On the Propagation of Flood Waves

Abstract

In natural rivers, the forms of the channels, --the bed slopes, the breadth, the forms of cross sections, etc.--are all very irregular and incessantly changing. It is impossible to grasp them defimtely. Yet the flow in rivers is steady and nearly uniform in the broad means. The disturbances on the flow caused by these irregularities damp away within a few kilometres and have certain limited dimensions and durations. The stochastic character of the collective of these elementary disturbances causes a large scale longitudinal mixing. The order of magnitude of the diffusion coefficient may be estimated to be 106 ~ 108 c. g. s. according to the scale of a river. Introducing the effect of longitudinal diffusion caused by the mixing into the equation of continuity and assuming the mean flow taken over a suitable range to be steady and uniform, the differential equation of flood waves was derived. It is an equation of diffusion containing a term of advection. As the equation is non-linear, an approximate method of solution was discussed and solutions were obtained under several conditions. They well explain the properties of flood waves. The approximate equation of flood waves is linear, a flood of any form is, therefore, supposed to be composed of many elementary flood waves of simple character, -- unit graph, or unit flood. A method of computing the unit graph was described and some numerical examples were shown. In the last, some of the results of observations made on an artificial unit flood in the Yedo River were compared with the theoretical computations. Their agreement is excellent.