Wave propagation in a semi‐infinite, nonuniform, elastic thin‐walled tube filled with incompressible, inviscid fluid is considered. A one‐dimensional problem, with the tube and fluid initially at rest and a pressure disturbance applied at the end, is solved for different axial variations of the tube properties. Formal asymptotic techniques are applied to the linearized partial differential equation governing the pressure and simple progressing wave and high‐frequency solutions are obtained for Heaviside and oscillatory‐pressure boundary conditions, respectively. These asymptotic series solutions terminate for certain parameter variations to give exact solutions for the pressure. Numerical results are presented and some interesting observations concerning the accuracy of the formal results are made.