A note on the vibrations of a nearly vertical hanging string: An asymptotic analysis

This paper analyzes the frequencies and normal vibration modes of a suspended string in a nearly vertical configuration, taking into account the influence of its self-weight and the variable tension induced by the catenary shape in equilibrium. The study is carried out through a perturbation analysis around the catenary configuration. The results show that when the catenary formed by the string is nearly vertical, the tension at the lowest point is nonzero, in contrast with the case of a perfectly vertical hanging string. This finding provides a physical interpretation of the well-known paradox of the vertical hanging string with both ends fixed.