This chapter discusses several basic analytical tools to be repeatedly exploited in the main part of the monograph for dealing with diffraction of acoustic and electromagnetic waves by wedges and cones with impedance boundary conditions. We formulate basic equations, boundary, edge, and radiation conditions in the first three sections. Then Section 1.4 outlines different integral transforms and representations, including the Sommerfeld-Malyuzhinets and Kontorovich-Lebedev transforms. Section 1.5 expounds the solution to the so-called Malyuzhinets problem. The theory of Malyuzhinets functional difference equations for one unknown function is detailed in the last section of the chapter.

Chapter Contents:

• 1.1 Equations for acoustic and electromagnetic waves
• 1.1.1 Acoustic waves
• 1.1.2 Electromagnetic waves
• 1.2 Boundary conditions
• 1.3 Edge and radiation conditions
• 1.3.1 Vicinity of the edge and Meixner's condition
• 1.3.2 On the behavior of solutions to the Helmholtz equation in the angular domain as r→0
• 1.3.3 Radiation conditions: Formulation of the problem
• 1.3.4 The limiting-absorption principle
• 1.4 Integral transformations
• 1.4.1 Fourier transform and the convolution theorem
• 1.4.2 The Sommerfeld integral
• 1.4.3 Malyuzhinets's theorem: Sommerfeld-Malyuzhinets (SM) transform
• 1.4.4 Kontorovich-Lebedev (KL) transform and its connection with the Sommerfeld integral
• 1.4.5 Watson-Bessel integral
• 1.5 Malyuzhinets's solution for the impedance wedge diffraction problem
• 1.5.1 Functional equations for the Malyuzhinets problem
• 1.5.2 The multiplication principle and the auxiliary solution Ψ0(Z) to the functional equations (1.104)
• 1.5.3 The Malyuzhinets function ΨΦ(Z) and its basic properties
• 1.5.4 Examination of (d/dZ) lnΨΦ(Z)
• 1.5.5 The Malyuzhinets function ΨΦ(Z)
• 1.5.6 Completion of the construction of Ψ0(Z) and of s(Z)
• 1.5.7 Far-field analysis of the exact solution
• 1.6 Theory of Malyuzhinets functional equations for one unknown function
• 1.6.1 General Malyuzhinets equations
• 1.6.2 Solution to the homogeneous Malyuzhinets equations
• 1.6.3 Solution to the inhomogeneous Malyuzhinets equations
• 1.6.4 Modified Fourier transform and S-integrals
• 1.6.5 The direct application of S-integrals

Inspec keywords: electromagnetic wave diffraction; boundary-value problems; acoustic wave diffraction; difference equations; functional equations; shapes (structures); transforms

Other keywords: Malyuzhinets functional difference equations; impedance boundary conditions; wedges; integral transforms; acoustic wave diffraction; electromagnetic wave diffraction; cones; Sommerfeld-Malyuzhinets transforms; Kontorovich-Lebedev transforms; integral representations; radiation conditions

Subjects: Electromagnetic wave propagation; Differential equations (numerical analysis); Integral transforms; Continuum mechanics; Electromagnetic waves: theory; Mathematical analysis; Function theory, analysis; Numerical approximation and analysis; Ultrasonics, quantum acoustics, and physical effects of sound