Application of the WKB method to catenary-shaped slender structures

The purpose of this work is the derivation of closed form expressions for the linear vibration modes of catenary-shaped slender structures. The dynamic behaviour of the structure is described using the Euler–Bernoulli beam formulation with variable tension and angle terms. The desired expressions are obtained by treating the governing fourth-order partial differential equation of dynamic equilibrium using the WKB method [J.D. Logan, Applied Mathematics, second ed., Wiley Interscience, 1997].