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Stationary Diffraction by Wedges: Method of Automorphic Functions on Complex Characteristics

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We present a complete solution to the classical problem of stationary diffraction by wedges with general boundary conditions (b.c.). For the Dirichlet and Neumann b.c., the solution was found by Sommerfeld in 1896 and for the impedance b.c. (or Leontovich and Robin b.c.) by Malyuzhinetz in 1958. Our approach relies on a novel “method of automorphic functions (MAF) on complex characteristics” which gives all solutions to boundary problems for second-order elliptic operators with general boundary conditions. This method is also applicable to the problems of guided water waves on a sloping beach, scattering of seismic waves, etc.

Author(s): Alexander Komech, Anatoli Merzon
Series: Lecture Notes in Mathematics 2249
Publisher: Springer
Year: 2019

Language: English
Pages: 157

Front Matter ....Pages i-xi
Introduction (Alexander Komech, Anatoli Merzon)....Pages 1-11
Front Matter ....Pages 13-13
The Early Theory of Diffraction (Alexander Komech, Anatoli Merzon)....Pages 15-17
Fresnel–Kirchhoff Diffraction Theory (Alexander Komech, Anatoli Merzon)....Pages 19-35
Stationary and Time-Dependent Diffraction (Alexander Komech, Anatoli Merzon)....Pages 37-41
The Sommerfeld Theory of Diffraction by Half-Plane (Alexander Komech, Anatoli Merzon)....Pages 43-61
Diffraction by Wedge After Sommerfeld’s Article (Alexander Komech, Anatoli Merzon)....Pages 63-68
Front Matter ....Pages 69-69
Stationary Boundary Value Problems in Convex Angles (Alexander Komech, Anatoli Merzon)....Pages 71-75
Extension to the Plane (Alexander Komech, Anatoli Merzon)....Pages 77-83
Boundary Conditions via the Cauchy Data (Alexander Komech, Anatoli Merzon)....Pages 85-87
Connection Equation on the Riemann Surface (Alexander Komech, Anatoli Merzon)....Pages 89-91
On Equivalence of the Reduction (Alexander Komech, Anatoli Merzon)....Pages 93-95
Undetermined Algebraic Equations on the Riemann Surface (Alexander Komech, Anatoli Merzon)....Pages 97-98
Automorphic Functions on the Riemann Surface (Alexander Komech, Anatoli Merzon)....Pages 99-100
Functional Equation with a Shift (Alexander Komech, Anatoli Merzon)....Pages 101-103
Lifting to the Universal Covering (Alexander Komech, Anatoli Merzon)....Pages 105-109
The Riemann-Hilbert Problem on the Riemann Surface (Alexander Komech, Anatoli Merzon)....Pages 111-115
The Factorization (Alexander Komech, Anatoli Merzon)....Pages 117-123
The Saltus Problem and Final Formula (Alexander Komech, Anatoli Merzon)....Pages 125-127
The Reconstruction of Solution and the Fredholmness (Alexander Komech, Anatoli Merzon)....Pages 129-137
Extension of the Method to Non-convex Angle (Alexander Komech, Anatoli Merzon)....Pages 139-148
Comments (Alexander Komech, Anatoli Merzon)....Pages 149-149
Back Matter ....Pages 151-167