Difficulty: Medium
Correct Answer: L = (E2 − E1) / (i_b − i_e)
Explanation:
Introduction / Context:
Gradually varied flow (GVF) profiles like backwater curves are evaluated using the specific-energy form of the GVF equation. Over a reach, changes in specific energy relate directly to the difference between bed slope and energy grade line slope, allowing estimation of the water-surface profile length.
Given Data / Assumptions:
Concept / Approach:
The differential form is dE/dx = i_b − i_e. Integrating from section 1 to section 2 gives E2 − E1 = ∫(i_b − i_e) dx ≈ (i_b − i_e) L if slopes are roughly constant. Hence L = (E2 − E1)/(i_b − i_e).
Step-by-Step Solution:
Verification / Alternative check:
Backwater curves have i_b > i_e; thus denominator is positive, giving a positive L for E2 > E1, consistent with a rise in the water surface along the flow.
Why Other Options Are Wrong:
(b) Changes sign incorrectly; (c) ignores bed slope; (e) is dimensionally inconsistent. (d) is unnecessary since (a) is correct.
Common Pitfalls:
Confusing specific energy with total head; overlooking that i_e includes both friction and minor losses where relevant.
Final Answer:
L = (E2 − E1) / (i_b − i_e)
Discussion & Comments