Elementary wave solutions of the equations describing the motion of an elastic string. (English) Zbl 0577.73033

Shearer, Michael

SIAM J. Math. Anal. 16, 447-459 (1985).

In the paper the solution of the Riemann problem for the planar motion of an elastic string is presented. As an example the solution of a finite plucked string is discussed on a time interval shorter than the time necessary for the waves to reflect from the ends of the string. The Riemann problem is solved under the hypotheses that the longitudinal waves always propagate faster than the transversal waves. It is proved that this problem has a unique solution for any initial data admissible by a string. This hypothesis is too restrictive to be met by most of the materials of practical interest.
The author seems not to know that the Riemann problem was also uniquely solved much earlier by M. Mihăilescu and I. Suliciu in J. Math. Anal. Appl. 52, 10-24 (1975; Zbl 0312.35023) and in Rev. Roum. Math. Pures Appl. 20, 551-559 (1975; Zbl 0324.35057) under more realistic constitutive assumption and for the three dimensional motion of an elastic string. However, in these papers a slightly restrictive condition on the initial data is imposed.

Reviewer: N.Cristescu

MSC:

74H45	Vibrations in dynamical problems in solid mechanics
74M20	Impact in solid mechanics
74K05	Strings
74G30	Uniqueness of solutions of equilibrium problems in solid mechanics
74H25	Uniqueness of solutions of dynamical problems in solid mechanics
35L67	Shocks and singularities for hyperbolic equations

Keywords:

strict hyperbolicity; Riemann problem; planar motion; elastic string; finite plucked string; longitudinal waves always propagate faster than the transversal; waves; unique solution for any initial data; longitudinal waves always propagate faster than the transversal waves

Citations:

Zbl 0312.35023; Zbl 0324.35057