Fine Structure of the Crossing Resonance Spectrum of Wavefields in an Inhomogeneous Medium

Abstract

The crossing resonance of two wavefields m(x, t) and u(x, t) of different natures in an inhomogeneous medium with zero mean value of the coupling parameter η between fields has been studied. The stages of formation of the fine structure of the crossing resonance have been analyzed. It has been shown within the model of independent crystallites that the removal of the degeneracy of eigenfrequencies of these fields at the crossing resonance point has a threshold character in the coupling parameter and occurs under the condition η > ηc, where ηc = |Γu – Γm|/2, Γu and Γm are the relaxation parameters of the corresponding wavefields. At η > ηc, each random implementation of the Green’s functions $$\tilde {G}_{{mm}}^{{''}}$$ and $$\tilde {G}_{{uu}}^{{''}}$$ of wavefields has the form of two resonance peaks with the same half-width (Γu + Γm)/2 spaced by the interval 2η; this form is standard for crossing resonances. At η Γm, the function $$\tilde {G}_{{mm}}^{{''}}$$ has the form of a narrow resonance peak at ω = ωr, whereas the function $$\tilde {G}_{{uu}}^{{''}}$$ has the form of a broader resonance peak split at the top by a narrow antiresonance. Averaging over regions where η > ηc leads to the formation of a broad resonance with a resonance line half-width of about 〈η2〉1/2 on the both averaged Green’s functions, which is due to the stochastic distribution of resonance frequencies. Averaging over regions where η $${{{v}}_{u}}$$ at the first and second crossing points, respectively.