##
**Approximate boundary conditions in electromagnetics.**
*(English)*
Zbl 0828.73001

IEE Electromagnetic Waves Series. 41. London: IEE. x, 353 p. (1995).

Technological contingencies force us to consider the interaction of electromagnetic waves with objects of various forms and constitution. These objects, one might say, are more varied and complex than ever before. From this obvious remark follows the need to consider effective or approximate boundary conditions while dealing with Maxwell’s equations. Indeed, the complexity of the reaction and interactions of incident waves with bodies such as nonmetallic materials, composite structures, etc., calls for the construction of such effective boundary conditions if, in the end, we want that only the external field is present in the required boundary conditions. The authors of this monograph call them approximate boundary conditions (for short, ABCs), and they present the first comprehensive treatment of ABCs in applied electromagnetics with both analytical solutions and numerical approaches. There is a little need to emphasize the interest of this presentation for various problems concerning scattering, propagation, and wave guides in engineering problems involving interfaces, coatings, layered structures and perturbed surfaces such as rough and corrugated ones, in particular, in the high-frequency range.

The exposition of the authors, two noted contributors to the field, is progressive, dealing first with the simplest cases, and then increasing the order of differentiation of the approximate boundary conditions. The treated applications evolve in the same fashion. Therefore, after a first introductory chapter, chapter 2 naturally considers so-called first-order boundary conditions including impedance and transition types, the related error evaluation, surface perturbations such as due to roughness or corrugation, and the standard uniqueness theorem (non-positivity of the net real power flow at the surface). Chapter 3 deals with the application of these topics to planar structures, giving the standard techniques (Wiener-Hopf method and dual-integral equations) and comparison between these two, with examples to resistive and conductive half-planes, sheets and “impedance” junctions, and multiple illustrations from the so-many works of the authors (in particular from the “senior” one whose career has already fruitfully spanned more than forty years). The application to impedance wedges is treated in chapter 4. This is necessary because a high-frequency simulation requires the knowledge of the diffraction coefficient for a wedge of arbitrary angle. The related analytical computations are obviously made in the complex plane and offer an opportunity to introduce the celebrated works of Maliuzhinets (1950s) and Ufimtsev (1970s), especially in relation to the so-called physical theory of diffraction and the role played by the incremental length diffraction coefficient. This is rather technical and opportunely complemented by appendices providing Fortran subroutines for the evaluation of certain functions.

Boundary conditions on the \(n\)-th order are those effective boundary conditions that involve \(n\)-th order derivatives of the fields at most. These are supposed to be more realistic and accurate than usual boundary conditions. The analytical complexity of such conditions can be said to hide or capture the physical complexity of the phenomena at work at the surface or interface. This notion goes back to Rytov (1940s), although it is only in the 1960s that this started to be implemented. Thus chapter 5 considers second-order conditions (SOBC) with applications to surfaces of high-contrast materials and metal-backed layers. Half-planes and wedges again provide sample problems of diffraction with SOBCs. With chapter 6 enter higher-order conditions generically referred to as generalized impedance boundary conditions (GIBC). A high price in analytical and numerical complications has naturally to be paid to account for this. Here the authors present a formulation of Babinet’s principle to this rather complex situation, while in chapter 7, they apply such GIBCs to the case of cylindrical bodies, e.g., coated metallic cylinders. Finally, but this appears to be less connected to the authors’ works, the case of absorbing boundary conditions is dealt with in chapter 8. We remind the reader that this extremely important case stands for the case where the boundary should not perturb a field incident upon it, and this corresponds, in the practice of simulation, to a simulation of a surface which is actually not there. This flavor of “Cheshire cat” is of utmost importance in numerical simulations per force performed over finite domains. Here the emphasis is placed on two-dimensional rectangular and circular regions with “ABC” boundaries and on three-dimensional problems, where the method of successive approximations and that of mode annihilation, respectively, are exploited. These eight chapters forming a core of some 300 pages are complemented with a set of useful appendices recalling Rytov’s analysis, special functions, the method of steepest descent, etc., and providing useful Fortran routines.

What percolates from such a remarkably useful book is the importance of the works of the Russian school at all critical steps with Rytov in the 1940s, Maliuzhinets in the 1950s, Kouzov in the 1960s, and Ufimtsev in the 1970s. This nice work is obviously also well documented about works from the electrical engineering community in the USA, and it clearly exhibits the important contributions of its authors to the field. But unfortunately very few western European references are given, although we know of many relevant and constructive works. In all, however, forgetting the obvious emphasis on applications to electromagnetics, this nicely produced book – a true work of applied mathematics –, not a bedtime reading in reason of its analytical look and practice, provides a firm and rigorous background to many practitioners in all fields of wave propagation, whether electromagnetics, antenna theory, acoustics, elastic waves, nondestructive testing, signal processing, or seismology, for each one will find here something to crunch, some useful powerful methods and material, if only by analogy, and perhaps some inspiration. Both students and professionals will undoubtedly benefit from either reading this book in an ordered manner or safely keeping it on one’s shelf for further reference.

