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

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 eta between fields has been studied. ThThe 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 and 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 < ηc, the functions and are different: if Γu > Γm, the function has the form of a narrow resonance peak at ω = ωr , whereas the function 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 < ηc results in the sharpening of a resonance peak on the function and an antiresonance peak on the function at the same frequency ω = ωr . As a result, a pattern of the crossing resonance in the inhomogeneous medium is formed, consisting of identical broad peaks on both functions with the narrow resonance peak of the fine structure on the function and the antiresonance peak on the function . Thus, the fine structure of the spectrum of any crossing resonance of two wavefields of different natures in the inhomogeneous medium is due to the contribution of random realizations corresponding to degenerate states of the natural oscillations of the system. In a ferromagnet with a spatially inhomogeneous coupling parameter, spin and elastic waves acquire damping parameters Γm(k) ∝ kc and Γu(k) ∝ kc proportional to the correlation wavenumber kc of inhomogeneities and to the velocities of the corresponding waves, which are summed with the homogeneous damping parameters Γm and Γu of the same waves. This situation has been considered in a new self-consisting approximation for the case where the contribution of homogeneous damping parameters is negligibly small. It has been shown that the form of the fine structure on the functions and at the second (high-frequency) crossing point of dispersion curves of spin and elastic waves changes to the opposite form: narrow resonance peaks of the fine structure appear on the function , and antiresonance peaks arise on the function because < and > at the first and second crossing points, respectively stages of formation of the fine structur...