Exact solutions of alternate depth ratio for three exponential channels

Abstract

In this paper, the characteristics of exponential channels are investigated towards obtaining explicit solutions for the alternate depth ratio in three exponential channel shapes. The governing equations of alternate depth ratio for exponential channel in general and its three special cases viz., rectangular, parabolic and triangular channels in particular, are obtained. These governing equations of alternate depth ratio are solved using exact analytical methods. In the case of rectangular channel, the governing equation is quadratic and hence is extremely simple to solve. However, in the case of parabolic and triangular channels, though the equations are cubic and quartic, they are amenable to exact solution. Two empirical equations have been derived for each of parabolic and triangular channel shapes for more easy and practical calculation of alternate depth ratio for given initial Froude number. The results of exact and empirical calculations are compared and the empirical equations are found to be accurate with absolute errors being less than 1% and 1.2% for parabolic and triangular channels, respectively. © 2018, © 2018 Indian Society for Hydraulics.