The exposition of the authors, two noted contributors to the field, is progressive, dealing first with the simplest cases, and then increasing the order of differentiation of the approximate boundary conditions. The treated applications evolve in the same fashion. Therefore, after a first introductory chapter, chapter 2 naturally considers so-called first-order boundary conditions including impedance and transition types, the related error evaluation, surface perturbations such as due to roughness or corrugation, and the standard uniqueness theorem (non-positivity of the net real power flow at the surface). Chapter 3 deals with the application of these topics to planar structures, giving the standard techniques (Wiener-Hopf method and dual-integral equations) and comparison between these two, with examples to resistive and conductive half-planes, sheets and “impedance” junctions, and multiple illustrations from the so-many works of the authors (in particular from the “senior” one whose career has already fruitfully spanned more than forty years). The application to impedance wedges is treated in chapter 4. This is necessary because a high-frequency simulation requires the knowledge of the diffraction coefficient for a wedge of arbitrary angle. The related analytical computations are obviously made in the complex plane and offer an opportunity to introduce the celebrated works of Maliuzhinets (1950s) and Ufimtsev (1970s), especially in relation to the so-called physical theory of diffraction and the role played by the incremental length diffraction coefficient. This is rather technical and opportunely complemented by appendices providing Fortran subroutines for the evaluation of certain functions.

Boundary conditions on the \(n\)-th order are those effective boundary conditions that involve \(n\)-th order derivatives of the fields at most. These are supposed to be more realistic and accurate than usual boundary conditions. The analytical complexity of such conditions can be said to hide or capture the physical complexity of the phenomena at work at the surface or interface. This notion goes back to Rytov (1940s), although it is only in the 1960s that this started to be implemented. Thus chapter 5 considers second-order conditions (SOBC) with applications to surfaces of high-contrast materials and metal-backed layers. Half-planes and wedges again provide sample problems of diffraction with SOBCs. With chapter 6 enter higher-order conditions generically referred to as generalized impedance boundary conditions (GIBC). A high price in analytical and numerical complications has naturally to be paid to account for this. Here the authors present a formulation of Babinet’s principle to this rather complex situation, while in chapter 7, they apply such GIBCs to the case of cylindrical bodies, e.g., coated metallic cylinders. Finally, but this appears to be less connected to the authors’ works, the case of absorbing boundary conditions is dealt with in chapter 8. We remind the reader that this extremely important case stands for the case where the boundary should not perturb a field incident upon it, and this corresponds, in the practice of simulation, to a simulation of a surface which is actually not there. This flavor of “Cheshire cat” is of utmost importance in numerical simulations per force performed over finite domains. Here the emphasis is placed on two-dimensional rectangular and circular regions with “ABC” boundaries and on three-dimensional problems, where the method of successive approximations and that of mode annihilation, respectively, are exploited. These eight chapters forming a core of some 300 pages are complemented with a set of useful appendices recalling Rytov’s analysis, special functions, the method of steepest descent, etc., and providing useful Fortran routines.

What percolates from such a remarkably useful book is the importance of the works of the Russian school at all critical steps with Rytov in the 1940s, Maliuzhinets in the 1950s, Kouzov in the 1960s, and Ufimtsev in the 1970s. This nice work is obviously also well documented about works from the electrical engineering community in the USA, and it clearly exhibits the important contributions of its authors to the field. But unfortunately very few western European references are given, although we know of many relevant and constructive works. In all, however, forgetting the obvious emphasis on applications to electromagnetics, this nicely produced book – a true work of applied mathematics –, not a bedtime reading in reason of its analytical look and practice, provides a firm and rigorous background to many practitioners in all fields of wave propagation, whether electromagnetics, antenna theory, acoustics, elastic waves, nondestructive testing, signal processing, or seismology, for each one will find here something to crunch, some useful powerful methods and material, if only by analogy, and perhaps some inspiration. Both students and professionals will undoubtedly benefit from either reading this book in an ordered manner or safely keeping it on one’s shelf for further reference.

Reviewer: G.A.Maugin (Paris)

### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

78A45 | Diffraction, scattering |

74J20 | Wave scattering in solid mechanics |

78A50 | Antennas, waveguides in optics and electromagnetic theory |

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