Nonlinear resonant oscillations in closed tubes of variable cross-section

An axisymmetric tube with a variable cross-sectional area, closed at both ends, containing a polytropic gas is oscillated parallel to its axis at or near a resonant frequency. The resonant gas oscillations In an equivalent tube of constant cross-section contain shocks. We show how cone, horn and bulb resonators produce shockless periodic outputs. The output consists of a dominant fundamental mode, where its amplitude and detuning are connected by a cubic equation - the amplitude-frequency relation. For the same gas, a cone resonator exhibits a hardening behaviour, while a bulb resonator may exhibit a hardening or softening behaviour. These theoretical results agree qualitatively with available experimental results and are the basis for resonant macrosonic synthesis (RMS).