Regime channels (Lacey): The regime (stable) mean velocity in an alluvial channel is proportional to which combination of hydraulic radius R and bed slope S?

Introduction / Context:
Velocity–slope–size relations in alluvial channels are often expressed through regime or resistance equations. Although Lacey introduced relations among velocity, silt factor, hydraulic radius, and slope for channels in regime, the resulting form reflects a square-root dependence on slope and a sub-linear dependence on hydraulic radius.

Given Data / Assumptions:

• Mean velocity V in a stable (regime) alluvial channel.
• Hydraulic radius R and bed slope S as governing variables in a proportionality form.

Concept / Approach:
For practical purposes, regime/roughness formulations lead to V being proportional to S^(1/2) and to a fractional power of R, commonly near 2/3. This matches the general resistance behavior also echoed by Manning-type expressions for wide channels.

Step-by-Step Solution:
Recall proportionality: V ∝ R^m * S^n with n ≈ 1/2 for uniform open-channel flow.Empirical fits/regime theory produce m in the range 0.5–0.7; 2/3 is a practical value.Therefore choose V ∝ R^(2/3) * S^(1/2).

Verification / Alternative check:
Comparisons with Manning’s V = (1/n) R^(2/3) S^(1/2) corroborate the selected exponents for regime-scale proportionality, noting Lacey’s additional dependence on sediment factor in full form.

Why Other Options Are Wrong:

• R^(1/2) * S^(3/4): Overstates slope influence and understates radius.
• Q^(3/4) * S^(1/3): Introduces discharge directly and a weaker slope exponent.
• R^(3/4) * S^(1/3): Slope exponent too small for regime resistance trends.

Common Pitfalls:
Confusing Lacey’s complete set (including silt factor f) with simplified proportionalities; mixing discharge-based and geometry-based forms.

Final Answer:
R^(2/3) * S^(1/2)