Summary: We investigate the optimal tuning in terms of modal damping of a viscous damper mounted near the end of a shallow cable. The damping properties of free vibrations are extracted from the complex wavenumber. The full solution for lower modes is evaluated numerically, and an explicit analytical approximation is obtained. It is found that the effect of the damper on nearly antisymmetric modes is independent of the sag and stiffness parameter. In contrast, the nearly symmetric modes develop regions of reduced motion near the ends, with increasing cable stiffness, and this reduces the effect of viscous damper. Explicit results are obtained for modal damping ratio and for optimal tuning of the damper.