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Short-wavelength diffraction theory. Asymptotic methods. Transl. from the Russian by E. F. Kuester. (English) Zbl 0742.35002

Springer Series on Wave Phenomena. 4. Berlin etc.: Springer-Verlag. xi, 445 p. (1991).
[For the Russian original edition (1972) see Zbl 0255.35002.]
The book under review presents in a systematic manner the most important modern methods and results in the study of diffraction problems.
After a short preface and an introduction, in Chapter 2 the ray method (the geometrical optics approximation) for the wave equation and for the Maxwell equations is presented. In Chapter 3 an analog of this method is considered in the case of an irregular ray field. In Chapter 4 the Keller-Rubinow method (a heuristic method for obtaining asymptotic expansions for the eigenvalues and eigenfunctions of the Helmholtz equation) is presented, while in Chapter 5 this method is extended to handle the case of “closed congruencies constructed from rays that satisfy Euler’s equation and the law of reflection only in the first approximation”.
Chapter 6 is devoted to the parabolic method of obtaining “eigenfunctions of the whispering gallery type and to find solutions of the wave equation concentrated in a neighborhood of a ray”. In Chapter 7 a method of finding asymptotic expansions of eigenvalues and eigenfunctions is described. In Chapters 8 and 10 the construction of the eigenfunctions concentrated in the neighborhood of an extremal ray region to the Helmholtz equation is carried out by “the etalon problem method” and by “the parabolic equation method” respectively, while Chapter 9 contains a construction of the eigenfunctions of the Laplacian in the neighborhood of a closed geodesic on an \(m+1\)-dimensional compact orientable Riemann manifold.
In Chapter 11 a construction of “the shortwave asymptotic expansion in the shadow zone for the two-dimensional problem of a point source located in the side of a curve \(S\) where whispering gallery waveforms cannot arise” is presented, while in Chapter 12 “the high-frequency scalar field of a line source inside an inhomogeneous infinite body close to its surface” is investigated. Finally, in Chapter 13 “the high-frequency asymptotic solution for the diffraction of a wave, given as a ray expansion, by a smooth surface \(S\) on which the Dirichlet condition \(u|_ S=0\) is satisfied” is presented. The book ends with an appendix containing some notion and results widely used in the text.
Reviewer: I.Vrabie (Iaşi)

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q60 PDEs in connection with optics and electromagnetic theory
35C20 Asymptotic expansions of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35L05 Wave equation
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs

Citations:

Zbl 0255.35002
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