arXiv subjects

A. V. Kisil

Publications and source records attributed to A. V. Kisil.

5 recordsLinked to original sources

arXiv2026

Embedding formulae for diffraction problems on square lattices

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.

math-ph↗

arXiv2025

On an analogy between the Wiener--Hopf formulations of discrete and continuous diffraction problems

This article is dedicated to unifying the framework used to derive the Wiener--Hopf equations arising from some discrete and continuous wave diffraction problems.The main tools are the discrete Green's identity and the appropriate notion of discrete normal derivative. The resulting formal analogy between the Wiener--Hopf equations allows one to effortlessly move between the discrete and continuous formulations. The validity of this novel analogy is illustrated through several famous two-dimensional canonical diffraction problems and extended to three-dimensional problems.

math-ph↗

arXiv2024

Recycling solutions of boundary value problems: the Wiener--Hopf perspective on embedding formula

Embedding formula allows to recycle solution of a family boundary value problems by expressing all the solutions in terms of a small number of solutions. Such formulas have been previously derived in the context of diffraction by applying a cleverly chosen operator to the solution and the construction of edge Green's functions which are introduced in an elaborate manner specific for each problem. We demonstrate that embedding formula naturally appears from a matrix Wiener--Hopf equation, and the embedding formula is derived from the canonical solution to this matrix Wiener--Hopf problem. This allows to drive the embedding formula in any context where the problem can be formulated as a Wiener--Hopf equation. We illustrate the effectiveness of this approach by revisiting known problems, such as the problem of diffraction by half-line, a strip and the problem of diffraction by a wedge. Additionally, a new matrix Wiener--Hopf formulation is derived for wedge problems.

math-ph↗

arXiv2022

Diffraction of acoustic waves by a wedge of point scatterers

This article considers the problem of diffraction by a wedge consisting of two semi-infinite periodic arrays of point scatterers. The solution is obtained in terms of two coupled systems, each of which is solved using the discrete Wiener--Hopf technique. An effective and accurate iterative numerical procedure is developed to solve the diffraction problem, which allows us to compute the interaction of thousands of scatterers forming the wedge. A crucial aspect of this numerical procedure is the effective truncation of slowly convergent single and double infinite series, which requires careful asymptotic analysis. A convergence criteria is formulated and shown to be satisfied for a large class of physically interesting cases. A comparison to direct numerical simulations is made, highlighting the accuracy of the method.

math.NA↗

arXiv2017

Aerodynamic noise from rigid trailing edges with finite porous extensions

This paper investigates the effects of finite flat porous extensions to semi-infinite impermeable flat plates in an attempt to control trailing-edge noise through bio-inspired adaptations. Specifically the problem of sound generated by a gust convecting in uniform mean steady flow scattering off the trailing edge and permeable-impermeable junction is considered. This setup supposes that any realistic trailing-edge adaptation to a blade would be sufficiently small so that the turbulent boundary layer encapsulates both the porous edge and the permeable-impermeable junction, and therefore the interaction of acoustics generated at these two discontinuous boundaries is important. The acoustic problem is tackled analytically through use of the Wiener-Hopf method. A two-dimensional matrix Wiener-Hopf problem arises due to the two interaction points (the trailing edge and the permeable-impermeable junction). This paper discusses a new iterative method for solving this matrix Wiener-Hopf equation which extends to further two-dimensional problems in particular those involving analytic terms that exponentially grow in the upper or lower half planes. This method is an extension of the commonly used "pole removal" technique and avoids the needs for full matrix factorisation. Convergence of this iterative method to an exact solution is shown to be particularly fast when terms neglected in the second step are formally smaller than all other terms retained. The final acoustic solution highlights the effects of the permeable-impermeable junction on the generated noise, in particular how this junction affects the far-field noise generated by high-frequency gusts by creating an interference to typical trailing-edge scattering. This effect results in partially porous plates predicting a lower noise reduction than fully porous plates when compared to fully impermeable plates.

physics.flu-dyn↗