Q = [Omega]u = constant.
Differentiating,
[Omega] du + u d[Omega] = 0;
.:du = -u d[Omega]/[Omega].
Let x be the width of the stream, then d[Omega] = xdh very nearly.
Inserting this value,
du = -(ux/[Omega]) dh. (3)
Putting the values of du and dz found in (2) and (3) in equation (1),
i ds - dh = -(u²x/g[Omega]) dh + ([chi]/[Omega])[zeta](u²/2g) ds.
dh/ds = {i - ([chi]/[Omega]) [zeta] (u²/2g)}/{1 - (u²/g)(x/[Omega])}. (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, [chi]h = [Omega], and
if [chi] is large compared with h, [Omega]/[chi] = xh/x = h nearly.
Then equation (4) becomes
dh/ds = i(1 - [zeta]u²/2gih)/(1 - u²/gh). (5)
§ 117. _General Indications as to the Form of Water Surface furnished
by Equation_ (5).--Let A0A1 (fig. 121) be the water surface, B0B1 the
bed in a longitudinal section of the stream, and ab any section at a
distance s from B0, the depth ab being h. Suppose B0B1, B0A0 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
B0B1. This tangent dh/ds will be positive, if the stream is increasing
in depth in the direction B0B1; negative, if the stream is diminishing
in depth from B0 towards B1. If dh/ds = 0, the surface of the stream
is parallel to the bed, as in cases of uniform motion. But from
equation (4)
dh/ds = 0, if i - ([chi]/[Omega])[zeta](u²/2g) = 0;
.: [zeta](u²/2g) = ([Omega]/[chi])i = mi,
which is the well-known 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.
[Illustration: FIG. 121.]