A mathematical model of the dynamics of a curved pipeline as a membrane shell filled with an ideal compressible fluid is constructed. A form of an approximate solution of the model equations is found in which the dimension of the problem reduces by one. An asymptotic analysis of the obtained differential equations is performed for various small parameters. Two systems of model equations with different characteristics are obtained. These systems are transformed into systems of first-order equations in canonical form with explicitly defined Riemann invariants. The characteristics of the obtained systems of equations are found and the main types of waves propagating in a curved pipe are identified. Numerical experiments have been performed. The results obtained correspond to the results from the literature sources.