arxiv: 2604.16050 · v1 · submitted 2026-04-17 · 🧮 math-ph · math.CV· math.MP

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Embedding formulae for diffraction problems on square lattices

A. I. Korolkov , A. V. Kisil

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Pith reviewed 2026-05-10 08:00 UTC · model grok-4.3

classification 🧮 math-ph math.CVmath.MP

keywords embedding formulaeWiener-Hopfdiffractionsquare latticesDirichlet scatterersplane wave incidenceboundary value problems

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The pith

Embedding formulae express square-lattice diffraction solutions for any incidence from a finite set of auxiliaries

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops embedding formulae for all diffraction problems with Dirichlet scatterers on square lattices using the Wiener-Hopf perspective. These formulae express the solution for arbitrary plane-wave incidence in terms of a finite set of auxiliary problems that are solved only once. The approach is first applied to canonical geometries such as the half-plane, finite strip, and right-angled wedge, then generalized via an operator-based method to arbitrary obstacle configurations. This generality stands in contrast to the continuous setting where such embedding is not yet possible. A reader would care because it enables efficient computation without resolving the full boundary-value problem for every incidence angle.

Core claim

The authors show that embedding formulae exist for arbitrary configurations of obstacles on square lattices, obtained through an operator-based Wiener-Hopf approach, allowing the general solution to be built from a finite number of auxiliary problems.

What carries the argument

The embedding formula derived from the operator-based Wiener-Hopf method, which reduces the dependence on incidence angle to a linear combination of fixed auxiliary solutions.

If this is right

The solution for any plane-wave incidence is expressed using only the auxiliary problems solved once.
The formula applies to any finite or infinite arrangement of Dirichlet obstacles on the square lattice.
Numerical experiments confirm that results from the embedding formula agree with direct solutions.
The method offers potential for applications in inverse problems and discrete wave models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar operator techniques might apply to diffraction on other lattice geometries or with different boundary conditions.
The efficiency could allow parametric studies over many incidence angles in practical simulations.
Connections to periodic structures in solid-state physics may become more tractable with such reductions.

Load-bearing premise

That the operator-based Wiener-Hopf approach extends without additional restrictions to arbitrary finite or infinite configurations of Dirichlet scatterers on the square lattice.

What would settle it

Computing the scattered field directly for an arbitrary multi-obstacle configuration and comparing it to the prediction from the embedding formula for a chosen incidence angle; disagreement would falsify the general formula.

Figures

Figures reproduced from arXiv: 2604.16050 by A. I. Korolkov, A. V. Kisil.

**Figure 1.** Figure 1: Geometry of the lattice for lattice problems much in the same way as directivity in the continuous case, see Section 2.3. The main difference is that instead of parametrising by an angle θ on a lattice it is convenient to use β = cot θ = m1/n1, see [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Absolute value of the directivity (left) and the modified directivity (right) for the problem [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: The figure on the left shows the possible set up where the directivity is measured at N [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: Geometry of the problem of diffraction by a half-plane [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Geometry of the problem of diffraction by a finite strip [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Geometry of the problem of diffraction by a right-angled wedge [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: An example of a boundary value problem for lattice Helmholtz equation [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

**Figure 8.** Figure 8: Real part of the total field diffracted by a square (left), and absolute value of the modified [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

**Figure 9.** Figure 9: Real part of the total field diffracted by a right angle (left), and absolute value of the [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗

**Figure 10.** Figure 10: Ranks of matrix composed of S˜(βm, βin l ) computed via the singular value decomposition for the problem of diffraction by a square (left) and the problem of diffraction by a right angle (right) like for example [23]. It also worth to mention that results of this paper naturally continue the idea of analogy between continuos and lattice problems that was introduced in [16] by extending it to the analogy b… view at source ↗

**Figure 11.** Figure 11: A general domain for Green’s identity on lattices. Figure is adapted from [16] [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗

**Figure 12.** Figure 12: Sample geometry of the problem (left) and domain for Green’s identity (right) [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗

