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On properties of large wave effect in classical problem of bead string vibration. (English) Zbl 1329.74069

Summary: We consider a classical problem of free oscillations of the elastic weightless string with \(N+1\) beads which has been originally studied by Lagrange. It is proved that for \(N\) being prime or a power of \(2\), the maximal displacement of the bead from its equilibrium position increases logarithmically to infinity.

MSC:

74F05 Thermal effects in solid mechanics
34D05 Asymptotic properties of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
39A12 Discrete version of topics in analysis
74H45 Vibrations in dynamical problems in solid mechanics
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References:

[1] Myshkis AD, Journal Matematicheskaya Fizika, Analiz i Geometriya
[2] Lagrange GL, Mechanique Analitique (1788)
[3] Kurchanov PF, Prikladnaya Matematika i Mekhanika 55 pp 989– (1991)
[4] Filimonov AM, Computes Rendus Acad. Sci. Parisv. 313 pp 961– (1991)
[5] DOI: 10.1080/10236199808808125 · Zbl 0907.34050 · doi:10.1080/10236199808808125
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