Geodesic
Direct and inverse problems of acoustic sounding in a~layered medium with discontinuous parameters
Sibirskij žurnal industrialʹnoj matematiki, Tome 17 (2014) no. 1, pp. 120-134

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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.
Keywords: direct and inverse problems, Sturm–Liouville operator, inverse spectral problem, acoustic impedance. 
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     author = {A. A. Sedipkov},
     title = {Direct and inverse problems of acoustic sounding in a~layered medium with discontinuous parameters},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {120--134},
     publisher = {mathdoc},
     volume = {17},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2014_17_1_a12/}
}
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A. A. Sedipkov. Direct and inverse problems of acoustic sounding in a~layered medium with discontinuous parameters. Sibirskij žurnal industrialʹnoj matematiki, Tome 17 (2014) no. 1, pp. 120-134. http://geodesic.mathdoc.fr/item/SJIM_2014_17_1_a12/