Analytical insights into semi-infinite plate scattering: the wiener–hopf technique and two-sided linear boundary conditions

Summary

The Wiener–Hopf Technique is popular throughout applied mathematics, particularly for wave scattering problems. One such problem is the scattering of an incident wave impinging upon a semi-infinite surface. For scattering problems in which one applies separate boundary conditions on each side of the diffraction medium, we must adapt the standard Wiener–Hopf approach via a generalised technique to deal with the resulting matrix system. Such problems are present in the literature, where the Wiener–Hopf problem is reduced to solving Hilbert equations, leading to the so-called Wiener–Hopf–Hilbert method. However, both boundaries tend to have the same mathematical form, which is overly simplistic when describing realistic compliant boundaries or surface coatings. In some other literature, the matrix factorization problem remains the focus and is solved for arbitrary entries, which may prove challenging to use in complex scattering problems. We solve a more general problem while keeping an applied scattering framework, drawing attention to the additional analytical considerations needed to adapt previous methods. More specifically, we study the Wiener–Hopf matrix kernel, which shows that it holds the key to distinguishing these problems from those studied in previous literature. Finally, we present an example of a sound wave scattering off a plate that has a one-sided surface coating and is in flow. We treat this as a two-sided boundary and model the upper side with the Ingard–Myers impedance boundary condition and the lower side with the rigid Neumann boundary condition. Results are presented using impedance values from existing literature that reflect real materials.

1 Introduction

The Wiener–Hopf technique [1] is an analytical method that is commonplace in applied mathematics. Areas of its use vary from scattering problems in both aeroacoustics and electromagnetism to finance, hydrodynamics and beyond [2]. This article aims to generalise existing work on the Wiener–Hopf technique in the context of solving scattering problems with a view to applications in aeroacoustics. In particular, we will focus on constructing a clear framework to solve scattering problems wherein an incident wave diffracts off a semi-infinite flat plate. We assume this boundary has different material properties on its upper and lower sides that can be mathematically modelled with some generalised linear boundary conditions. By developing a general and precise mathematical method, we may be able to study new materials and surfaces that permit aerodynamic noise reduction. This noise reduction is of particular appeal to the scientific community due to the health hazards associated with aircraft noise, particularly for those living near airports. Moreover, noise is a drawback of implementing wind turbines that produce excessive low-frequency noise [3].

This article positions itself at the interface of two subtopics in applied mathematics: the theoretical matrix Wiener–Hopf factorisation problem and the direct application of the Wiener–Hopf technique to wave-scattering problems. Considering the former, considerable research has been conducted to understand what sort of matrices can be factorised as a standard matrix multiplicative product where one factor is analytic and non-zero in some upper region of the complex plane and the other is analytic and non-zero in the lower region. This procedure has been reviewed thoroughly and excellently in [2], where the most common approaches to this factorisation problem that arise in wave scattering problems are outlined. This approach usually revolves around considering a general class of matrices to which a specific factorisation technique may be applied, such as those of Daniele–Khrapkov type [4–7], Jones type [8–11] or Rawlins type [12?–15]. Regarding the second subtopic, problems involving a plane wave impinging upon a semi-infinite boundary are common in the literature due to their importance to aeroacoustics and beyond. The most common representation of a compliant boundary (a ‘non-rigid’ boundary in the no-slip Neumann sense) is using an impedance boundary condition, such as in [16–22]. In these examples, the boundary is taken as ‘two-sided’, meaning we prescribe distinct boundary conditions on each side of the plate, a matrix Wiener–Hopf kernel arises of a specific type, as described in [12, 13]. Reference [12] discusses the problem of multiplicatively factorising the particular class of matrices that arises in more general scattering problems; however, this study is purely theoretical and does not provide a constructive method to solve the diffraction problem.

Conversely, numerous papers [18–21] study two-sided boundary condition problems in which the same ‘type’ of boundary condition occurs on each side. The only mathematical difference between the two boundary conditions is the change in some parameter values, not the differential operator’s order or structure. Moreover, although theoretical work on generalised linear boundary conditions such as [23] exists, there are still caveats as to which boundary conditions this applies. Similarly, some conditions still exist for the entries to Rawlins generalised matrix in [12], such as the assumed non-zero nature of the functions. Amending all these omissions is an arduous yet worthwhile task; we aim to lay the groundwork for this and present a method that can tackle various problems in the future.

When dealing with two-sided conditions, the matrix Wiener–Hopf equation must be handled carefully. This is mainly due to the branch cuts that arise when solving the initial Helmholtz equation and their impact on the analyticity of factors in preselected regions of the complex plane. With this in mind, our analysis will act as a bridge between these two approaches. We still consider a general setup as in [12]; however, the underlying scattering problem directly motivates our approach. A specific example not covered by the previous literature is a semi-rigid plate; we define this as a plate in which one side is rigid, and the other has some coating modelled by an impedance boundary condition. It was shown in [24] that introducing velvety structures on the upper side of a plate can reduce aerodynamic noise, mimicking the surface of an owl’s wing. A theoretical model for this would be to solve a scattering problem in which we consider a boundary that is rigid below and has an impedance boundary condition above to represent the coating on top. However, it is unclear what condition is best. Impedance boundary conditions are prevalent in the literature and have many forms with varying intricacy [25–27].