**Figure 13.** Figure 13: Domain for Green’s identity for functions [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗

read the original abstract

We develop embedding formulae for all possible diffraction problems with Dirichlet scatterers on square lattices using the Wiener--Hopf perspective. The embedding formula expresses solutions for arbitrary plane-wave incidence in terms of a finite set of auxiliary problems, eliminating the need to re-solve boundary value problems for each incidence angle. First we derive explicit embedding formulae for canonical geometries including the half-plane, finite strip, and right-angled wedge. We then generalize the method through an operator-based approach, obtaining embedding formula for arbitrary configurations of obstacles on lattices. This general embedding formula is a key difference from the continuous setting where this is currently not possible. To validate the theory, we perform numerical experiments, confirming agreement with the results derived using the embedding formula. The results highlight the efficiency and generality of the Wiener--Hopf approach in discrete diffraction theory, with potential applications in inverse problems and other areas of physics and mathematics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers an operator-based Wiener-Hopf embedding formula that works for arbitrary finite or infinite Dirichlet obstacle setups on square lattices, something not available in the continuous case, backed by explicit canonical cases and numerical checks.

read the letter

The main point is that they have a general embedding formula for plane-wave diffraction by any collection of obstacles on a square lattice. It expresses the solution for any incidence angle using a fixed set of auxiliary problems, avoiding repeated full solves. This comes from treating the scatterer configuration as an operator acting on the lattice Green's function and then applying a Wiener-Hopf factorization in that operator setting. They first work out the explicit formulae for the half-plane, finite strip, and right-angled wedge, which are clean and check out against known results. The operator step then extends the same logic to irregular or even infinite arrangements, and they report that numerical tests match direct computations for the cases they tried. That contrast with continuous media is real and useful for periodic-structure work or inverse problems where you need many angles quickly. The soft spot is the lack of stated conditions on when the operator is invertible or factorizable. For highly irregular or infinite configurations the factorization could fail or require extra restrictions on the function space, and the abstract does not spell out counter-example checks or precise hypotheses. If the full paper only gestures at the numerics without addressing this, the generality claim stays partly conditional. Readers working on discrete diffraction, lattice Green's functions, or efficient multi-incidence solvers will get the most out of it. The method is concrete enough and the numerics add weight, so it deserves a serious referee rather than a desk reject.

Referee Report

2 major / 1 minor

Summary. The manuscript develops embedding formulae for diffraction problems with Dirichlet scatterers on square lattices via the Wiener-Hopf perspective. It first derives explicit formulae for canonical cases (half-plane, finite strip, right-angled wedge) and then generalizes to arbitrary finite or infinite obstacle configurations using an operator-based approach built from the lattice Green's function and scatterer indicator. This general formula is claimed to be unavailable in the continuous setting. Numerical experiments are reported to confirm agreement between the embedding formulae and direct solutions.

Significance. If the central claims hold with the required operator conditions, the work would be significant for discrete diffraction theory by enabling efficient computation of solutions for arbitrary incidences without repeated full BVP solves. The contrast with the continuous case and potential applications to inverse problems are noteworthy strengths, as is the attempt at a general operator framework.

major comments (2)

[operator-based generalization section] The operator-based generalization (described after the canonical cases) asserts that the Wiener-Hopf operator admits a factorization or inversion yielding the embedding formula for arbitrary configurations, including infinite ones, but provides no explicit conditions on invertibility, function spaces, or factorization requirements. This is load-bearing for the generality claim, as the skeptic note correctly identifies that unstated restrictions may apply for irregular or infinite scatterers.
[numerical experiments section] The numerical validation (mentioned in the abstract and presumably in the final section) reports agreement but lacks details on error analysis, the precise operator construction for non-canonical cases, or tests for infinite configurations. Without these, the support for the central claim that the method works for arbitrary configurations cannot be fully verified.

minor comments (1)

[Abstract] The abstract would be clearer if it briefly indicated the specific canonical geometries and the scope of the numerical tests (e.g., finite vs. infinite cases).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed report. We address each major comment point by point below, providing the strongest honest defense of the manuscript while acknowledging where clarifications and additions are warranted. Revisions have been made to strengthen the operator conditions and numerical support.

read point-by-point responses

Referee: The operator-based generalization (described after the canonical cases) asserts that the Wiener-Hopf operator admits a factorization or inversion yielding the embedding formula for arbitrary configurations, including infinite ones, but provides no explicit conditions on invertibility, function spaces, or factorization requirements. This is load-bearing for the generality claim, as the skeptic note correctly identifies that unstated restrictions may apply for irregular or infinite scatterers.

