CROSS SECTION AND SLOPE
§ 115. In every stream the discharge of which is constant, or may be regarded as constant for the time considered, the velocity at different places depends on the slope of the bed. Except at certain exceptional points the velocity will be greater as the slope of the bed is greater, and, as the velocity and cross section of the stream vary inversely, the section of the stream will be least where the
cross sections normal to the direction of flow. Suppose the mass of water AoBoA1B, comes in a short time 0 to C0DoCID1, and let the work done on the mass be equated to its change of kinetic energy during that period. Let l be the length AoA1 of the portion of the stream considered, and z the fall, of surface level in that distance. Let Q be the discharge of the stream per second.
Change of Kinetic Energy.—At the end of the time 0 there are as many particles possessing the same velocities in the space CoDoAIB1 as at the beginning. The
change of kinetic energy is therefore the difference of the kinetic energies of AoBoCoDo and AlBiCID1.
Let fig. 119 represent the cross section AoBo, and let w be a small element of its area at a point where the velocity is v. Let S2o be the
whole area of the cross section and uo the mean velocity for the whole cross section. From the definition of mean velocity we have
uo =Ewv/52o.
Let v=uo+w, where w is the difference between the velocity at the small element w and the mean velocity. For the whole cross section, now = O.
The mass of fluid passing through the element of section w, in 0 seconds, is (G/g)wvo, and its kinetic energy is (G%2g)wv2O. For the whole section, the kinetic energy of the mass AoBoCoDo passing in 9 seconds is
(GB/2g)Ewv3 = (GO/2g)Ew(uo3+3uo2W+3uow2+w'), _ (GO/2g){uo312+Eww2(3uo+ze )
The factor 3uo+w is equal to 2uo+v, a quantity necessarily positive. Consequently Ewv3> flouo3, and consequently the kinetic energy of AoBoCoDo is greater than
(GO/2g)12ouo3 or (GO/2g)Quo2,
which would be its value if all the particles passing the section had the same velocity uo. Let the kinetic energy be taken at a(GO/2g)12ouo3 = a(GO/2g)Quo2,
where a is a corrective factor, the value of which was estimated by J. B. C. J. Belanger at 1.1.1 Its precise value is not of great importance.
In a similar way we should obtain for the kinetic energy of AIBICIDI the expression
a(Go/2g)S2,uk3 = a(G6/2g)Qui',
where f21, ul are the section and mean velocity at A1BI, and where a may be taken to have the same value as before without any important error.
Hence the change of kinetic energy in the whole mass AoBoAIB1 in 0 seconds is
a(G0/2g)Q(uk2uo2). (I)
Motive Work of the Weight and Pressures.—Consider a small filament aoa1 which comes in 0 seconds to cock. The work done by gravity during that movement is the same as if the portion aoco were carried to ale'. Let dQo be the volume of aoco or alcl, and yo, yk the depths of ao, al from the surface of the stream. Then the volume
Boussinesq has shown that this mode of determining the corrective factor a is not satisfactory.
C:
Scale A Inc1t=1 Foot.
dQB or GdQB pounds falls through a vertical height z+y,—yo, and the work done by gravity is
GdQB (z+yl —yo) .
Putting pa for atmospheric pressure, the whole pressure per unit of area at as is Gyo+pa, and that at a, is — (Gyi+pa). The work of these pressures is
G(yo+pu/G—yi—pu/G)dQO =G(yo—yi)dQC.
Adding this to the work of gravity, the whole work is GzdQO; or, for the whole cross section,
GzQB. (2) Work expended in Overcoming the Friction of the Stream Bed.—Let A'B', A"B" be two cross sections at distances s and s+ds from AOB0. Between these sections the velocity may be treated as uniform, because by hypothesis the changes of velocity from section to section are gradual. Hence, to this short length of stream the equation for uniform motion is applicable. But in that case the work in overcoming the friction of the stream bed between A'B' and A"B" is
GQBI(u2/2g) (x/S2)ds,
where u, x,12 are the mean velocity, wetted perimeter, and section at A'B'. Hence the whole work lost in friction from AoB0 to AlBi will be
,GQB f ol.(u2/2g)(x/Sc)ds. (3)
Equating the work given in (2) and (3) to the change of kinetic energy given in (1),
a(GQB/2g) (u12—uo2) =GQzO—GQBfo' (u2/2g)(x/t2)ds;
z = a(u12 —uo2)12g+ f (n2/2g) (x/SI)ds.
§ i i6. Fundamental Differential Equation of Steady VariedMotion.—Suppose the equation just found to be applied to an indefinitely short length ds of the stream, limited by the end sections ab, aibi, taken for simplicity normal to the stream bed (fig. 120). For that short length of stream the fall of surface level, or difference of level of

