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.

Reviewer: G.Bruckner (Berlin)

##### MSC:

35R30 | Inverse problems for PDEs |

35L05 | Wave equation |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

##### Keywords:

wave velocity; inverse problem; piecewise constant coefficient; one- dimensional wave equation; Lamé equation; Helmholtz equation
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\textit{M. M. Lavrent'ev jun.}, Sib. Math. J. 33, No. 3, 101--111 (1992; Zbl 0784.35121); translation from Sib. Mat. Zh. 33, No. 3, 101--111 (1992)

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##### References:

[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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