Resonant oscillations of an inhomogeneous gas between concentric spheres

Abstract

We investigate the effects of nonlinearity, geometry and stratification on the resonant motion of a gas contained between two concentric spheres. The emphasis is on whether the motion is continuous, and on how the inhomogeneity, geometry or nonlinearity can move the motion to a shocked state. Linear undamped theory yields a standing wave of arbitrary amplitude and an eigenvalue equation in which the higher eigenvalues are not integer multiples of the fundamental; the system is said to be dissonant. Higher modes, generated by the nonlinearity, are not resonant and consequently shocks may not form. When the output is shockless, the amplitude is two orders of magnitude greater than that of the input. When the eigenvalues for a homogeneous gas are not sufficiently dissonant and shocks form, the introduction of a stratification in the gas can restore dissonance and allow a continuous output. Similarly, the introduction of an inhomogeneity can change a continuous motion to a shocked one, as can an increase in the input Mach number, or an increase in the geometrical parameter. Various limits of the eigenvalue equation are considered and previous results for simpler geometries are recovered; e.g. a full sphere, a cone and a straight tube.