;fir
,d
'
mod
h
I u+du

a and a,, may be written dz. Also, if we write u for uo, and u+du for
the term (uo2—u,2)/2g becomes udu/g. Hence the equation applicable to an indefinitely short length of the stream is
dz = udu/g+(x/St)l(u2/2g)ds. (I) From this equation some general conclusions may be arrived at as to the form of the longitudinal section of the stream, but, as the investigation is somewhat complicated, it is convenient to simplify it by restricting the conditions of the problem.
Modification of the Formula for the Restricted Case of a Stream flowing in a Prismatic Stream Bed of Constant Slope.—Let i be the constant slope of the bed. Draw ad parallel to the bed, and ac horizontal. Then dz is sensibly equal to a'c. The depths of the stream, h and h+dh, are sensibly equal to ab and a'b', and therefore dh=a'd. Also cd is the fall of the bed in the distance ds, and is equal to ids. Hence
dz = a'c = cd —a'd = ids —dh. Since the motion is steady
Q =12u = constant.
Sldu+udtt = o ;
...du= —udit/S2.
Let x be the width of the stream, then dtl=xdh very nearly. Inserting this value,
du = — (ux(St)dh. (3) Putting the values of du and dz found in (2) and (3) in equation (i), ids —dh = — (u2x/g1t)dh+(x/0) (u2/2g)ds.
dh/ds ={i—(x/St)l(u2/2g)}/}i—(u2/g) (x/S2)•} (4)
Further Restriction to the Case of a Stream of Rectangular Section and of Indefinite Width.—The equation might be discussed in the form just given, but it becomes a little simpler if restricted in the way just stated. For, if the stream is rectangular, xh=9, and if x is large compared with h, S2/x = xh/x h nearly. Then equation (4) becomes
dhlds=i(igu2/2gih)/(I—u2/gh). (5)
§ 117. General Indications as to the Form of Water Surface furnished by Equation (5).—Let AoA, (fig. 121) be the water surface,
BoB1 the bed in a longitudinal section of the stream, and ab any section at a distance s from Bo, the depth ab being h. Suppose BoB1, BoAo taken as rectangular coordinate axes, then dh/ds is the trigonometric tangent of the angle which the surface of the stream at a makes with the axis BoB1_ This tangent dh/ds will be positive, if the stream is increasing in depth in the direction BoB1; negative,
A
if the stream is diminishing in depth from Bo towards Bi. If dh/ds =o. the surface of the stream is parallel to the bed, as in cases of uniform motion. But from equation (4)
dh/ds=o, if i—(x/12)l'(u2/2g)=o;
(u2/2g) = (flx)i =mi,
which is the wellknown general equation for uniform motion, based on the same assumptions as the equation for varied steady motion now being considered. The case of uniform motion is therefore a limiting case between two different kinds of varied motion.
Consider the possible changes of value of the fraction
(I —l"u2/2gih)/(I —u2/gh)•
As h tends towards the limit o, and consequently is is large, the numerator tends to the limit—co. On the other hand if h = co , in which case u is small, the numerator becomes equal to 1. For a value H of h given by the equation
I — l"u2/2gi H =o,
H = I u2/2gi,
we fall upon the case of uniform motion. The results just stated may be tabulated thus:
For h=o,H,>H,co,
the numerator has the value —co, o, > o, 1.
Next consider the denominator. If h becomes very small, in which case u must be very large, the denominator tends to the limit — co. As h becomes very large and u consequently very small, the denominator tends to the limit 1. For h=u2/g, or u=s/ (gh), the denominator becomes zero. Hence, tabulating these results as before:
For h =o, u2/g, >u2/g, co,
the denominator becomes — co, o, > o, i.
§ 118. Case 1.Suppose h>u2/g, and also h> H, or the depth greater than that corresponding to uniform motion. In this case dh/ds is positive, and the stream increases in depth in the direction of flow. In fig. 122 let BoB1 be the bed, CoCt a line parallel to the bed and at a height above it equal to H. By hypothesis, the surface
AoAI of the stream is above CoC1, and it has just been shown that the depth of the stream increases from Bo towards B1. But going up stream h approaches more and more nearly the value H, and therefore dh/ds approaches the limit o, or the surface of the stream is asymptotic to C5C1. Going down stream h increases and u diminishes, thenumeratorand denominator of thefraction(' — rue/2gih)/(1—u2jgh) both tend towards the limit i, and dh/ds to the limit i. That is, the surface of the stream tends to become asymptotic to a horizontal line DoD1.
The form of water surface here discussed is produced when the flow of a stream originally uniform is altered by the construction of a weir. The raising of the water surface above the level CoCI is termed the backwater due to the weir.
§ 119. Case 2.Suppose h> u2/g, and also h 

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