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 twodimensional curve into a three-dimensional surface. What's more, damping caused wave dissipation over time.