The evolution of macrosonic standing waves in a resonator

The evolution of resonant oscillations in closed containers shaped like a cone or a bulb, as in the defining experiments of Lawrenson et al. [C.C. Lawrenson, B. Lipkens, T.S. Lucas, D.K. Perkins, T.W. Van Doren, Measurements of macrosonic standing waves in oscillating closed cavities, J. Acoust. Soc. Am. 104 (2) (1998) 623-636], is investigated theoretically. These shapes ensure the resonant frequencies are incommensurate, and hence that the resonant motions are shockless. The containers are oscillated parallel to the axis of symmetry at or near the fundamental frequency. The relationship between shocks and inhomogeneities in elastic panels is a related problem. A perturbation scheme that combines the method of multiple scales with a Duffing-type expansion yields a set of coupled nonlinear ordinary differential equations for the slow variation of the amplitude and phase of the fast resonant oscillations. The inclusion of damping ensures the oscillations approach a steady state, and the parameters in the steady state amplitude-frequency relation appear in the evolution equations.