The Riemann problem for an elastic string with a linear Hooke’s law

Abstract

We solve the Riemann problem for vibrations of an infinite, planar, perfectly elastic string with a linear relation between the string tension and the local stretching, i.e., with a linear Hooke’s law. The motion is governed by a 4 × 4 4 \times 4 system of conservation laws that is linearly degenerate in all wave families. Conservation of energy leads to L 2 {L^{2}} estimates that are used in the analysis.