Application of the theory of reversible discontinuities to the investigation of equations describing waves in tubes with elastic walls

Abstract

Equations which describe the propagation of waves in tubes with elastic walls are investigated, methods for calculating them are developed, and solutions containing reversible discontinuity structures are analysed in the case of a fluid-filled tube. The model of a tube with elastic walls constructed on the basis of the complete model of a membrane and the non-linear theory of elasticity is generalized. The viscosity and compressibility of the material, the possibility of filling the tube with a gas, and the flexural rigidity of the tube walls are taken into account. The problem of the decay of an arbitrary discontinuity is solved numerically in the case of a fluid-filled tube. The results obtained correspond to the previously developed theory of reversible discontinuities. Simplified hyperbolic equations of long waves, as well as equations for small-amplitude waves which do not take into account longitudinal elastic waves and are similar to the Boussinesq equations, are derived for cases when a tube is filled with a liquid and a gas. The possibility of overturning of the waves is analysed. A procedure for correcting the numerical schemes by adding terms with high-order derivatives to the equations is developed, and the order of approximation of the numerical scheme remains unchanged, enabling the performance of calculations with low schematic dissipation.