Nonlinear mode coupling of elastic waves

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Abstract
The theory of nonlinear elasticity is applied to a study of mode coupling of elastic waves at a particular frequency, called the critical frequency, in a long circular wire. It is assumed that the wire is of homogeneous isotropic elastic material and that the nonlinearity of medium (the effects of higher-order elasticity) is primary rather than that involved in the Lagrangian stress and strain tensors. The latter is suggested from the fact that the third-order elastic constants are of larger order of magnitude than the Lamé constants. The method of multiple scales is employed to obtain a system of equations which describes the behavior of the amplitudes involved in the mode coupling. The analysis of the equations shows that nonlinear mode coupling can occur at the critical frequency and that, except at this frequency, the wave undergoes only a phase shift. Further, progressive wave solutions show that the two wave amplitudes can be expressed in terms of Jacobian elliptic functions, and energy exchange between two modes takes place. Under special conditions, these periodic solutions degenerate to the solitary or shocklike solutions.

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