Nonlinear Wave Equation for a String under Time-Dependent Nonuniform Tension

The classical one-dimensional wave equation for a vibrating string relies on restrictive assumptions such as small slopes, planar motion, uniform tension, uniform mass per unit length, and constant cross-sectional area. In this work, we derive a nonlinear vector equation of motion for an extensible, perfectly flexible string evolving from an arbitrary initial configuration within the elastic limit. The present formulation determines the tension locally from the strain field. Finite elastic deformations are described using logarithmic (Hencky) strain, while transverse contraction is incorporated through the Poisson effect, leading to spatially and temporally varying cross-sectional area and linear mass density. Moreover, the string is allowed to have spatially varying mass per unit length and cross-sectional area in the strain-free configuration. A geometry-dependent damping model for motion in a resistive medium is also developed. The resulting equation reduces to the classical wave equation in the small-strain limit. Numerical simulations illustrate the spatiotemporal evolution of the cross-sectional area, tension, mass per unit length, and damping-force components governed by the nonlinear equation of motion. We also identify a nonsmooth peakon-type traveling wave solution in a local weak sense for the equation of motion restricted to pure longitudinal deformation.