Understanding the wave propagation is crucial for dynamic analysis of long elastic cables prone to high-order vibrations. The coupling of elastic waves in cables is still not sufficiently understood. However, wave coupling produces attenuation frequency bands, which is useful for vibration control. In this study, the coupled waves equation for a damped cable with a small curvature was proposed. Using the equation, wave frequencies and velocities were studied. Then the cable responses in the stopbands and passbands were examined. Finally, the free responses were analysed in the spatial domain. The study focused on the influence of curvature and damping on wave propagation. The results showed that the wave coupling caused by curvature was critical for producing the attenuation bands and dispersion of waves. It generated the stopbands and passbands for the wave propagation, in which the waves respectively attenuated or propagated. Furthermore, the coupling effect induced pronounced dispersion in the in-plane longitudinal waves. As for damping, it not only affected the frequencies of stopbands and passbands, but also caused wave attenuation. Damping was found to shift the cut-off frequency from the real axis to the imaginary axis and transform the stopband from a line segment into a quadrilateral area. Additionally, the relationship between frequency and wave-number evolved from a two-dimensional curve into a three-dimensional surface. What's more, damping caused wave dissipation over time.

中文翻译：

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弹性波在小曲率、阻尼、无限电缆中的自由传播


了解波传播对于容易发生高阶振动的长弹性电缆的动态分析至关重要。电缆中弹性波的耦合仍然没有得到充分的了解。然而，波耦合会产生衰减频带，这对于振动控制很有用。在这项研究中，提出了小曲率阻尼电缆的耦合波方程。利用该方程，研究了波的频率和速度。然后检查阻带和通带中的电缆响应。最后，在空间域中分析自由响应。研究重点是曲率和阻尼对波传播的影响。结果表明，曲率引起的波耦合对于产生波的衰减带和色散至关重要。它生成波传播的阻带和通带，波在阻带和通带中分别衰减或传播。此外，耦合效应引起面内纵波的明显色散。至于阻尼，它不仅影响阻带和通带的频率，而且会引起波的衰减。研究发现阻尼可以将截止频率从实轴转移到虚轴，并将阻带从线段转变为四边形区域。此外，频率与波数之间的关系从二维曲线演变为三维曲面。更重要的是，随着时间的推移，阻尼会导致波耗散。