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**Asymptotic methods in electromagnetism.
(Méthodes asymptotiques en électromagnétisme.)**
*(French)*
Zbl 0817.35110

Mathématiques & Applications (Berlin). 16. Paris: Springer-Verlag. xvii, 416 p. (1994).

The object of this book is to discuss problems in the field of diffraction of high frequency electromagnetic waves and to consider the relationship between the boundary layer approach and the geometrical optics approach.

There are eight chapters. The first is concerned with the geometrical theory of diffraction and sets out the concepts involved. The second considers the idea of solutions in the form of asymptotic expansions and introduces the concept of the eikonal. The third chapter is entitled the boundary layer method. Amongst the topics discussed are creeping waves, whispering-gallery waves, caustics, diffraction by wedges and the matching of fields. The fourth, on the spectral theory of diffraction, introduces the idea of building up fields by means of integrating a spectrum of plane waves.

The fifth chapter, on uniform solutions, is a long one, comprising about a third of the main portion of the book. After a discussion of the principles of uniform expansions, there follow treatments of subjects such as diffraction by a wedge, the problem of a sudden change in curvature of the surface of a diffracting object, the shadow region, creeping waves on curved wedges, and caustics. The sixth chapter, integral methods, is mainly concerned with Maslov’s method, which provides solutions in the form of integrals, valid in the neighbourhood of caustics. The seventh chapter deals with surface fields and physical diffraction theory. In this approximations to diffracted fields are developed by using radiation from current elements on the illuminating part of the surface of the diffracted object. The eighth and final chapter, a short one, is concerned with the idea of surface impedance of a diffracting object, and the effect upon the diffraction fields.

The main portion of the book is followed by six appendices: canonical problems, differential geometry, asymptotic developments of integrals, complex rays, Fock functions and the reciprocity principle. These appendices provide some of the necessary mathematics for the main portion of the book.

This is a source book, rather than a textbook in that it generally provides results with references, rather than proving theorems. Much of the book is concerned with two dimensional fields, although there is also some work on three-dimensional fields. Each chapter is followed by an up to date (the latest is 1993) list of references, over 200 in all. Unfortunately, in one or two cases, these are incomplete and it would be difficult to trace them. A few printing errors were noticed, but they are such as to be easily spotted. This is an interesting book and should prove useful as a reference source for workers in the field.

There are eight chapters. The first is concerned with the geometrical theory of diffraction and sets out the concepts involved. The second considers the idea of solutions in the form of asymptotic expansions and introduces the concept of the eikonal. The third chapter is entitled the boundary layer method. Amongst the topics discussed are creeping waves, whispering-gallery waves, caustics, diffraction by wedges and the matching of fields. The fourth, on the spectral theory of diffraction, introduces the idea of building up fields by means of integrating a spectrum of plane waves.

The fifth chapter, on uniform solutions, is a long one, comprising about a third of the main portion of the book. After a discussion of the principles of uniform expansions, there follow treatments of subjects such as diffraction by a wedge, the problem of a sudden change in curvature of the surface of a diffracting object, the shadow region, creeping waves on curved wedges, and caustics. The sixth chapter, integral methods, is mainly concerned with Maslov’s method, which provides solutions in the form of integrals, valid in the neighbourhood of caustics. The seventh chapter deals with surface fields and physical diffraction theory. In this approximations to diffracted fields are developed by using radiation from current elements on the illuminating part of the surface of the diffracted object. The eighth and final chapter, a short one, is concerned with the idea of surface impedance of a diffracting object, and the effect upon the diffraction fields.

The main portion of the book is followed by six appendices: canonical problems, differential geometry, asymptotic developments of integrals, complex rays, Fock functions and the reciprocity principle. These appendices provide some of the necessary mathematics for the main portion of the book.

This is a source book, rather than a textbook in that it generally provides results with references, rather than proving theorems. Much of the book is concerned with two dimensional fields, although there is also some work on three-dimensional fields. Each chapter is followed by an up to date (the latest is 1993) list of references, over 200 in all. Unfortunately, in one or two cases, these are incomplete and it would be difficult to trace them. A few printing errors were noticed, but they are such as to be easily spotted. This is an interesting book and should prove useful as a reference source for workers in the field.

Reviewer: Ll.G.Chambers (Bangor)

### MSC:

35Q60 | PDEs in connection with optics and electromagnetic theory |

78-02 | Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory |

78A45 | Diffraction, scattering |

35C20 | Asymptotic expansions of solutions to PDEs |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |