Resonant radial oscillations of an inhomogeneous gas in the frustum of a cone

The effects of nonlinearity, geometry and inhomogeneity on the resonant motion of a gas contained in the frustum of a cone are investigated. The motion is radially symmetric, and the inhomogeneity arises from a body force term. We show how to construct a variable density, containing an arbitrary parameter μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu}$$\end{document} , that can be used to approximate a given density ρ(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho(r)}$$\end{document} . The approximate density allows us to solve exactly the eigenvalue equation associated with linear theory. This is the basis for continuous resonant solutions. There is a critical value of the parameter μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu}$$\end{document} which separates when the system behaves like a hard or soft spring. When motions are shocked, they may be represented by the superposition of oppositely traveling modulated simple waves. In all cases, approximate solutions are compared with exact numerical solutions.