One-dimensional reflection by a semi-infinite periodic row of scatterers

Abstract

h i g h l i g h t s• A canonical one-dimensional scattering problem is solved exactly in 3 ways. • First approach: shift semi-infinite row by one period. • Second approach: solve for finite row and then take limit. • Third approach: solve semi-infinite problem (discrete Wiener-Hopf). a b s t r a c t Three methods are described in order to solve the canonical problem of the onedimensional reflection by a semi-infinite periodic row of identical scatterers. The exact reflection coefficient R is determined. The first method is associated with shifting the domain by a single period and subsequently considering two scatterers, one being a single scatterer and the second being the entire semi-infinite array. The second method determines the reflection coefficient R N associated with a finite array of N scatterers. The limit as N → ∞ is then taken. In general R N does not converge to R in this limit, although we summarize various arguments that can be made to ensure the correct limit is achieved. The third method considers direct approaches. In particular, for point masses, the governing inhomogeneous ordinary differential equation is solved using the discrete Wiener-Hopf technique.

Method of singular integral equations in diffraction by semi-infinite grating: $H$-polarization case

“…Scattered field is expressed with the use of the reflection operator obtained by the operator method from the nonlinear operator equation. In [20], different approaches to diffraction by one-dimensional semi-infinite array are discussed.…”

The Wiener–Hopf technique, its generalizations and applications: constructive and approximate methods

“…For example, this happens in the Sommerfeld half plane problem for discrete Helmholtz equation [35]. This is also the case when an semi-infinite discrete array of point scatterers is considered [36–39]. Additionally, this type of problems is common in crack propagation problems [40].…”

The Wiener--Hopf Technique, its Generalisations and Applications: Constructive and Approximate Methods

“…For example, this happens in the Sommerfeld half plane problem for discrete Helmholz equation [162]. This is also the case when an semi-infinite discrete array of point scatterers is considered [42,87,121,172]. Additionally this type of problems is common in crack propagation problems [131].…”