In recent years, catenary channel has been widely used. There is no analytical solution to the transcendental equation for the calculation of critical depth and normal depth of catenary linear section. Firstly, the uniform flow and critical flow equations of the catenary linear section are derived by introducing appropriate dimensionless parameters in combination with the geometric characteristics of the catenary linear section, the hydraulic elements, and the basic equations of uniform flow and critical flow. After mathematical transformation, the Newtonian iterative formulae of normal and critical water depths of catenary linear section are obtained, and the initial value calculation formulas of them are obtained by using the optimization fitting principle, and the direct calculation formulas of normal and critical water depths are obtained after an iteration. Finally, the error analysis and comparison of the formula show that within the scope of engineering application, the maximum absolute relative error of the direct calculation formula of normal water depth and critical water depth is 5.33×10^-5% and 5.05×10^-5%, respectively. After the second iteration, the accuracy can be improved by 10⁸ times and 10⁶ times, respectively.