A brief historical perspective of the Wiener–Hopf technique

Lawrie, Jane B.; Abrahams, I. David

doi:10.1007/s10665-007-9195-x

Cited by 116 publications

(104 citation statements)

References 53 publications

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“…The general question of constructive matrix Wiener-Hopf factorisation is open: (Lawrie & Abrahams 2007;Rogosin & Mishuris 2016). In this paper we are concerned with a class of Wiener-Hopf equations with triangular matrix functions containing exponential factors.…”

Section: Introductionmentioning

confidence: 99%

Aerodynamic noise from rigid trailing edges with finite porous extensions

Kisil

Ayton

2017

J. Fluid Mech.

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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 new method is validated by comparing the iterative solutions for acoustic scattering by a finite impermeable plate against a known solution (obtained in terms of Mathieu functions). 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.

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Section: Introductionmentioning

confidence: 99%

Aerodynamic noise from rigid trailing edges with finite porous extensions

Kisil

Ayton

2017

J. Fluid Mech.

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“…For geometries comprising semi-infinite duct sections the Wiener-Hopf technique [15] can prove to be a powerful tool. The method is most appropriate for the solution of 2-D (or 3-D) boundary value problems involving a governing equation together with a two-part boundary condition imposed along one infinite coordinate line [16,4] (for example, one condition for x < 0, y = 0 and a different condition for x > 0, y = 0).…”

Section: Introductionmentioning

confidence: 99%

On acoustic propagation in three-dimensional rectangular ducts with flexible walls and porous linings

Lawrie

2012

The Journal of the Acoustical Society of America

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The focus of this paper is towards the development of hybrid analytic-numerical modematching methods for model problems involving three-dimensional ducts of rectangular cross-section and with flexible walls. Such methods require firstly closed form analytic expressions for the natural fluid-structure coupled waveforms that propagate in each duct section and secondly the corresponding orthogonality relations. It is demonstrated how recent theory [Lawrie, Proc. R. Soc. A. 465, 2347-2367] may be extended to a wide class of three-dimensional ducts, for example, those with flexible walls and a porous lining (modelled as an equivalent fluid) or those with a flexible internal structure such as a membrane (the "drum-like" silencer). Two equivalent expressions for the eigenmodes of a given duct can be formulated. For the ducts considered herein, the first ansatz is dependent on the eigenvalues/eigenfunctions appropriate for wave propagation in the corresponding two-dimensional duct with flexible walls, whilst the second takes the form of a Fourier series. The latter has several advantages: in particular, no "rootfinding" is involved and the method is appropriate for ducts in which the flexible wall is orthotropic. The first ansatz, however, provides important information about the orthogonality properties of the eigenmodes.

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“…Mixed BVPs (where the boundary conditions change on part of the boundary) cannot be solved in general by the classic transform method. Nevertheless, for some of these problems it is possible, after using a suitable transform, to express their solution in terms of a Wiener-Hopf (or Riemann-Hilbert) problem [83], [70]. The application of the new method to these types of problems is work in progress; preliminary investigations have shown that the global relation yields immediately the relevant Wiener-Hopf problem, thus eliminating the need to first choose a suitable transform.…”

Section: Extensions Of the Method And Connections With Other Techniqmentioning

confidence: 99%

Synthesis, as Opposed to Separation, of Variables

Fokas¹,

Spence²

2012

SIAM Rev.

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Abstract. Every applied mathematician has used separation of variables. For a given boundary value problem (BVP) in two dimensions, the starting point of this powerful method is the separation of the given PDE into two ODEs. If the spectral analysis of either of these ODEs yields an appropriate transform pair, i.e., a transform consistent with the given boundary conditions, then the given BVP can be reduced to a BVP for an ODE. For simple BVPs it is straightforward to choose an appropriate transform and hence the spectral analysis can be avoided. In spite of its enormous applicability, this method has certain limitations. In particular, it requires the given domain, PDE, and boundary conditions to be separable, and also may not be applicable if the BVP is non-self-adjoint. Furthermore, it expresses the solution as either an integral or a series, neither of which are uniformly convergent on the boundary of the domain (for nonvanishing boundary conditions), which renders such expressions unsuitable for numerical computations. This paper describes a recently introduced transform method that can be applied to certain nonseparable and non-selfadjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane that is uniformly convergent on the boundary of the domain. The starting point of the method is to write the PDE as a one-parameter family of equations formulated in a divergence form, and this allows one to consider the variables together. In this sense, the method is based on the "synthesis" as opposed to the "separation" of variables. The new method has already been applied to a plethora of BVPs and furthermore has led to the development of certain novel numerical techniques. However, a large number of related analytical and numerical questions remain open. This paper illustrates the method by applying it to two particular non-self-adjoint BVPs: one for the linearized KdV equation formulated on the half-line, and the other for the Helmholtz equation in the exterior of the disc (the latter is non-self-adjoint due to the radiation condition). The former problem played a crucial role in the development of the new method, whereas the latter problem was instrumental in the full development of the classical transform method. Although the new method can now be presented using only classical techniques, it actually originated in the theory of certain nonlinear PDEs called integrable, whose crucial feature is the existence of a Lax pair formulation. It is shown here that Lax pairs provide the generalization of the divergence formulation from a separable linear to an integrable nonlinear PDE.

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A brief historical perspective of the Wiener–Hopf technique

Cited by 116 publications

References 53 publications

Aerodynamic noise from rigid trailing edges with finite porous extensions

Aerodynamic noise from rigid trailing edges with finite porous extensions

On acoustic propagation in three-dimensional rectangular ducts with flexible walls and porous linings

Synthesis, as Opposed to Separation, of Variables

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