A benefit of our generalised approach is that we can consider any linear differential operator within our boundary conditions. In our example, we apply the more commonly used Ingard–Myers boundary condition that accounts for the presence and influence of constant mean flow [28] on a traditional impedance surface instead of the standard Robin condition used in [19–21]. Although this boundary condition was addressed in [16], no results were presented, and only one condition was applied on the plate.

The article is structured as follows: Section 2 discusses the governing equations and how they may be used to construct a matrix Wiener–Hopf equation. We discuss the analytic requirements for boundary conditions due to their importance when modelling realistic behaviour and provide an overview of the tasks that will be needed to find the solution. This overview will include brief discussions on edge conditions, zeros in the scalar kernels and their influence on regions of analyticity and surface waves. Section 3 formally solves this problem by addressing these tasks and discusses how the theory developed in previous work can be extended to our general linear boundary conditions. It particularly focuses on solving the Hilbert equations that arise when using the Wiener–Hopf–Hilbert approach. Section 4 demonstrates this theory by solving an example related to acoustic diffraction; a sound wave scatters off a rigid edge with some theoretical upper coating, which we model with the Ingard–Myers impedance boundary condition [28]. We demonstrate the significant changes in the diffracted field when altering one side of a boundary while using realistic parameter values for impedance and free stream Mach number.

2 Constructing Wiener–Hopf equations for scattering problems in flow

We begin with a more abstract set of governing equations to form the type of Wiener–Hopf equations we seek to solve using a new factorisation process. The setup is primarily for scattering problems in which the scattering medium is a semi-infinite plane; however, the method can be altered for other setups thanks to the versatility of the Wiener–Hopf technique. We include constant background flow $U_{\infty}$ ⁠, which can be set to zero for examples without flow.

While constructing our governing equations for a more general wave problem, we must not only consider an incidental plane wave but also account for the possibility of a wake forming downstream from the plate. This is a complication of diffraction problems in flow and is modelled in [19, 29, 30] using a jump condition in x > 0 with some specific exponential forcing, we choose to implement this below with a scaling constant that can be set to zero if no wake is needed.

2.1 Governing equations and initial assumptions

Our problem will consist of an incident wave $φ_{I} (x, y)$ scattering off a two-sided semi-infinite boundary that occupies the negative x-axis, as seen in aerodynamic noise problems. We assume each wave has a (suppressed) $e^{i ω t}$ term, thereby converting the linear 2D wave equation to a 2D Helmholtz equation. Our scattered variable $φ_{s}$ will therefore satisfy the Helmholtz equation with some wavenumber k, which may or may not be the acoustic wavenumber $\frac{ω}{c_{0}}$ (ω the angular frequency and c₀ the speed of sound), depending on the context. Due to the influence of constant mean flow, a convective transform of the governing equations is required to shift this wavenumber from the Helmholtz number k₀ to some convective wavenumber that we denote k for simplicity. We also include a source term on the positive x-axis in the form of an aerodynamic wake, as implemented in [18, 19]. To begin our analysis, we assume there exist constants $δ_{1}, δ_{2}, δ_{3}$ such that our incident wave is of the form $e^{- i k (δ_{1} x + δ_{2} y)}$ and our wake can be modelled by a source term of the form $C_{K} e^{i k δ_{3} x}$ downstream from the plate. The purpose of this is to permit the method to apply to any incident waves with $δ_{1} = cos (θ_{0}), δ_{2} = sin (θ_{0}), δ_{3} = 1 / M$ ⁠, where $M = \frac{U_{\infty}}{c_{0}}$ is the Mach number of the constant mean flow.

We may also include incidental surface waves in which $δ_{1, 2}$ could be arbitrary, possibly imaginary, constants. A final possible choice relevant to aeroacoustics is hydrodynamic incident waves- known as ‘gusts’. These waves represent turbulence and require real constants δ₂ and $δ_{1} \propto M$ ⁠.

We outline this setup in Fig. 1.

Fig. 1

Illustration of the general scattering problem we will solve with the Wiener–Hopf technique.

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We include the common assumption in wave scattering problems: a physical solution. This is usually accounted for by including the Sommerfeld radiation condition. Mathematically speaking, we avoid ab initio the possibility of surface waves that would blow up exponentially along the plate. If these functions did not decay, the common procedure is to give the wavenumber a small imaginary part ϵ, which is taken to zero at the end of the analysis. When this parameter is taken to zero, our problem transitions from a Wiener–Hopf to a Riemann–Hilbert problem [31].

The upper and lower surfaces give rise to two boundary conditions for the total field

φ_{I} + φ_{s}

⁠. They are formulated as linear differential operators that will solve

L_{1, 2} (φ_{I} + φ_{s}) = 0

⁠. We impose a forcing condition for the jump in

φ_{s}

across the wake x > 0, that is, we set

φ_{s} (x, 0^{+}) - φ_{s} (x, 0^{-}) : = {[φ_{s}]}_{-}^{+} = C_{K} e^{i k δ_{3} x} .

