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Acoustic scattering of a plane wave by a circular impedance cone

Acoustic scattering of a plane wave by a circular impedance cone

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This chapter considers the diffraction of a plane acoustic wave illuminating completely the surface of a right-circular impedance cone. Formulation of the problem is also given for an arbitrary convex cone and uniqueness of the solution is discussed. By separating the radial variable the problem at hand is then reduced to that for a spectral function. Further separation of the angular variables for the circular cone leads to integral equations for the Fourier coefficients. The use of the Sommerfeld integral enables one to describe the far-field asymptotics for, in particular, the reflected wave, the spherical wave from the vertex of the cone, and the surface waves.

Chapter Contents:

  • 5.1 Formulation of the problem and uniqueness
  • 5.1.1 Formulation of the problem
  • 5.1.2 On uniqueness of the classical solution
  • 5.2 Kontorovich-Lebedev (KL) transform and incomplete separation of variables
  • 5.2.1 Integral representation of the solution
  • 5.2.2 Formulation of the problem for the spectral function u?
  • 5.3 The boundary value problem for the spectral function u? (ω,ω0)
  • 5.3.1 Separation of the angular variables for the circular cone
  • 5.3.2 Study of the integral equation for R(?, n)
  • 5.4 Diffraction coefficient in the oasis M for a narrow cone
  • 5.4.1 Problems for the leading terms and for the first corrections
  • 5.4.2 Calculation of V 1 and B 2j
  • 5.4.3 Basic formula for the diffraction coefficient of the spherical wave from the vertex of a narrow cone
  • 5.5 Numerical calculation of the diffraction coefficient in the oasis M
  • 5.5.1 Numerical aspects
  • 5.5.2 A perturbation series for ≫ 1
  • 5.5.3 Examples
  • 5.6 Sommerfeld-Malyuzhinets transform and analytic continuation
  • 5.6.1 Analytic properties of Φtilde;(α, ω, ω0) and Φ(α, ω, ω0)
  • 5.6.2 Problems for the Sommerfeld transformants
  • 5.6.3 The singularity corresponding to the wave reflected from the conical surface
  • 5.7 The reflected wave
  • 5.8 Scattering diagram of the spherical wave from the vertex
  • 5.9 Surface wave at axial incidence
  • 5.9.1 Ray solution for the surface wave
  • 5.9.2 Singularities of the Sommerfeld transformants corresponding to the surface wave
  • 5.9.3 Asymptotic evaluation of the surface wave
  • 5.10 Uniform asymptotics of the far field and the parabolic cylinder functions
  • 5.11 Appendices
  • 5.11.1 Appendix A
  • 5.11.2 Appendix B. Reduction of integrals
  • 5.11.3 Appendix C. Derivation of the constant C 0

Inspec keywords: integral equations; Fourier analysis; acoustic wave propagation

Other keywords: plane acoustic wave illumination diffraction; right-circular impedance cone; arbitrary convex cone; spectral function; plane wave; far-field asymptotics; surface waves; Fourier coefficients; radial variable; acoustic scattering; angular variable separation; Sommerfeld integral equation

Subjects: Mathematical analysis; Nonlinear acoustics and macrosonics; Function theory, analysis; Electromagnetic waves: theory; Electromagnetic wave propagation

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