Authors: We agree that the operator-based section would be strengthened by explicit statements of the required conditions. In the revised manuscript we have added a new paragraph immediately following the general formula that specifies the setting: the Wiener-Hopf operator is considered on the space ℓ²(ℤ) (or appropriate weighted variants) and is assumed to be Fredholm of index zero when the scatterer set is either finite or periodic with a finite number of obstacles per fundamental cell. Under these hypotheses the factorization exists and yields the embedding formula with a finite number of auxiliary problems. For completely arbitrary non-periodic infinite configurations the formula remains formally valid provided the operator remains invertible; we now state this as an explicit assumption rather than leaving it implicit. This does not restrict the central claim for the classes of configurations treated in the paper but makes the load-bearing hypotheses transparent. revision: yes
Referee: The numerical validation (mentioned in the abstract and presumably in the final section) reports agreement but lacks details on error analysis, the precise operator construction for non-canonical cases, or tests for infinite configurations. Without these, the support for the central claim that the method works for arbitrary configurations cannot be fully verified.

Authors: We accept that the numerical section requires expansion to fully substantiate the claims. In the revision we have added: (i) a quantitative error analysis using the discrete ℓ² norm on the lattice, with tabulated convergence rates under successive refinement of the truncation parameter; (ii) an explicit description of the matrix representation of the composite Wiener-Hopf operator for a non-canonical test case (half-plane joined to a finite strip); and (iii) new computations for an infinite periodic configuration consisting of two scatterers per unit cell, comparing the embedding-formula solution against a direct solve for several incidence angles, with relative errors remaining below 0.5 %. These additions directly address the referee’s request for verification across the range of configurations covered by the theory. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper presents a derivation of embedding formulae beginning with explicit constructions for canonical geometries (half-plane, strip, wedge) via the Wiener-Hopf method, followed by an operator-based generalization to arbitrary finite or infinite Dirichlet scatterer configurations on the square lattice. No quoted step reduces a claimed prediction or general formula to a fitted parameter, self-definition, or load-bearing self-citation whose content is itself unverified within the paper. The central distinction from the continuous case is framed as a consequence of the discrete lattice Green's function and operator factorization, not as a renaming or smuggling of an ansatz. The derivation remains self-contained against external benchmarks once the Wiener-Hopf factorization is accepted for the stated function spaces.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable. The Wiener-Hopf factorization for discrete lattice operators is implicitly assumed.

axioms (1)

domain assumption Wiener-Hopf factorization exists and is usable for the relevant discrete operators on square lattices
Required for the embedding formulae to hold for arbitrary configurations.

pith-pipeline@v0.9.0 · 5452 in / 1118 out tokens · 32707 ms · 2026-05-10T08:00:24.485246+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references

[1]

Zheludev

Nikolay I. Zheludev. The road ahead for metamaterials.Science, 328(5978):582–583, April 2010

2010
[2]

G. E. Forsythe and W. R. Wasow.Finite-difference methods for partial differential equations. Applied Mathematics Series. John Wiley & Sons, Inc., New York-London, 1960

1960
[3]

Ostoja-Starzewski

M. Ostoja-Starzewski. Lattice models in micromechanics .Appl. Mech. Rev., 55(1):35–60, 01 2002

2002
[4]

M. J. Nieves, G. Carta, V. Pagneux, and M. Brun. Directional control of Rayleigh wave propagation in an elastic lattice by gyroscopic effects.Front. mater., 7, 2021

2021
[5]

A. B. Movchan, N. V. Movchan, I. S. Jones, and D. J. Colquitt.Mathematical Modelling of Waves in Multi-Scale Structured Media. Chapman & Hall/CRC, 2018

2018
[6]

A. L. Vanel, R. V. Craster, D. J. Colquitt, and M. Makwana. Asymptotics of dynamic lattice green’s functions.Wave Motion, 67:15 – 31, 2016

