An Iterative Wiener--Hopf method for triangular matrix functions with exponential factors

This paper introduces a new method for constructing approximate solutions to a class of Wiener{Hopf equations. This is particularly useful since exact solutions of this class of Wiener{Hopf equations, at the moment, cannot be obtained. The proposed method could be considered as a generalisation of the \pole removal" technique and Schwarzschild's series. The criteria for convergence is proved. The error in the approximation is explicitly estimated, and by a su cient number of iterations could be made arbitrary small. Typically only a few iterations are required for practical purposes. The theory is illustrated by numerical examples that demonstrate the advantages of the proposed procedure. This method was motivated by and successfully applied to problems in acoustics. 1.I acknowledge support from the Sultan Qaboos Research Fellowship at Corpus Christi College at University of Cambridge

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.Comment: LaTeX, 20 pp., 19 graphics in 6 figure

Applying an iterative method numerically to solve n × n matrix Wiener–Hopf equations with exponential factors

This paper presents a generalization of a recent iterative approach to solving a class of 2 × 2 matrix Wiener–Hopf equations involving exponential factors. We extend the method to square matrices of arbitrary dimension n, as arise in mixed boundary value problems with n junctions. To demonstrate the method, we consider the classical problem of scattering a plane wave by a set of collinear plates. The results are compared to other known methods. We describe an effective implementation using a spectral method to compute the required Cauchy transforms. The approach is ideally suited to obtaining far-field directivity patterns of utility to applications. Convergence in iteration is fastest for large wavenumbers, but remains practical at modest wavenumbers to achieve a high degree of accuracy.This work was supported by EPSRC DTP grant no. EP/N509620/1 (M.J.P.), the Sultan Qaboos Research Fellowship at Corpus Christi College at University of Cambridge (A.V.K.) and by EPSRC early career fellowship grant no. EP/P015980/1 (L.J.A.). The authors thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the WHT programme where some work on this paper was undertaken (EPSRC grant no. EP/R014604/1)

Numerical solution of scattering problems using a Riemann--Hilbert formulation

A fast and accurate numerical method for the solution of scalar and matrix Wiener--Hopf problems is presented. The Wiener--Hopf problems are formulated as Riemann--Hilbert problems on the real line, and a numerical approach developed for these problems is used. It is shown that the known far-field behaviour of the solutions can be exploited to construct numerical schemes providing spectrally accurate results. A number of scalar and matrix Wiener--Hopf problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane are solved using the approach

Mathematically modelling the deformation of frictional elastic half-spaces in contact with a rolling rigid cylinder

In this thesis we derive an analytical model of the deformation of an elastic half-space caused by a cylindrical roller. The roller is considered rigid, and is forced into the half-space and rolls across its surface, with contact modelled by Coulomb friction. In general, portions of the surface of the roller in contact with the half-space may slip across the surface of the half-space, or may stick to it. In this thesis, we consider the contact surface to have a central sticking region as well as a simplifying regime where the entire contact surface is fully slipping. This results in two mixed boundary value problem, which are formulated into a 4_4 matrix Wiener{Hopf problem for the stick-slip regime and a 2_2 matrix Wiener{Hopf problem for the full-slip regime. The exponential factors in the Wiener{Hopf matrix allows a solution by following the iterative method of Priddin, Kisil, and Ayton (Phil. Trans. Roy. Soc. A 378, p. 20190241, 2020) which is implemented numerically by computing Cauchy transforms using a spectral method following Slevinsky and Olver (J. Comput. Phys. 332, pp. 290{315, 2017). The limits of the contact region and stick-slip transitions are located a posteriori by applying an free-boundary method based on the secant method. The solution is illustrated with several examples, and the frictional regimes are analysed