Using pipes made of composite materials is to modify the performance of a fluid transporting system by combining materials with different properties to create lightweight, strong, corrosion-resistant, and durable devices. The wave propagation characteristics of composite pipes are of special interest when subjected to highly transient loads. However, transient loads can produce dynamic waves that travel through the pipe material. In such circumstances, composite pipes must show appropriate characteristics in managing the wave propagations generated by the applied loads. This paper attempts to study the wave propagation in composite pipes under the effects of internal flow velocity. The Hamilton principle and Euler-Bernoulli beam assumptions are used to derive the governing differential equations of pipes conveying fluids with clamped-free and clamped-clamped boundary conditions. The finite element method with the Newmark computational scheme is then utilized to discretize and solve the equations and find the time-dependent response of the system. The effects of boundary conditions and fluid flow velocity on the dynamic behavior of composite pipes will also be investigated. The results demonstrate that both fluid and boundary conditions significantly contribute to how composite pipes respond when subjected to highly transient impact loads.