# Theoretical study of unstable waves in the rapids of the river

*by*H. Kyotoh,

**Abstract:**This paper examines how the velocity distribution in free-surface flow affects the kinematic and dynamic velocities of disturbance waves on hillslopes or in high-gradient rivers, thereby assessing the role of free-surface instability (or roll waves) in modeling steep-channel flows. For clear-water or lightly sediment-laden flow in alluvial channels or gravel-bed rivers, the Lacey (or one-fourth power) formula is often found to model the flow better than the Manning (or one-sixth power) formula because the applicable range of the relative roughness for the Lacey formula is one order-of-magnitude larger (or 'relative smoothness' one order-of-magnitude smaller) than that for the Manning formula. However, a stability analysis of steep-channel flow based on linear stability theory shows that flow with a one-fourth power velocity distribution tends to generate roll waves more easily than that with a one-sixth power velocity distribution. Use of the Lacey formula in modeling steep-channel flows must therefore proceed with caution. When the flow becomes unstable, the effect of the Froude number on flow resistance can no longer be ignored. This paper aims to incorporate this Froude-number effect in the flow resistance formula, such as Manning's and Lacey's, and address the significance of the Vedernikov number in steep-channel flow modeling.

**Subject Headings:**Wave velocity | Velocity distribution | Flow resistance | Surface waves | Flow distribution | River flow | Linear flow

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