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 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 > η_c, where η_c = Γ_u – Γ_m /2, Γ_u and Γ_m are the relaxation parameters of the corresponding wavefields. At η > η_c, 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 (Γ_u + Γ_m)/2 spaced by the interval 2η; this form is standard for crossing resonances. At η < η_c, the functions ${\tilde{G}}_{mm}^{{}^{″}}$ and ${\tilde{G}}_{uu}^{{}^{″}}$ are different: if Γ_u > Γ_m, the function ${\tilde{G}}_{mm}^{{}^{″}}$ has the form of a narrow resonance peak at ω = ω_r, 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 > η_c 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 < η_c results in the sharpening of a resonance peak on the function $G_{mm}^{{}^{″}}$ and an antiresonance peak on the function $G_{uu}^{{}^{″}}$ at the same frequency ω = ω_r. 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 $G_{mm}^{{}^{″}}$ and the antiresonance peak on the function $G_{uu}^{{}^{″}}$. 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 Γ_m(k) ∝ k_c$v_m$ and Γ_u(k) ∝ k_c$v_u$ proportional to the correlation wavenumber k_c of inhomogeneities and to the velocities of the corresponding waves, which are summed with the homogeneous damping parameters Γ_m and Γ_u 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 $G_{mm}^{{}^{″}}$ and $G_{uu}^{{}^{″}}$ 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 $G_{uu}^{{}^{″}}$, and antiresonance peaks arise on the function $G_{mm}^{{}^{″}}$ because $v_m$ < $v_u$ and $v_m$ > $v_u$ at the first and second crossing points, respectively.