River Bed Degradation due to Abrupt Outfall Lowering

Lower reaches of tributaries to the Missouri River were straightened in the late 19th and early 20th century. Channel degradation has progressed upstream as a result. Channel deepening and widening have caused problems at stream crossings and have resulted in gully encroachment into cultivated fields. A diffusion model and an hyperbolic model, each describing channel degradation, are solved using a Laplace transform approach. A closed‐form solution is obtained for the diffusion model, but numerical methods are necessary for evaluation of the inverse transform of the hyperbolic model. A closed‐form asymptotic solution is found for the hyperbolic case. This asymptotic solution is shown to contain the solution to the diffusion model as a special case. The asymptotic solution is compared with observed degradation of West Tarkio Creek, a small river in northwestern Missouri and southwestern Iowa, with good results.