Taking into account damping effects, the propagation-distance dependence of the amplitude of nonlinear guided waves arising from an internal resonance was investigated. In the analysis of Lamb-wave propagation, we introduced nonlinear quadratic effects derived from both geometric and material nonlinearities, referred to as classical nonlinearities, and damping effects by expressing the Lamé constants as complex numbers. Assuming a small detuning from the phase matching condition with respect to wavenumber and frequency, and using the method of multiple scales and self-adjointness of the governing equations with boundary conditions, the amplitude equations describing the nonlinear coupling of the two wave modes are derived as solvability conditions. This derivation also provides both physical and mathematical insights into both the phase and group velocity matching conditions. Numerical results obtained from these amplitude equations show that because of the nonlinear coupling an internal resonance between the two modes can occur under the above matching conditions. Without detuning, one mode increases monotonically in amplitude while the other decreases monotonically. Nevertheless, their changes gradually decrease with propagation distance, and each amplitude converges to a finite value. However, with wavenumber and frequency detuning, both amplitudes increase and decrease cyclically with propagation distance, resulting in a weakening of this resonance. Furthermore, results with damping effects show an attenuation of both amplitudes with propagation distance, which limits the distance resonance waves can propagate.

考虑到阻尼效应，研究了由内部共振引起的非线性导波振幅对传播距离的依赖性。在Lamb波传播的分析中，我们引入了从几何和材料非线性（称为经典非线性）得到的非线性二次效应，以及通过将Lamé常数表示为复数来获得阻尼效应。假设相位匹配条件在波数和频率方面略有失谐，并使用具有边界条件的控制方程的多尺度和自伴随方法，得出描述两个波模非线性耦合的振幅方程为可溶性条件。此推导还提供了相位和组速度匹配条件的物理和数学信息。从这些振幅方程获得的数值结果表明，由于非线性耦合，在上述匹配条件下，两种模式之间可能会发生内部共振。在不失谐的情况下，一种模式的振幅单调增加，而另一种模式的单调减少。然而，它们的变化随着传播距离逐渐减小，并且每个幅度收敛到一个有限值。然而，随着波数和频率的失谐，两个振幅都随着传播距离而周期性地增加和减少，从而导致该共振的减弱。此外，具有阻尼效应的结果表明，两个振幅都随传播距离而衰减，