Abstract The hierarchy of one-dimensional equations and numerical methods describing the motion of floods and disturbances in streams is studied, critically reviewed, and a number of results obtained. Initially the long wave equations are considered. When presented in terms of discharge and cross-sectional area they enable the development of simple fully-nonlinear advection-diffusion models whose only approximation is that disturbances be very long, easily satisfied in most flood routing problems. Then, making the approximation that changes in surface slope are relatively small such that diffusion terms in the equations are small, various advection-diffusion and Muskingum models are derived. Several well-known Muskingum formulations are tested; one is found to be in error. The three families of governing equations, the long wave equations, and the advection-diffusion and the Muskingum approximations, are linearised and analytical solutions obtained. A dimensionless diffusion-frequency number measures the accuracies of the approximate methods. Criteria for practical use are given, which reveal when they have difficulties, for streams of small slope, for fast-rising floods, and/or when shorter period waves are present in an inflow hydrograph. They can probably be used in most flood routing problems with an idealised smooth inflow. However the fact that they cannot be used for all problems requires a general alternative flood routing method, for which it is recommended to use the long wave equations themselves written in terms of discharge and cross-sectional area, when a surprisingly simple physical stream model can be used. An explicit finite difference numerical method is presented that can be used with different inflow specifications and downstream boundary conditions, and is recommended for general use.

中文翻译：

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洪水路由方法

摘要 研究了描述河流中洪水和扰动的一维方程的层次结构和数值方法，并对其进行了批判性审查，并获得了一些结果。首先考虑长波方程。当以流量和横截面面积表示时，它们能够开发简单的完全非线性对流扩散模型，其唯一近似值是扰动很长，在大多数洪水演算问题中很容易满足。然后，近似地表坡度变化较小，方程中的扩散项较小，推导出各种对流-扩散模型和Muskingum模型。测试了几种著名的 Muskingum 配方；一个被发现是错误的。三族控制方程，长波方程，对流扩散和 Muskingum 近似是线性化和解析解。无量纲的扩散频率数衡量近似方法的准确性。给出了实际使用的标准，当它们遇到困难时，对于小坡度的溪流，快速上升的洪水，和/或当流入水位过程线中出现较短周期的波浪时，这些标准都会给出。它们可能可用于大多数具有理想化平滑流入的洪水路由问题。然而，它们不能用于所有问题的事实需要一个通用的替代洪水演算方法，为此建议使用长波方程本身以流量和横截面积写成，当一个令人惊讶的简单物理流模型可以使用。