Stochastic Вiscrete Wafe Differential Equations and the Discrete String with Random Masses

The problem under consideration is a modern development of the age-old problem of mathematics and mechanics on investigation of the vibrations of a string all the mass of which is concentrated at a finite number of equidistant points. We study the situation where the masses of points are the realizations of a sequence of random variables. On the basis of investigation of corresponding stochastic differential equation the explicit asymptotic expressions for the frequencies and amplitudes of vibrations and their probabilistic characteristics are found for the string of finite size and in the case where number of points ends to infinity. The central limit theorems for frequencies of vibrations are proved.