On Properties of Large Wave Effect in Classical Problem of Bead String Vibration

Abstract

In Honor of Professor Saber Elaydi on His 60th Birthday.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 as W -• oo.Keywords: Oscillations of the elastic string with beads; Large wave effect; Asymptotic properties; Classical problem In Ref.[1], we discuss a difference-differential equation describing free vibrations of an elastic weightless string with beads:zo(O = O; ZNit) = O; 0 < f < oo; (2) Zj{O) = L (\<j^q<N-l) ZjiO) = O (q<j^N-\), (L > 0); (3) Zj(O) = O {j=l,...,N-l).This problem was studied by Lagrange [2]. Nevertheless, some new rather surprising results in this area were recently obtained [3][4][5]. Let M = {2,3,4,5,7,8,11,...} be a set of natural numbers consisting of all prime numbers and powers of 2.By virtue of the law of energy conservation for total energy variables Zj(t) (V/) have finite upper bound for t ^ 0, and therefore variableŝ f:= sup z"(t) (N>2, l<y<A^-l) are finite for each A' and ;.

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“…Such strings are widely used as simple models in physics (see e.g. [2][3][4]); in fact, the same type of equations arises e.g. in elasticity theory for systems of masses joined by springs (see e.g.…”

On inverse problem for tree of Stieltjes strings

“…Finite-dimensional spectral problems on an interval were considered in [5] (some recent results see in [16,17] and applications in [1,4]). Finitedimensional spectral problems on graphs occur in various fields of physics: mechanics of transverse vibrations of strings [6][7][8], longitudinal vibrations of point masses joined by springs [11], synthesis of electrical circuits [3,9].…”

Dirichlet–Neumann inverse spectral problem for a star graph of Stieltjes strings

“…Such strings are widely used as simple models in physics (see e.g. [25], [13], [14]). The same type of equations arises in elasticity theory for systems of masses joined by springs (see e.g.…”