This paper focuses on the optimal operation of water distribution networks. We model water distribution networks using physical and hydraulic constraints, and formulate a joint pump scheduling and water flow problem using the hydraulic characteristics of variable speed pumps. The optimal pump scheduling and water flow problem is a mixed integer nonlinear program. This problem is generally non-convex, and hence NP-hard. We propose a second-order cone relaxation for this problem, and analytically show that the proposed relaxation is exact for a wide class of water network topologies. The proposed problem is a mixed integer nonlinear program with a linear objective function and quadratic constraints. This problem can be solved with a commercial solver such as CPLEX. Finally, we consider a real-world water network, and demonstrate the effectiveness of the proposed relaxation in computing the optimal pump schedules and water flows.