On the transition between continuous and shocked solutions for resonant gas oscillations in the frustum of a closed cone

Abstract

Resonant gas oscillations in a closed straight tube contain shocks for frequencies near the linear resonant frequency. As the tube geometry changes from straight to a cone with slope $a,$ the shock strength decreases until, for large enough $a,$ the motion is continuous. The analytical result for small $a$ follows from a nonlinearization of the linear resonant response, while the result for large $a$ is a dominant single mode approximation. The connection between these two forms is analysed numerically. The shocked solutions change to multimode continuous solutions as $a$ increases to cross the curve $a_{s}\doteqdot 12.8M^{1/3}$ , where $M\ll 1$ is the Mach number of the input. Then the amplitudes of the higher modes decrease as $a$ continues to increase until, having crossed $a_{m}\doteqdot 30M^{1/3}$ , the single mode solution emerges.