Wave scattering by a transversal defect in a discrete waveguide

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.

💡 Research Summary

The paper investigates wave scattering in a discrete square‑lattice waveguide that contains a finite transverse Dirichlet screen. The geometry mimics the classic continuous waveguide with a screen, but the underlying medium is a lattice of point masses connected by springs, leading to a discrete Helmholtz equation. The authors first formulate the total field as the sum of a known incident duct mode and an unknown scattered field, imposing Dirichlet conditions on both the waveguide walls and the screen. By exploiting the symmetry of the problem (the scattered field is even in the longitudinal coordinate), the domain is reduced to the half‑space m ≥ 0.

A discrete Fourier transform is taken along the longitudinal direction, yielding two half‑transforms that are analytic inside and outside the unit circle, respectively. From these, even (Φ) and odd (Ψ) transforms are constructed, analogous to sine and cosine transforms in the continuous case. Applying the transform to the discrete Helmholtz equation produces a Wiener–Hopf system: a homogeneous difference equation for Φ in the regions outside the screen and an inhomogeneous equation for Ψ in the screen region. The kernel of the Wiener–Hopf problem is Λ(x)=ω²−4+x+x⁻¹, and the associated dispersion relation introduces a function y(x) that satisfies |y|≤1.

The general solution of the difference equations is expressed as a linear combination of yⁿ and y⁻ⁿ with unknown coefficient functions A(x), B(x), C(x), D(x). The inhomogeneous term is captured by a particular solution R(x,n)=Υ(x)sₚ(n+N₁), where sₚ denotes the sine shape of the incident mode and Υ(x)=1+Π(x) with Π(x)=(x²−1)/