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Inverse problem for a wave equation with a piecewise-constant coefficient. (English. Russian original) Zbl 0784.35121
Sib. Math. J. 33, No. 3, 452-461 (1992); translation from Sib. Mat. Zh. 33, No. 3, 101-111 (1992).
The paper concerns the determination of \(C(x)\) in the one-dimensional wave equation \(u_{tt}=u_{xx}/c^ 2(x)\) under the initial condition \(u|_{t<0}=0\) and boundary condition \(u_ x(0,t)=H(t)\), and analogous problems in the Lamé equation and Helmholtz equation. Considering an isotropic layered medium in the half-space the coefficient \(c\) can assumed to be piecewise constant. The additional information is the knowledge of \(\int^ \infty_{-\infty}u_ t(0,t) e^{-i \omega t}dt\), \(\omega\) from a finite interval.

35R30 Inverse problems for PDEs
35L05 Wave equation
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI
[1] K. Aki and P. G. Richards, Quantitative Seismology: Theory and Methods, Vols. 1 and 2, W. H. Freeman (1980).
[2] M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskii, Uncorrected Problems of Mathematical Physics and Analysis [in Russian], Nauka, Moscow (1980).
[3] V. G. Romanov, Inverse Problems of Mathematical Physics [in Russian], Nauka, Moscow (1984). · Zbl 0576.35001
[4] V. G. Yakhno, Inverse Problems for Differential Equations of Elasticity [in Russian], Nauka, Novosibirsk (1990). · Zbl 0787.35124
[5] A. V. Baev, ?On the solution of the inverse boundary problem for a wave equation with a discontinuous coefficient,? Zh. Vychisl. Mat. Mat. Fiz.,28, No. 11, 1619-1633 (1988).
[6] M. L. Gerver, Inverse Problem for a Wave Equation with an Unknown Source of Vibration [in Russian], Nauka, Moscow (1974). · Zbl 0307.35002
[7] M. I. Belishev and N. V. Kupriyanova, ?Reflection of plane inclined wave on a layered half-space of periodic profile,? Akust. Zh.,29, No. 6, 733-735 (1983).
[8] M. I. Belishev, ?Reduction of velocity profile in an inhomogenous layer based on the low-frequency asymptote of the reflection coefficient,? Akust. Zh.,32, No. 1, 8-14 (1986).
[9] M. A. Lavrent’ev and B. V. Shabat, Methods in the Theory of Functions of a Complex Variable [in Russian], Fizmatgiz, Moscow (1958).
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