On the vibrations of a string with a concentrated mass and rapidly oscillating density

The paper is devoted to the vibrations of a string I   with a concentrated mass ε−1ϰQ(ε−1x)ε−1ϰQ(ε−1x) and rapidly oscillating density q(x,μ−1x)q(x,μ−1x), where q(x,ζ)q(x,ζ) is a 1-periodic in ζ   function, Q(ξ)Q(ξ) is a function with compact support, the integral of which is equal to one, 0∈I0∈I, μ,εμ,ε are small positive parameters, ϰ∈Rϰ∈R. By combining homogenization and the method of matched asymptotic expansions, we construct solutions to the problems up to O(ε+μ)O(ε+μ).