(3)
Equating the work given in (2) and (3) to the change of kinetic energy given in (1),
α (GQθ / 2g) (u12 − u02) = GQzθ − GQθ ∫10 ζ (u2 / 2g) (χ / Ω) ds;
∴ z = α (u12 − u02) / 2g + ∫10 ζ (u2 / 2g) (χ / Ω) ds.
| Fig. 120. |
§ 116. Fundamental Differential Equation of Steady Varied Motion.—Suppose the equation just found to be applied to an indefinitely short length ds of the stream, limited by the end sections ab, a1b1, 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 a and a1, may be written dz. Also, if we write u for u0, and u + du for u1, the term (u02 − u12)/2g becomes udu/g. Hence the equation applicable to an indefinitely short length of the stream is
dz = u du/g + (χ/Ω) ζ (u2/2g) ds.
(1)
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