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Marine acoustics. Direct and inverse problems. (English) Zbl 1055.35134

Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 0-89871-547-4/pbk). xii, 336 p. (2004).
The book deals with direct and inverse problems concerning marine acoustics. More exactly, the authors examine the case of an ocean occupying a 3D-strip and containing a (bounded) body. They assume that the body is a sound-soft obstacle not touching the seabed. Usually a time-harmonic sound - the incident wave - is sent through the water to the body - the scatterer - that, according to Huygens’s principle, scatters acoustic waves that can be collected by hydrophones suitably dislocated.
Before dealing with their problems related to marine acoustics, the authors devote a (well written) chapter to highlighting the fundamental principles governing mechanics of continua.
Assuming initially that the the incident wave is time-harmonic and is generated by a point source, the equation governing the scattered wave propagation is elliptic and of Helmholtz type with a distributional right-hand side. This equation is actually Helmholtz when the ocean is homogeneous, while is of Helmholtz type when the ocean is stratified. In this case the index of acoustic refraction is assumed to depend on depth, only.
Direct acoustic problems consist in analyzing the propagation of the scattered waves, satisfying a radiation condition at infinity, generated by the merged body. The study of asymptotics at infinity leads naturally to the determination of the so-called far-field pattern.
The authors examine also the more general case of the propagation of scattered waves when the seabed is assumed to be an elastic or poroelastic medium. In this case the seabed interacts with ocean acoustic waves.
Further, suitable models for shallow oceans with poroelastic seabeds are derived and validated comparing the theoretic results with experimental data. After that, some direct problems are analyzed by introducing homogeneization of the (periodic) seabed.
The first inverse problem dealt with in the book consists in recovering the shape of an unknown scatterer from a suitable set of information concerning scattered waves.
Since such a problem is, in general, ill posed, Tikhonov regularization techniques are required. Moreover, for a constant-depth ocean the information involving the far-field pattern operator \(F\) does not suffice, since \(F\) is not an injection in \(L^2(\partial \Omega)\), \(\Omega\) denoting the scatterer.
Although this trouble can be overcome by introducing suitable families of incident waves, yet the shape of \(\Omega\) cannot be, in general, uniquely recovered.
Another interesting problem consists in determining seamounts in a constant-depth ocean. For this problem the authors provide a uniqueness result and an algorithm for the reconstruction of the unknown seamount.
Assume now that a possibly non-homogeneous ocean lies on an elastic, but unknown, seabed. The authors deal with the fundamental, but mathematically very hard, problem consisting in recovering the elastic coefficients of the seabed. For such a problem only some numerical computations are provided.
We conclude by noting that this research book is, of course, not self-contained so that its reading is somewhere not simple. Yet, the arguments treated in the book are very interesting, although they may seem hard also to people working in elliptic PDE’s, due to their intrinsic computational difficulties. However, they are fundamental for a lot of practical applications such as shipping, fishing, extraction of natural resources (e.g. petroleum) and so on.
So, I recommend this book to specialists in direct or inverse problems for PDE’s.

MSC:

35R30 Inverse problems for PDEs
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35R25 Ill-posed problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35C15 Integral representations of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35P25 Scattering theory for PDEs
86A22 Inverse problems in geophysics
86A05 Hydrology, hydrography, oceanography
76Q05 Hydro- and aero-acoustics

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