The motion of a planar string is modeled by a system of four first order conservation laws. For simplicity, we consider mostly a string satisfying Hooke's law, i.e., a linear stress/strain relationship. When the tension of the string is positive, the system is strictly hyperbolic, with a wave structure given by fast longitudinal waves and slow transversal waves. When the tension is negative, the system becomes non-hyperbolic and unstable. However, in the generic situation, the string seems to avoid this regime, instead folding back on itself in a highly singular manner, with the tension dropping down to, but never below, zero. I explain how an attempt at using Glimm's random choice method fails to throw any light on the problem, while simulations using front tracking yield highly suggestive results indicating that weak solutions may exist even for the case when the string goes slack.

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Helge Holden <holden@math.ntnu.no>

2000-09-20 08:32