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The generalized Wiener-Hopf equations for the elastic wave motion in angular regions.

Vito G Daniele1, Guido Lombardi1

  • 1DET-Poltecnico di Torino, 10129 Torino, Italy.

Proceedings. Mathematical, Physical, and Engineering Sciences
|February 14, 2022
PubMed
Summary

This study presents a novel method for deriving generalized Wiener-Hopf equations (GWHEs) in elasticity, enabling wave motion analysis in angular regions. The approach extends electromagnetic techniques to elasticity for the first time.

Keywords:
Wiener–Hopf methodelasticityintegral equationsspectral domainwave motionwedge

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Area of Science:

  • Solid Mechanics
  • Wave Propagation
  • Mathematical Physics

Background:

  • Spectral functional equations are crucial for analyzing wave phenomena.
  • Existing methods for elasticity are limited, especially for complex geometries.
  • General Wiener-Hopf equations (GWHEs) have proven effective in electromagnetic wave analysis.

Purpose of the Study:

  • To introduce a general method for deducing spectral functional equations in elasticity.
  • To derive generalized Wiener-Hopf equations (GWHEs) for wave motion in angular regions.
  • To establish a complete theory for GWHEs in elastic wave propagation.

Main Methods:

  • Developed a vector differential equation of first-order for elastic wave motion.
  • Utilized a matrix dependent on the medium's properties within the angular region.
  • Projected reciprocal vectors of the matrix onto the elastic field on the region's faces.
  • Applied boundary conditions to functional equations to obtain GWHEs.

Main Results:

  • Successfully deduced spectral functional equations in elasticity.
  • Formulated a complete theory for deriving generalized Wiener-Hopf equations (GWHEs) in elasticity.
  • Extended the methodology from electromagnetic applications to elastic wave motion.
  • Applied the theory to the canonical problem of elastic scattering in angular regions.

Conclusions:

  • The proposed method provides a general framework for spectral functional equations in elasticity.
  • The derived GWHEs offer a powerful tool for analyzing elastic wave motion in angular domains.
  • This work bridges a gap in the theoretical understanding of elastic wave scattering.