Abstract:
It is well known that the problem of linear acoustic wave propagation can be reduced to a boundary value problem for the Helmholtz equation, which can be solved analytically only for a bunch of simple geometries, and is usually treated numerically in more complicated cases with the help of BEM and FEM, for example. However, one key advantage of analytical solutions over numerical ones is that the dependence on parameters is explicit. This explicit dependence is particularly useful for optimizing regimes and addressing inverse problems. Here, we explore an intermediate method known as embedding, where complete analytical solutions are not derived, yet it is still possible to obtain some analytical dependence of the solution on specific parameters. A systematic approach to deriving embedding formulas for plane wave diffraction problems was outlined by Craster in 2003. The steps involve: The steps involve: identifying an operator H that “annihilates” the incident plane wave, creates singularities near the edges of the scatterer, preserves boundary conditions, and also commutes with the Laplace operator; introducing the auxiliary oversingular solutions; studying the combination of the modified field H[u] and the auxiliary solutions with some unknown coefficients. It is then shown, through uniqueness arguments, that the solution to the plane diffraction problem can be expressed in terms of these auxiliary solutions. In this talk, we propose an alternative approach. Simply put, we argue that every embedding formula is rooted in a matrix Wiener–Hopf equation, and the embedding formula is essentially the canonical solution to this matrix Wiener–Hopf problem. We demonstrate the effectiveness of this approach by revisiting well-known problems, such as the problem of diffraction by a strip and the problem of diffraction by a wedge. Additionally, we derive new embedding formulas for problems in discrete diffraction theory.
The talk is based on the joint work with A. V. Kisil:
Korolkov, A. I., Kisil, A. V. (2024). Recycling solutions of boundary value problems: the Wiener–Hopf perspective on embedding formula. arXiv:2410.08684v1
Andrey Korolkov
I am a research associate in the mathematics department at the University of Manchester. I am part of the Mathematics of Waves and Materials Group, where I primarily collaborate with Dr. Anastasia Kisil and Dr. Raphael Assier.
I completed my PhD in mathematical physics under the supervision of DSc Andrey Shanin at Lomonosov Moscow State University, receiving my degree in 2016. Following that, I held a research position in the Acoustics Department from 2016 to 2023.
My research interests include analytical methods in linear wave theory, wave propagation on lattices, and computational acoustics
