Bed, hydraulic and energy slopes

Definitions

• Bed slope ($S_0$): slope of the channel bottom, $S_0=-\dfrac{dz}{dx}=\sin\theta\approx\tan\theta$.
• Hydraulic slope ($S_w$): slope of the hydraulic grade line, which is the free water surface: $S_w=-\dfrac{d(z+y)}{dx}$.
• Energy slope ($S_f$): slope of the energy grade line (EGL), the rate at which total head $H=z+y+\alpha V^2/2g$ is lost: $S_f=-\dfrac{dH}{dx}$.

      EGL  -----\ ____  V1^2/2g
                   \    -----\
      WSL  ----------\        \------
                       \  y1       y2
      Bed  ----------------\--------\---
      datum ......................

Why they are not parallel in non-uniform flow

1. In uniform flow depth $y$ and velocity $V$ are constant, so $\dfrac{dy}{dx}=0$ and $\dfrac{d}{dx}\left(\dfrac{V^2}{2g}\right)=0$. Then the fall of the water surface equals the fall of the bed, and the fall of the EGL equals both: $S_0=S_w=S_f$. All three lines are parallel.
2. In non-uniform flow, depth and velocity change along the channel. Therefore $z$, $z+y$ and $z+y+V^2/2g$ fall at different rates.
3. The relations are $S_w=S_0-\dfrac{dy}{dx}$ and $S_f=S_w-\dfrac{d}{dx}\left(\dfrac{\alpha V^2}{2g}\right)$.
4. For a rising depth (backwater) $dy/dx>0$ so $S_w<S_0$; for a falling depth $S_w>S_0$. Velocity head also changes, so $S_f\neq S_w$.
5. Friction loss $S_f$ is always positive, so the EGL always falls in the flow direction.

Only when $dy/dx=0$ and $dV/dx=0$ do the three slopes become equal.