In this study, flume experiments were conducted under conditions where alternate bars occur, develop, and migrate, to understand the existence and scale of the spatial distribution of the migrating speed of alternate bars and their dominant physical quantities. In the flume experiment, the bed level and water level during the development of alternate bars were measured with high frequency and high spatial resolution. By comparing the geometric variation of the bed shape, the results showed that the migrating speed of the alternate bars is spatially distributed and changes with time. Next, to quantify the spatial distribution of the migrating speed of the alternate bars, a hyperbolic partial differential equation for the bed level and an calculating equation the migrating speed based on the advection term of the same equation were derived. Subsequently, the derived equation was shown to be applicable by comparing it with the measurements obtained in the flume experiments described above. The migrating speed of the alternate bars was calculated using above formulas, and it was found to have a spatial distribution that changed with the development of the alternate bars over time. The mathematical structure of the equation showed that the three dominant physical quantities of the migrating speed are the particle size, Shields number, and energy slope. In addition, our method is generally applicable to actual rivers, where the scale and hydraulic conditions are different from those in the flume experiments.

The hydrological quantities governing the generation of riverbed waves (formed spontaneously on the bottoms of rivers) have been elucidated through geomorphological methods, laboratory experiments, stability analyses, numerical analyses, and other research methods.Recently, numerical analysis was performed with a fine spatial resolution.However, numerical analysis cannot always describe the real phenomena because it is based on assumptions.Therefore, understanding the physical phenomena by measurements with the same resolution as the numerical analysis is necessary.Measurement data with high resolution enable the construction of a duplicate of the measurement target on a computer, called a “digital twin”.To construct a digital twin of the process of riverbed wave generation and development, the geometries of the water surface and the river bottom must be measured simultaneously.We developed and verified a measurement method for the construction of a digital twin during the generation and development of riverbed waves.The measurement system uses two cameras and a line laser to simultaneously measure the water surface and river bottom.Accurate refraction correction at the water surface is possible by acquiring the shape of the water surface, allowing the bottom shape to be determined by geometric processing.The method provides submillimeter-accurate measurements of the water surface and bottom with a spatial resolution of 0.95 cm longitudinally and 0.038 cm transversely in a 12 m $\times$ 0.45 m channel and takes only one minute per measurement.This method can provide measurement results that contribute to the understanding of the formation and development of riverbed waves.

• The spatial distribution of the migrating speed of the free alternate bars that occur in rivers was determined. • The spatial distribution of the migrating speed M of bars was discovered. The formula of M was proposed and its applicability was showed. • The main dominant physical quantity of the migrating speed of alternate bars was found to be the energy slope.

Alternate bars can spontaneously occur and develop in rivers. They are considered to be a wave phenomenon due to their geometrical features and propagation characteristics. Presently, there is insufficient knowledge about their propagation, which is an important wave phenomenon property. In this study, a flume experiment was conducted under the condition that alternate bars occur and develop. This investigation aims to understand the existence and the scale of migrating speed of these alternate bars. The bed and water levels during the occurrence and development of the alternate bars were measured frequently with a high spatial resolution. By comparing the geometrical changes in the bed shape, the migrating speed of the alternate bars has a spatial distribution that changes with time. To quantify the spatial distribution of the migrating speed of the alternate bars, a hyperbolic partial differential equation for the bed level and migrating speed formula were derived. A comparison of the measured values for the flume experiment showed that the derived formula is applicable. Using the formula of the migrating speed in this hyperbolic partial differential equation, the migrating speed was verified to have a spatial distribution. In addition, the distribution changes with the development of the alternate bars over time. This study demonstrates that the dominant physical quantity of the migrating speed is the energy slope from the experimental results and the migrating speed formula.