs off as fast as it comes, or until the ve locity in the contracted water-way be to that in the open channel, reciprocally as the relative sections of the stream.
Granting that the velocities of the running water arc such as would be produced by falling from a certain height above the stream, a principle which is at any rate sufficiently just for our purpose, it follows that the fall, or accumulation produced by the obstacle, will be measured by the difference between the heights which would be requisite for producing the two velocities, viz. of the river in general, and of the current just under the bridge. But if a body fall a Sect in a second of time, it acquires a velocity of 2 a feet per second, and the heights are as the squares of the velocities; wherefore, in order to produce the velocity v, we must suppose a fall from a 4 height =— or — expressing v in feet, and neglecting 4a 16 that small fraction, whereby the fall of a heavy body in one second exceeds 16 feet.
But when water is forced from a larger channel through a smaller passage, it is observed, that the stream through this passage is contracted. This con traction has been variously stated. It is probable, that the ratio of the diameter of the contracted stream to that of the passage, is that of 4 to 5, or .8 to I, according to Bossut, Alichelotti,andVenturi.
Bunt gives . 6 to 9, or .66fi to 1.
And Newton 21 to 25, or .84 to I.
Consequently, if c express the breadth of the water way between the piers, 4 c is the water-way contract ed ; take b the breadth of the channel, 4. e : b v : 5 b v the velocity in the contraction ; and to produce / 6 2 this velocity, we need a height or fall of 5 — ÷ 64, 4 c and the difference between that and the former, or that which prodnces the original velocity v, will be v 2 2 2 - 4 c the fall sought ; or b will also be a 8 theorem for it, which may be thus expressed.
Add one fourth to the breadth of the river, and divide the sum by the water-way under the arches ; from the square of the quotient subtract unity ; and multiply the remainder by the square of one eighth part of the mean velocity of the streamfor the fall in feet.
Upon this principle, the following Table is construct ed. It is not so complete as could be wished, for a great deal depends on the depth of the river; the effect of which is not so easily ascertained, and the due conside ration of it would extend our investigations much fur ther than the present subject would warrant. We may find another opportunity to communicate some further researches on this matter; and, in the mean time, what we give here will, we think, be of no small value to the engineer and practical bridge-huilder. We have given a separate column for the usual designation of the stream, and for the nature of the bottom, which will just bear the velocities expressed in the first column, that the use of the Table may be extended and facilitated. For, by this means, a look at the bottom will determine the state and velocity of the river, without the necessity of measuring it. We next give the head or fall pro duced by various obstructions, and the velocity thereby acquired, from whence we are enabled to form an idea of the action likely to take place upon the bottom.
We have already admitted, that this Table is incom plete, yet it will, in all probability, answer many useful purposes The science of hydraulics is as yet so em pirical, that we can seldom predict with certainty what will be the result of a proposed combination. It is pro bable that the fall or head, and consequently the velocity acquired, is always stated too high, at least in the ear lier part of the Table. For the contraction of one fifth of the breadth, is nearly as much as is observed in a nar now pipe. We have, indeed, made no allowance for the contraction or diminution of effect which may be sup posed to arise from the friction and other causes in the original bed of the river. Or, what is the same thing, the additional head which is requisite to overcome this friction, over and above that which is due to the assum ed velocity of the stream. And again, the friction in creasing under the piers, from the increased velocity of the stream, will require also an additional head of water to overcome it. This, in small velocities, and with small obstructions, is a very great part of the whole rise. Its piopen • lion diminishes in the latter part of our Table.