Elementary Wave Solutions of the Equations Describing the Motion of an Elastic String

Abstract

The equations of planar motion of an elastic string form a $4 \times 4$ system of first order conservation laws. Two of the characteristic fields correspond to genuinely nonlinear longitudinal shocks and rarefaction waves, involving changes in the tension in the string, but not the slope. The other two fields correspond to contact discontinuities, across which the slope of the string jumps, reflecting the absence of any resistance to bending. Here, the tension T is related to the local elongation $\xi > 1$ in such a way as to ensure strict hyperbolicity: $T''(\xi ) > T{{(\xi )} / {\xi \geq 0}}$. The other main assumption is chosen to reflect properties of typical materials such as nylon and rubber. That is, $T'(\xi) $ is negative for $\xi > \xi _I $ and positive for $\xi > \xi _I $ for some $\xi _I > 1 $. The principal result of the paper is that the Riemann problem has a unique solution among combinations of centered waves, with a natural entropy condition placed on shocks. That is, any initial jump discontinuity in the tension, slope and velocity of the string can be resolved into combinations of longitudinal waves and contact discontinuities. This is illustrated for the plucked string problem, whose solution (valid until a wave first hits an end of the string) necessarily involves longitudinal waves.