On the Migrating Speed of Free Alternate Bars

Abstract

AbstractFree alternate bars exhibit wave properties. However, there has been a significant lack of research regarding these wave properties, including the migration speed. In this study, we quantified the migration speed M for bars and identified the dominant variable exerting control over M. Experiments were performed in a laboratory flume under controlled steady state conditions to ensure the development of alternate bars; in addition, the bed level and flow depth were continuously measured. We derive a hyperbolic partial differential equation (HPDE) for the bed level by assuming that the bed level is a continuous function and the advection velocity of the function equals M. Then, to verify the HPDE, we show that it adequately describes the temporal variation in the bed level that is measured experimentally. The advection velocity of the HPDE can be used to calculate M from the energy slope and the Shields number, enabling the calculation of the spatial and temporal variations of M in the experiments. The formula for M showed that the magnitude of M is three to four orders of magnitude smaller than the velocity of the uniform flow. Then, we show that M obtained from the proposed formula is in agreement with those obtained from an instability analysis. Furthermore, we demonstrate that the proposed formula is applicable to actual rivers, in which the scale and conditions differ from those in the experiments.