Mathematical Derivation
Begin with the Bernoulli equation for a stream tube in steady, incompressible flow. The total energy head H measured from a fixed horizontal datum is
$$ H \;=\; z_b \;+\; h \;+\; \frac{v^{2}}{2g} $$
where zb is the bed elevation above the datum, h is the depth of flow, v is the cross-sectional mean velocity, and g is gravitational acceleration. The specific energy E is the part of H measured from the channel bed itself — equivalently, H with zb subtracted:
$$ E \;=\; h \;+\; \frac{v^{2}}{2g} \;=\; h \;+\; \frac{Q^{2}}{2\,g\,A^{2}} $$
because v = Q/A. For a prismatic channel of fixed shape, the cross-sectional area A is a known function of depth h. Substituting that function gives E entirely as a function of h, with Q as a parameter. The result is the E-h curve.
Area and top-width functions for each prismatic section
The two geometric quantities that drive the calculation are the wetted area A(h) and the surface top width T(h) = dA/dh — the latter being what determines critical depth.
where b is the bottom width, z is the side-slope ratio (horizontal : 1 vertical), D is the pipe diameter, and $\theta = 2\,\arccos(1 - 2h/D)$ is the angle subtended at the centre by the free water surface.
Minimum specific energy and critical depth
The minimum of E with respect to h occurs where the derivative dE/dh vanishes. Differentiating $E = h + Q^{2}/(2gA^{2})$ with respect to h and noting that dA/dh = T:
$$ \frac{dE}{dh} \;=\; 1 \;-\; \frac{Q^{2}\,T}{g\,A^{3}} \;=\; 0 \qquad\Longleftrightarrow\qquad \frac{Q^{2}\,T}{g\,A^{3}} \;=\; \mathrm{Fr}^{2} \;=\; 1 $$
So the minimum of the specific energy curve occurs exactly at the critical depth — the condition Fr = 1 and the condition dE/dh = 0 are mathematically equivalent. For a rectangular channel the result has the particularly tidy form
$$ E_\mathrm{min} \;=\; \frac{3}{2}\,h_c \qquad\text{(rectangular only)} $$
For non-rectangular sections Emin is obtained by substituting the numerically-found hc back into the specific-energy equation.
Interpreting the Curve
The shape of the E-h curve is the key teaching tool of open-channel hydraulics. Read it from the bottom upward:
The supercritical branch — the lower portion, below hc — is the regime of shallow, fast flow. Reducing the depth in this branch increases the specific energy, because the velocity head Q²/(2gA²) grows faster than h shrinks. This is the regime of chutes, spillway aprons, and the tail of a hydraulic jump.
The subcritical branch — the upper portion, above hc — is the regime of deep, slow flow. Here the velocity head is negligible and E ≈ h, so the curve asymptotes to the 45° line E = h as depth grows. This is the regime of irrigation canals, lowland rivers, and most natural channels.
The critical point — at the leftmost tip of the curve, where the two branches meet — is the depth and energy at which Fr = 1. Flow at exactly critical depth is unstable: small disturbances generate standing waves and the surface becomes choppy. Engineering designs almost always avoid running channels at critical depth.
Alternate depths — for any specific energy E greater than Emin, the horizontal line at that energy crosses the curve at two depths: one subcritical, one supercritical. Both depths can carry the same discharge with the same specific energy. They are alternate depths — a quite different concept from conjugate or sequent depths, which conserve momentum across a hydraulic jump rather than energy.
What this curve tells the designer. Pick a discharge, draw the E-h curve, and you can read off (a) the minimum energy required to pass that flow at all, (b) the critical depth at which it would just barely pass, (c) the two depths it could run at for any given energy, and (d) what depth-change to expect across a transition such as a smooth bump on the channel floor — the new depth lies on the same curve at an energy reduced by the obstacle height.