It has been observed that the shape of a meandering river is roughly circular, not sinusoidal (Leopold and Wolman 1960). It has also been observed that the following mathematical relationships tends to hold:.
$$\lambda \sim 11 w$$
$$r \sim 2.3w$$
Where $r$ is the radius of the circular bend, $w$ is the width of the river, and $\lambda$ is the length of the meander (wavelength), as labelled in the diagram below.
Since this pattern was first observed in 1960, has a theoretical model of meandering rivers been obtained that explains the above observed relationships and if so, what is the theoretical explanation of this relationship?
Note: Given $r \sim 2.3 w$, it can be shown that $\lambda \sim 11 w$ from geometrical principles, or vice versa. For example $\lambda$ should be equal to
$$0.5w + 2r + w + 2r + 0.5w = 2w + 4r = 2w + 4(2.3w) = 11.2 w$$
So since the two equations are not completely independent, the question is how could either of these relations be determined from a theoretical basis.
- Leopold, Luna B. and M. Gordon Wolman, 1960, River Meanders, Geological Society of America Bulletin, no. 6;769-793, https://www.usu.edu/jackschmidt/files/uploads/Fluvial_2013_Labs/Leopold_Wolman_1960.pdf