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On the relation between coefficient and boundary value for solutions of Webster’s horn equation. (English) Zbl 0629.35009

Author’s abstract: Webster’s horn equation is a normalized version of the one-dimension linear acoustic wave equation. It has been used extensively as a simple model for plane wave propagation in layered systems, and particularly as the arena for much work on the relation between the acoustic impedance (coefficient) and the surface response or seismogram (boundary value) in theoretical seismology (this is the simplest so- called seismic reflection in inverse problem). The question of continuous dependence of the solution on the coefficient arises naturally in this context, particularly in connection with perturbational techniques.
We study the dependence of boundary values for solutions of Webster’s horn equation on its coefficient. For suitable choice of topologies (Sobolev spaces), we show that the map from coefficient to boundary values is a \(C^ 1\)-diffeomorphism.
Reviewer: B.Sleeman

MSC:

35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35R25 Ill-posed problems for PDEs
78A40 Waves and radiation in optics and electromagnetic theory
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