scholarly journals A Surprising Observation in the Quarter-Plane Diffraction Problem

2021 ◽  
Vol 81 (1) ◽  
pp. 60-90
Author(s):  
Raphael C. Assier ◽  
I. David Abrahams
Keyword(s):  
Diffraction Problem ◽  
Quarter Plane ◽  
Plane Diffraction ◽  
Surprising Observation

scholarly journals On the asymptotic properties of a canonical diffraction integral

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200150
Author(s):  
Raphaël C. Assier ◽  
I. David Abrahams
Keyword(s):  
Unknown Function ◽  
Diffraction Problem ◽  
Asymptotic Properties ◽  
Complex Field ◽  
Cauchy Integral ◽  
Decay Properties ◽  
Diffraction Integral ◽  
Quarter Plane ◽  
Plane Diffraction ◽  
Appl Math

We introduce and study a new canonical integral, denoted I + − ε , depending on two complex parameters α 1 and α 2 . It arises from the problem of wave diffraction by a quarter-plane and is heuristically constructed to capture the complex field near the tip and edges. We establish some region of analyticity of this integral in C 2 , and derive its rich asymptotic behaviour as | α 1 | and | α 2 | tend to infinity. We also study the decay properties of the function obtained from applying a specific double Cauchy integral operator to this integral. These results allow us to show that this integral shares all of the asymptotic properties expected from the key unknown function G +− arising when the quarter-plane diffraction problem is studied via a two-complex-variables Wiener–Hopf technique (see Assier & Abrahams, SIAM J. Appl. Math. , in press). As a result, the integral I + − ε can be used to mimic the unknown function G +− and to build an efficient ‘educated’ approximation to the quarter-plane problem.

The half-plane diffraction problem for harmonic time dependence

1959 ◽  
Vol 252 (1270) ◽  
pp. 411-417 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Time Dependence ◽  
Boundary Value Problems ◽  
Half Plane ◽  
Mixed Type ◽  
Boundary Value ◽  
Diffraction Problem ◽  
Two Dimensional ◽  
Conducting Screen ◽  
Plane Diffraction

Green’s functions are obtained for the boundary-value problems of mixed type describing the general two-dimensional diffraction problems at a screen in the form of a half-plane (Sommerfeld’s problem), applicable to acoustically rigid or soft screens, and to the full electromagnetic field at a perfectly conducting screen.

Diffraction of a pulse by a resistive half-plane. I. Normal incidence

1960 ◽  
Vol 255 (1283) ◽  
pp. 538-549 ◽  
Keyword(s):  
Time Dependence ◽  
Half Plane ◽  
Wave Diffraction ◽  
Diffraction Problem ◽  
Normal Incidence ◽  
Step Function ◽  
Two Dimensional ◽  
Dynamic Similarity ◽  
Function Time ◽  
Dimensional Wave

The two-dimensional wave diffraction problem, acoustic or electromagnetic, in which a pulse of step-function time dependence is diffracted by a resistive half-plane is solved by assuming dynamic similarity in the solution.

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