Second-harmonic generation in a multilayered structure with nonlinear spring-type interfaces embedded between two semi-infinite media

The acoustic second-harmonic generation behavior in a multilayered structure with nonlinear spring-type interlayer interfaces is analyzed theoretically to investigate the frequency dependence of second-harmonic amplitudes in the reflected and transmitted fields when the structure is subjected to the normal incidence of a monochromatic longitudinal wave. The multilayered structure consists of identical linear elastic layers and is embedded between two identical linear elastic semi-infinite media. The layers are bonded to each other by spring-type interfaces possessing identical linear stiffness but different quadratic nonlinear parameters. By combining a perturbation analysis with the transfer-matrix method, analytical expressions are derived for the second-harmonic amplitudes of the reflected and transmitted waves. The second-harmonic amplitudes due to a single nonlinear interface are shown to vary remarkably with the fundamental frequency, reflecting the pass and stop band characteristics of the Bloch wave in the corresponding infinitely extended layered structure. By calculating the spatial distribution of second-harmonic amplitude inside the multilayered structure, the influence of the position of the nonlinear interface as well as the number of layers on the frequency dependence of second-harmonic amplitudes of the reflected and transmitted waves is elucidated. When all interlayer interfaces possess the identical nonlinearity, the second-harmonic amplitudes on both sides of the structure are shown to increase monotonically with the number of layers in the frequency ranges where both fundamental and double frequencies are within the pass bands of Bloch wave. The influence of two non-dimensional parameters, i.e., the relative linear compliance of the interlayer interfaces and the acoustic impedance ratio between the layer and the surrounding semi-infinite medium, on the second-harmonic amplitudes is elucidated.