Cross-modulation in guided wave propagation: how does it relate to the Luxemburg-Gorky effect?

The well-known Luxemburg-Gorky effect in radio waves has also been observed in elastic waves recently, which points to new possibilities for incipient damage detection. However, how the cross-modulation phenomenon of guided waves in a weakly nonlinear medium is related to the Luxemburg-Gorky effect remains an open question. This issue is investigated in this paper. Considering the third-order nonlinear elasticity of a plate waveguide, a theoretical framework is proposed to analyze the influence of the mode combination and mixing direction of a pair of single-frequency and modulated waves on the cross-modulated component generation. In particular, a codirectional shear-horizontal wave mixing scheme is highlighted which enables the generation of internally-resonant cross-modulated components at all frequencies. After verification by finite element simulation, mechanisms underpinning the cross-modulated components in the codirectional shear-horizontal wave mixing scheme are revealed through tactical tuning of the higher-order material elastic constants. Experiments are conducted to further confirm the phenomena and substantiate their relevance to the Luxemburg-Gorky effect. It is established that the cross-modulated components of guided waves can be generated and practically measured in a weakly nonlinear plate via both pure and mixed mechanisms as a result of the cubic nonlinearity instead of the quadratic nonlinearity. Compared with the conventional two-wave mixing methods based on quadratic nonlinearity, the cross-modulated components exhibit higher sensitivity to material microstructural changes, which is conducive to incipient damage detection. Although the observed nonlinear cross-modulation in guided waves shows similarities with the Luxemburg-Gorky effect, they stem from different mechanisms: the former from nonlinear elasticity and the latter nonlinear dissipation.