Nonlinear wave propagation in cylindrical air-filled tubes

Abstract

Three dimensional conservation laws are reformulated in a dimensionless form for a cylindrical air-filled tube. Three nondimensional numbers therefore appear, and are used to evaluate the order of magnitude of each term. Simplifying leads to two quasi-unidimensional generalized Burgers equations, which take into account both nonlinear phenomena and viscothermal losses. Special attention is paid to the validity of the different approximations we proposed. The method of multiple scales indicates that the interaction between the two opposite propagative waves is noncumulative and remains a local effect. The two Burgers equations are therefore globally independent. A numerical computation in the frequency domain solves the Burgers equations in the case of a stationary wave. The signal is supposed to be known at the two ends of the tube (or impedance instead of signal for one extremity). The computation uses the harmonic balance method to calculate the two opposite propagative waves which makes it possible to verify the right conditions in the two extremities. The signal everywhere in the tube is so pieced together. Some experiments are performed with a siren as source in one extremity, and measurements are compared with theoretical results.