negative, and the stream is diminishing in depth in the direction of
flow. In fig. 123 let BOB, be the stream bed as before; CDC, a line
drawn parallel to B0B1 at a height above it equal to H. By hypothesis
the surface AQA1 of the stream is below CUC1, and the depth has
just been shown to

Cf; diminish from Bu

A towards Bl. Going

0, up stream 11 ap-4

I, Q proaches the limit

f J! T H, and dh/ds tends

go 4' to the limit' zero.

A ' . That is, u stream

/M / if VB AOA, is ag/mptotic

1' to COC1. Going down

I, -IG 123 ' stream h diminishes

and u increases; the

inequality h>u2/g diminishes; the denominator of the fraction (1I-(15/zcgilz)/(1-1¢2/'g}r) tends to the limit zero, and cons ”

equent y 1, 5 tends to oo . That is, down stream AOA, tends t o a direction perpendicular to the bed. Before, however, this limit was reached the assumptions on which the general equation is based would cease to be even approximately true, and the equation would cease to be applicable. The filaments would have a relative motion. which would make the influence of internal friction in the fluid too important to be neglected. A stream surface of this form may be produced

if there

is an abrupt

fall in the bed

of the stream

(fig. 124).

On the Ganges

canal, as originally

con-

structed, there

w e r e a b r u p t

falls precisely

of this kind,

and it appears

that the lowering of the water surface and increase of velocity which such falls occasion, for a distance of some miles up stream, nas not foreseen. The result was that, the velocity above the falls being greater than was intended, the bed was scoured and l

consir erabe damage was done to the works. “ ¥Vhen the canal was first opened the water was allowed to pass freely over the crests of the over falls, which were laid on the level of the bed of the earthen channel; erosion of bed and sides for some miles stream is of indefinitely great width, so that m =H, then § 'u2/2g =iH, and H = fu”/2gi. Consequently the condition stated above involves that

5142/2gi<u2/g, or that f> 5°/2.

If such a stream is interfered with by the construction of a weir which raises its level, so that its depth at the weir becomes h1> u2, 'g, then for a portion of the stream the depth h will satisfy the conditions h <u2/g and h> H, which are not the same as those assumed in the two previous cases, At some point of the stream above the weir the depth h becomes equal to 142/g, and at that point dh/ds becomes infinite, or the surface of the stream is normal to the bed. It is obvious that at that point the influence of internal friction will be too great to be neglected, and the general equation will cease to represent the true conditions of the motion of the water. It is known that, in cases such as this, there occurs an abrupt rise of the free surface of the stream, or a standing wave is formed, the conditions of motion in which will be examined presently. It appears that the condition necessary to give rise to a standing wave is that i> § '/2. Now § ' depends for different channels on the roughness of the channel and its hydraulic mean depth. Bazin calculated the values of § ' for channels of different degrees of roughness and different depths given in the following table, and the corresponding minimum values of i for which the exceptional case of the production of a standing wave may occur. Slope below

which a Stand-Nature

of Bed of Stream. ing Vlfave is

impossible in

Standing Wave Formed,

Slope in feet Least Depth

feet per foot. per foot. in feet.

0-002 0-262

Very smooth cemented surface 0'00147 0-003 -098 0-004 -065

0'003 '394

Ashlar or brickwork 0-00186 0-004 -197 0-006 -098

0-004 ' 1-181

Rubble masonry 0'OO235 0-006 -525 0-010 ~262

0-006 3-478

Earth 0-00275 0-010 I'542

0-015 -919

must

ingly

up rapidly followed, and 1t soon became apparent that means be adopted for raising the surface of the stream at those points (that is, the crests of the falls). Planks were accord fixed in th - 'e

grooves above the bridge arches, or temporary weirs were formed over which the water was allowed to fall; in some cases the surface of the water was thus raised above its normal height, causing a backwater in the channel above" (Cr0fton's Report on the Ganges Canal, p. 14). Fig. 125 represents in an exaggerated form what probably occurred, the diagram being intended Q-.e

STANDING WAVES

§ 121. The formation of a standing wave was first observed by Bidone. Into a small rectangular masonry channel, having a slope of O°O23 ft. per foot, he admitted water till it flowed uniformly with a depth of O'2 ft. He then placed a plank across the stream which raised the level just above the obstruction to 0-95 ft. He found that the stream above the obstruction was sensibly unaffected up to a point 15 ft. from it. At that point the depth suddenly increased from 0-2 ft. to 0-56 ft. The velocity of the stream in the part unaffected by the obstruction was 5-54 ft. per second. Above the point where the abrupt change of depth occurred u2=5-542=30-7, and gh=32-2 X0-2=6-44; hence u2 was>gh. ]ust below the abrupt change of depth u=5-54><o-2/0-56=1-97; u2=3-88; and gh-32'2X0'56= I8~03; hence at this point u2<gh. Between these two B ~ " points, therefore, u2=gh; and the condition for the production of a ""~- s C standing wave occurred.

"'r~-' ' /1 fl Theghange of level at a staardilig wave may be found thgs. bLe§ 1 17, ~~ g. 12 represent the ongitu ma section 0 a stream an a, c %4/%%y%y, ,-

2, if B

— /// /. / I//./4/y /, ~, fy 2/ . Q "H i -aflit

= = = J 2 z>>>»»-

5 5 5 5 5 5

/ "'0 Eb gb, 5 di

Fla 125

FIG. 126.

to represent some miles' length of the canal bed above the fall. AA parallel to the canal bed is the level corresponding to uniform motion with the intended velocity of the canal. In consequence of the presence of the ogee fall, however, the water surface would take some such form as BB, corresponding to Case 2 above, and the velocity would be greater than the intended velocity, nearly in the inverse ratio of the actual to the intended depth. By constructing a weir on the crest of the fall, as shown by dotted lines, a new water surface CC corresponding to Case I would be produced, and by suitably choosing the height of the weir this might be made to agree approximately with the intended level AA. § 120. Case 3.-Suppose a stream flowing uniformly with a depth h<u'/g. For a stream in uniform motion fu'/2g=mi, or if the cross sections normal to the bed, which for the short distance considered may be assumed horizontal. Suppose the mass of water abcd to come to a'b'c'd' in a short time t; and let ug, ul be the velocities at ab and 611,520,521 the areas of the cross sections. The force causing change of momentum in the mass abcd estimated horizontally is simply the difference of the pressures on ab and cd. Putting ho, hi for the depths of the centres of gravity of ab and cd measured down from the free water surface, the force is G(h0S74;-hlfll) pounds, and the impulse in t seconds is G (hnflo-hlfll) I second pounds. The horizontal change of momentum is the difference of the mornenta of cdc'd' and aba'b'; that is,

(G/gl (91“1”-9o“0”)¢-