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Diverse areas of physics and its applications deal with wave scattering at rough surfaces: optics, acoustics, remote sensing, radio astronomy, physics of solids, diffraction theory, radio and wave-propagation techniques, etc. The mathe matical description of relevant problems is rather universal and theoretical methods analysing this phenomenon form a definite area of mathematical physics. It was founded by Lord Rayleigh at the beginning of the 20th century and has been intensively developed since 1950 in response to practical needs. Modern theory involves, along with the classical methods, some new ap proaches with much more extensive possibilities. The theoretical methods describing wave scattering at rough surfaces repre sent the subject of this book. Having studied this area over many years, the author came to the conclusion that most of the results found in this theory can easily be obtained and comprehended in the framework of a rather universal scheme and this became the motive for writing the present monograph. The first half of the book deals with the classical results. However, several new problems are considered in connection with the methods used here (for example, some applications of the Rayleigh hypothesis and its relation to the Lippman argument). The second half is fully devoted to recent results presented in papers but not yet in books. For this reason the author hopes that the present book will be of interest to both newcomers and experts in this area.
1. Introduction.- 2. Scattering Amplitude and Its Properties.- 2.1 Formulation of Problem and Some Auxiliary Relations.- 2.2 Scattering Amplitude.- 2.3 Reciprocity Theorem and Unitarity Condition.- 2.4 Scattering Amplitude for Statistical Case.- 3. Helmholtz Formula and Rayleigh Hypothesis.- 3.1 Helmholtz Formula.- 3.2 Rayleigh Hypothesis.- 3.3 Analytical Properties of Surface Sources Density function.- 3.4 Rayleigh Hypothesis and Lippman Argument: Completeness of Set of the Plane Waves.- 4. Small Perturbation Method.- 4.1 Dirichlet Problem.- 4.2 Small Perturbation Method and the Rayleigh Hypothesis.- 4.3 Neumann Problem.- 4.4 Scattering of Electromagnetic Waves at a Perfectly Conducting Boundary.- 4.5 Scattering of Sound Waves at the Interface Between Liquid Half-Spaces.- 4.6 Electromagnetic Wave Scattering at the Interface Between Two Dielectrics.- 5. Kirchhoff`—Tangent Plane Approximation.- 5.1 Tangent Plane Approximation.- 5.2 Kirchhoff Approximation.- 5.3 Corrections to the Kirchhoff Approximation: Deterministic Case.- 5.4 Corrections to the Kirchhoff Approximation: Statistical Case.- 5.5 Two-Scale Model.- 6. “Nonclassical” Approaches to Wave Scattering at Rough Surfaces.- 6.1 Small Slope Approximation.- 6.2 Small Slope Approximation and Rayleigh Equation. Numerical Experiments.- 6.3 Phase Perturbation Technique.- 6.4 Phase Operator Method for Dirichlet Problem.- 6.5 Meecham— Lysanov Approach.- 6.6 Relation Between Bahar’s Full-Wave Approach and Small Slope Approximation.- 7. Waveguide with Statistically Rough Boundary in Noncorrelated Successive Reflections Approximation.- 7.1 Directional Source in the Waveguide with Rough Boundary: General Solution.- 7.2 Average Field.- 7.3 Correlation Function of the Field.- 7.4 Radiative Transport Theory.- 7.5 Problem of Noise Surface Sources: Diffusion Approximation.- References.