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This study explores wave fields on branched surfaces, showing they can be analytically continued into complex coordinates. A finite basis of these wave fields is identified, simplifying diffraction problem analysis.

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Area of Science:

  • Mathematical physics
  • Wave propagation theory
  • Diffraction theory

Background:

  • Wave fields on branched surfaces (Sommerfeld surfaces) arise from diffraction problems with ideal boundary conditions.
  • These surfaces are relevant to canonical diffraction problems like those involving half-lines or segments.

Purpose of the Study:

  • To investigate the analytical continuation of wave fields on Sommerfeld surfaces into complex coordinates.
  • To characterize the branch sets and basis functions of these continuations.

Main Methods:

  • Analytical continuation of wave fields into two complex coordinates.
  • Detailed study of the resulting branch sets.
  • Derivation of explicit Green's integral expressions for basis functions.

Main Results:

  • Wave fields on Sommerfeld surfaces admit analytical continuation into two complex coordinates.
  • The branch sets of this continuation are explicitly identified and analyzed.
  • For generic scattering problems, the multi-valued analytical continuation has a finite basis.
  • Basis functions are expressed using Green's integrals along double-eight contours.

Conclusions:

  • The finite basis property simplifies the analysis of wave fields on branched surfaces.
  • This finding is crucial for the application of coordinate equations in diffraction theory, particularly for segment diffraction.