Propagation of standing waves in axisymmetric cavities enclosing an inhomogeneous medium

Abstract

This paper studies how gradients in speed of sound c and density ρ influence standing waves in an axisymmetric cavity. The c and ρ gradients are assumed independent of radial and circumferential coordinates of the cavity and to vary only along its axis. The effects on propagation of the c and ρ gradients are studied separately. First, the problem with a step function as c gradient in a cylindrical cavity is treated in detail since it is amenable to analytical solution. A smooth c gradient described by exponential functions is considered. In this case, a transfer matrix coupled to a numerical integration procedure is adapted to the solution of the differential equation governing the axial dependence. The quasilinear c2 gradient follows and it also yields to analysis. The poor energy transmission characteristics of transverse standing waves in cylindrical cavities with c gradient prompted consideration of a geometry that includes two cylindrical segments with different cross sections joined by a conical connector. The transfer matrix method couples the acoustics of various segments of cavities. The radial dependence in the conical segment is determined by the method of shooting while the axial dependence follows from numerical integration. The analysis is then extended to cavities with concentric center bodies. An approximation considers piecewise conical frustra and leads to discontinuous pressure gradients at the junction of conjoined segments. The approximate analysis is tested against numerical results from an accurate simulation that invokes a Green's function and surface elements with account made for medium inhomogeneity.