The behavior of the wave field in a resonator containing a cubically nonlinear medium is studied. The field is constructed as a linear superposition of two counter-propagating and strongly distorted waves. As distinct from a quadratic nonlinear medium the waves traveling in opposite directions are connected through their averaged (over the period) intensities. Both free and forced standing waves are studied. Profiles of discontinuous vibrations containing shock fronts of both compression and rarefaction are constructed. Resonant curves depicting the dependence of mean intensity on the difference between the frequency of vibration of the boundary and the natural frequency of one of the resonator’s mode are calculated. The structure of temporal profiles of strongly distorted forced waves is analyzed. It is shown, that shocks can appear only if the difference between the mean intensity and the discrepancy takes on definite negative values. The discontinuities are studied as jumps between the different solutions of a nonlinear integro-differential equation degenerating at weak dissipation to a third order algebraic equation with an undetermined coefficient. The dependence of the intensity of shocked vibrations on the frequency of vibration of the boundary is found. Nonlinear saturation is shown to appear: the intensity of wave field inside the resonator does not depend on the amplitude of boundary vibration when the amplitude is large. If the amplitude tends to infinity, the intensity tends to its limiting value determined by the nonlinear absorption at shock fronts. This maximum can be reached by smooth increase in frequency above the linear resonance. A hysteresis area and bistability appears, in analogy with the nonlinear resonance phenomena in localized vibration systems described by ordinary differential equations.

Resultat av stående akustiska vågor i en kubisk olinjär resonator presenteras.