Application of Analytical and Numerical Methods to the Sequent Depths Problem in Civil Engineering

A classical problem on hydraulic research is to provide explicit equations for evaluating the sequent depths of open channels whose geometry is not rectangular since barring for this section configuration, other solutions are nearly inexistent. Exponential section is the category which includes triangular, parabolic and rectangular channel shapes. For rectangular channel section, as pointed before, it is possible to express the sequent depths analytically by radicals. For other sections, the sequent depths are presently obtained by computational methods. In this paper, in order to find the sequent depths of channels whose sections are exponential or trapezoidal, we apply two different methods: The Lagrange’s Inversion Theorem, which is analytical and provides an exact solution by means of an infinite series; and the Householder’s Methods, which are numerical and provide approximations of the solutions by using an iterative algorithm. In general, the series obtained from Lagrange’s theorem have fast convergence. Otherwise, if the convergence rate is low, we use the Householder’s methods. Practical examples are also included.