Linear and nonlinear propagation of higher order modes in hard-walled circular ducts containing a real gas

Abstract

This paper deals with the linear and nonlinear propagation of sound waves through a real gas contained in a circular tube with rigid, isothermal walls. Special emphasis is placed on the asymptotically correct treatment of the higher order modes and their interaction with the acoustic boundary layer. In the first part, a linear perturbation analysis is carried out to calculate the correction terms arising from the viscothermal damping mechanisms present in the system. In extension to previous work, the propagation length is assumed to be so large that the exponentially growing boundary layer effects do not only affect the second order terms of the sound pressure but also the leading order terms. The series expansions derived for the propagation parameters extend the results given in the literature with additional terms resulting from viscosity and heat conduction in the core region. The second part is concerned with the nonlinear modulation of a wave packet transmitted through a real gas. A damped nonlinear Schrödinger equation is derived and its solutions for positive as well as negative values of the nonlinearity parameter are studied. In particular, the case of wave propagation in ducts containing a so-called BZT fluid is discussed.

Sound travellingwaves in wind instruments as solutions to non linear homogeneous gas dynamics equations

“…This construction of a wave should also feed the discussion between linear or non linear models (see for instance [5]). It seems, as a conclusion, that the difference is not so important when a source term is present.…”