EXACT, QUASI-EXACT, AND EXPLICIT UNIFIED SOLUTION FOR CRITICAL FLOW DEPTH IN TRAPEZOIDAL CHANNELS

Abstract

This paper provides an exact foundation and uniformly high-accuracy formulas for the critical depth in trapezoidal channels. Beginning with the dimensionless critical-flow condition, the problem is recast as an exact map η = z(ξ) [Eq. (5)], whose unique physical root can be characterized either as the positive zero of a trinomial sextic [Eq. (5a)] or through rigorous inverse-function representations, i.e., residue integrals and Lagrange–Bürmann expansions, thereby establishing a machine-precision “ground truth” and clarifying why elementary radicals are unavailable. To convert this exact theory into a practical computation, a fifth-root normalization is introduced [Eqs. (5i) – (5j)], reducing the problem to a strictly monotone, single-unknown relation [Eq. (5k)]. From this normalized equation, the authors derive a reduced form [Eqs. (5n) – (5o)] and construct a closed-form rational seed u₀(C) [Eq. (5p) with coefficients in Eq. (5q)] that approximates the exact root with sub-0.002% worst-case deviation over the broad range ξ ∈ [0.001,1000]; the worst case occurs at ξ = 0.001; a single Newton/Halley correction [Eq. (5r)] then recovers near machine precision across the full range, as documented by the tabulated examples. This yields a quasi-exact, one-step workflow that requires neither charts nor special functions and is robust at both shallow and deep ξ limits. In parallel, for non-iterative spreadsheet use, a single blended explicit formula for z(ξ) is developed by asymptotically correct small- and large-ξ branches merged with a smooth Hill-type switch [Eqs. (6) – (9)]. The base blend achieves uniform, sub-6.5×10⁻⁵ % deviations on the restricted range ξ ∈ [0.001,100] and remains highly accurate within the broad range ξ ∈ [100,1000]; a minimal deep-branch enrichment further flattens the far tail.

For readers who prefer a direct depth formula, the paper introduces an explicit η(ξ) relationship obtained via a well-conditioned change of variables and a controlled asymptotic–Taylor construction; the final expression [Eq. (37)] is quasi-exact for ξ ∈ [0,1000], with a worst-case relative deviation of about 7.1×10⁻⁶ % at ξ = 50. An analytic mapping quantifies how errors in z propagate to η and design variables [Eq. (17)], ensuring that numerical accuracy translates transparently to hydraulically meaningful quantities.

Collectively, the exact map, and proofs of uniqueness, the normalized one-unknown route culminating in the explicit seed u₀ [Eq. (5p)] plus a single corrective step [Eq. (5r)], the blended z(ξ) approximation, and the direct explicit η(ξ) formula [Eq. (37)] deliver a verification-driven, implementation-ready toolkit that replaces legacy trial-and-error procedures with fast, stable, and reproducible computation over the entire operating range.