Direct and inverse problems of acoustic sounding in a layered medium with discontinuous parameters

We consider the direct problem of finding a solution to a one-dimensional acoustic equation with discontinuous coefficients on the whole line $y\in\mathbb R$ with boundary conditions of special kind at the interior point $y=0$. We prove that the direct problem is uniquely solvable in the corresponding function space and obtain a special presentation for its solution. Along with the direct problem, we study the inverse problem of recovering the acoustic impedance of the medium from known one-sided limits of the solution to the direct problem and its derivative at the point $y=0$. It is shown that, with the use of the obtained special representation of the direct problem, the inverse problem can be reduced to a inverse spectral problem for a Sturm–Liouville operator with discontinuous coefficients.