Wave scattering by a transversal defect in a discrete waveguide

TL;DR

This work studies wave scattering by a finite transversal screen in a discrete square lattice waveguide with Dirichlet walls. It develops a Wiener Hopf framework that yields a 4x4 matrix kernel, which reduces to a 2x2 kernel in the symmetric case, and solves the problem exactly via pole removal, contrasting with the continuous case where exact solutions are elusive. The authors compute reflection and transmission coefficients with high accuracy and validate the results against a Boundary Algebraic Equations approach using a tailored lattice Green function. The study clarifies the discrete to continuous correspondence in waveguide diffraction, evaluates matrix factorisation prospects, and outlines future directions for extending the method to more general lattices and boundary conditions.

Abstract

We study wave scattering by a finite transversal strip in a discrete square-lattice waveguide with Dirichlet boundary conditions imposed on the strip and the waveguide walls. The setting is motivated as a discrete analogue of the classical continuous waveguide problem with a screen. The corresponding Wiener--Hopf formulation leads to an equation with a $4 \times 4$ matrix kernel, which reduces to a $2 \times 2$ matrix kernel under some symmetry assumptions. The factorisation prospects of this kernel are discussed, but this route is not followed. Instead, an exact analytical solution is obtained using the pole removal technique. This contrasts with the continuous case, where only approximate solutions are currently available. The reflection and transmission coefficients resulting from an incident duct mode are computed with an accuracy up to $10^{-13}$, showing consistency with theoretical predictions from continuous waveguide theory. In particular, full reflection and zero transmission are recovered as the frequency approaches the cut-off value for the incident mode. Finally, the solution is validated against a numerical computation of the diffraction problem via the Boundary Algebraic Equations method with a tailored lattice Green's function.

Figures (4)

• Figure 1: A discrete waveguide of a square lattice with a transversal Dirichlet screen and Dirichlet waveguide walls, $u^\text{tot}_{m,n} = 0$ for $(m,n)\in\Gamma',\Gamma"$
• Figure 2: Real part of the incident, scatter and total fields for the discrete waveguide with a transversal Dirichlet screen, where the number of the incident mode is $p=1$, defined by (\ref{['eq_u_in_duct']}), for two configurations: $\Omega=0.5$, $N_1=15$, $N_2=13$, $n_1=0, n_2=9$ (a), (c), (e) and $\Omega=0.5$, $N_1=10$, $N_2=19$, $n_1=0, n_2=9$ (b), (d), (f). The location of the screen is marked by black dots for the incident and scattered fields (a)-(d). On figures (e), (f), one can observe that the Dirichlet boundary condition on the screen is satisfied.
• Figure 3: Real parts $\text{Re}[T_p]$ and $\text{Re}[R_p]$ as functions of the lattice frequency $\Omega$ for $\ell=10$ and $\ell_0=10$, with incident mode $p=1$. The black line shows the imaginary part $\text{Im}[T_p]=\text{Im}[R_p]$. The solid line shows the analytic results obtained via the residue theorem from (\ref{['eq_R_res']}) and (\ref{['eq_RT_def']}), while the circles denote the numerical coefficients computed by the inner products in (\ref{['eq_RT_numerical']}) from the wavefield obtained by the boundary algebraic equations system by (\ref{['eq_solution_BAE']}).
• Figure 4: Frequency dependence of the absolute values of the magnitudes of (\ref{['eq_RT_def_weight']}) for $\ell=10$, $\ell_0=10$, with incident mode $p=1$. The solid line shows the analytic results obtained via the residue theorem from (\ref{['eq_R_res']}) and (\ref{['eq_RT_def']}), while the circles denote the numerical coefficients computed by the inner products in (\ref{['eq_RT_numerical']}) from the wavefield obtained by the boundary algebraic equations system by (\ref{['eq_solution_BAE']}). The black line shows the total energy flux.

Theorems & Definitions (6)

• Remark 3.1
• Remark 3.2
• Remark 4.1
• Remark 5.1
• Remark 6.1
• Remark 7.1