Determination of local inhomogeneity of the medium from the natural frequencies of string oscillations

The paper considers the problem of determining the local inhomogeneity of the medium from the natural frequencies of string oscillation. The inhomogeneity is modeled in three sections: in the first and third sections medium is homogeneous, and on average section the elastic characteristics are modeled by a quadratic function. This model is implemented using the conjugation conditions at boundary between media. It is shown that to identify the center of an inhomogeneity and determine its size, two natural frequencies are enough, and in the case of rigid fixing of both ends of the string, the solution of the problem is dual. The problem is solved by expanding the fundamental system of solutions into a power series in the variables $x$ and $\lambda$. The estimates of the error of the method are given.