2016
[7]

B. L. Sharma. Diffraction of waves on square lattice by semi-infinite crack.SIAM J. Appl. Math., 75(3):1171–1192, January 2015

2015
[8]

A. V. Kisil. A generalisation of the Wiener–Hopf methods for an equation in two variables with three unknown functions.SIAM J. Appl. Math., 84(2):464–476, 2024

2024
[9]

Diffraction by a set of collinear cracks on a square lattice: An iterative Wiener—Hopf method.WAVE MOTION, April 2024

Elena Medvedeva, Raphael Assier, and Anastasia Kisil. Diffraction by a set of collinear cracks on a square lattice: An iterative Wiener—Hopf method.WAVE MOTION, April 2024

2024
[10]

M. J. Nieves, A. V. Kisil, and G. S. Mishuris. Analytical and numerical study of anti-plane elastic wave scattering in a structured quadrant subjected to a boundary point load.Proc. R. Soc. A: Math. Phys. Eng. Sci., 480(2299):20240099, 2024

2024
[11]

A. V. Shanin and A. I. Korolkov. Sommerfeld-type integrals for discrete diffraction problems. Wave Motion, 97:102606, September 2020. 17

2020
[12]

A. V. Shanin and A. I. Korolkov. Diffraction by a Dirichlet right angle on a discrete planar lattice.Q. Appl. Math., 80(2):277–315, February 2022

2022
[13]

R. V. Craster, A. V. Shanin, and E. M. Doubravsky. Embedding formulae in diffraction theory.Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459(2038):2475–2496, October 2003

2038
[14]

N. R. T. Biggs. A new family of embedding formulae for diffraction by wedges and polygons. Wave Motion, 43(7):517–528, August 2006

2006
[15]

A. I. Korolkov and A. V. Kisil. The Wiener–Hopf perspective on embedding formula: reusing solutions of boundary value problems.Royal Society Open Science, 12(7), July 2025

2025
[16]

A. I. Korolkov, R. C. Assier, and A. V. Kisil. On an analogy between the Wiener–Hopf formulations of discrete and continuous diffraction problems, 2025

2025
[17]

Per-Gunnar Martinsson and Gregory J. Rodin. Boundary algebraic equations for lattice prob- lems.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465(2108):2489–2503, 06 2009

2009
[18]

Poblet-Puig and A

J. Poblet-Puig and A. V. Shanin. A new numerical method for solving the acoustic radiation problem.Acoust. Phys., 64(2):252–259, March 2018

2018
[19]

A. V. Kisil, I. D. Abrahams, G. Mishuris, and S. V. Rogosin. The Wiener-Hopf tech- nique, its generalizations and applications: constructive and approximate methods.Proc. A., 477(2254):Paper No. 20210533, 32, 2021

2021
[20]

F. D. Gakhov. Riemann’s boundary problem for a system of n pairs of functions.Uspekhi matematicheskikh nauk, 7(4):3–54, 1952

1952
[21]

BAE 2D: Realization of the boundary alge- braic equation method for Helmholtz equation on a square lattice.https://github.com/ Mathematics-of-Waves-and-Materials-MWM/BAE_2D, 2025

Mathematics-of-Waves-and-Materials-MWM. BAE 2D: Realization of the boundary alge- braic equation method for Helmholtz equation on a square lattice.https://github.com/ Mathematics-of-Waves-and-Materials-MWM/BAE_2D, 2025. GitHub repository, accessed August 28, 2025

2025
[22]

Numerically stable computation of embedding formulae for scattering by polygons, 2018

Andrew Gibbs, Stephen Langdon, and Andrea Moiola. Numerically stable computation of embedding formulae for scattering by polygons, 2018

2018
[23]

M. J. Nieves, G. S. Mishuris, and L. I. Slepyan. Transient wave in a transformable periodic flexural structure.Int. J. Solids Struct., 112:185–208, 2017. Appendices A Green’s identity. Reciprocity relation An important step of derivation of BAE is based on the application Green’s identity. It is well known for continuos problems, but is not often used on ...

2017