This results in the following system of governing equations for

φ_{s}

⁠:

\frac{\partial^{2} φ_{s}}{\partial x^{2}} + \frac{\partial^{2} φ_{s}}{\partial y^{2}} + k^{2} φ_{s} = 0,

(2.1a)

L_{1} [φ_{s} + φ_{I}] = 0, x < 0, y = 0^{+},

(2.1b)

L_{2} [φ_{s} + φ_{I}] = 0, x < 0, y = 0^{-},

(2.1c)

{[φ_{s}]}_{-}^{+} = C_{K} e^{i δ_{3} k x}, x > 0,

(2.1d)

{[\frac{\partial φ_{s}}{\partial y}]}_{-}^{+} = 0, x > 0,

(2.1e)

where each

L_{j} [φ_{s}]

is a linear differential operator of the form

L_{j} = \sum_{m = 0}^{M} a_{m}^{(j)} \frac{\partial^{m}}{\partial x^{m}} + \sum_{m = 0}^{M} b_{m}^{(j)} \frac{\partial^{m + 1}}{\partial x^{m} \partial y}, a_{m}^{(j)}, b_{m}^{(j)} \in C, 0 \leq m \leq M, j = 1, 2,

(2.2)

and we use the terminology

0^{\pm}

to indicate the boundary conditions are being evaluated along the lines

θ = \pm π

in the

(r, θ)

plane.

2.2 Constructing the Wiener–Hopf equation

To solve the general problem with the Wiener–Hopf technique, we take a solution of the form

φ_{s} (x, y) = \frac{1}{2 π} {\begin{matrix} \int_{- \infty}^{\infty} A (α) e^{- i α x - γ y} d α, y > 0, \\ \int_{- \infty}^{\infty} B (α) e^{- i α x + γ y} d α, y < 0, \end{matrix}

(2.3)

where

γ = \sqrt{α^{2} - k^{2}}

is a complex function of α whose branch cuts

Γ^{\pm}

emanating from

\pm k

are preselected to extend to

\pm k \infty

respectively as in Fig. 1. This choice ensures

γ (0) = - i k

and the function has a positive real part.

Notation and definitions

Before proceeding, we clarify the key notation and important terms for the rest of the article.

We rely on two defined regions of the complex plane D_U and D_L in which functions are analytic. These are the upper and lower regions, respectively, overlapping in some non-zero areas where our integration contour will lie. They are chosen under the assumption that $D_{U} \cup D_{L} = C$
The additive factorisation of some scalar function or matrix of functions refers to an additive decomposition $f = f_{+} + f_{-}$ ⁠, where $f_{+}$ is analytic in D_U and $f_{-}$ is analytic in D_L. Additive factors will have lower case symbols to refer to which region they are analytic in.
The multiplicative factorisation of some scalar function or matrix of functions refers to a multiplicative decomposition $f = f^{+} f^{-}$ ⁠, where f⁺ is analytic in D_U and f^– is analytic in D_L. Multiplicative factors will have upper case symbols to refer to which region they are analytic in.
The following analysis is dependent upon branch cuts arising from the function $γ = \sqrt{α^{2} - k^{2}}$ ⁠. These are labelled $Γ^{k}$ and $Γ^{- k}$ referring to which branch point of γ they emanate from. They extend to $\pm \infty$ at an angle ϑ satisfying $\arg k = ϑ$ ⁠.
The discontinuity in γ across the branch cuts is important to exploit when solving our problem. When evaluating functions on the upper side of (for example) the $+ k$ branch cut, we use the notation $f |_{k, U}$ ⁠. Similarly, when we evaluate a function on the lower side of this branch cut, we denote this $f |_{k, L}$ ⁠.

To begin the method for solving (2.1), we Fourier transform our continuity conditions (2.1d) in x:

\begin{matrix} A (α) - B (α) & = \int_{- \infty}^{\infty} [\frac{1}{2 π} \int_{- \infty}^{\infty} A (α) - B (α) e^{- i α x} d α] e^{i α x} d x \\ = \int_{- \infty}^{0} {[φ_{s}]}_{-}^{+} e^{i α x} d x + \int_{0}^{\infty} C_{K} e^{i x (α + δ_{3} k)} d x \\ = l_{1} (α) + \frac{C_{K} i}{α + δ_{3} k} . \end{matrix}

(2.4)

We set the half Fourier transform of the jump in $φ_{s}$ ⁠, including any possible junction condition that arises from this, equal to some unknown function l₁ that we will solve for in our Wiener–Hopf equation. From (2.4), we can see that l₁ will be analytic in D_L and $C_{K} i {(α + δ_{3} k)}^{- 1}$ ¹ will be analytic in D_U.

For (2.1e), we define the half-range transform of the unknown jump as

l_{2} = \int_{- \infty}^{0} {[\frac{\partial φ_{s}}{\partial y}]}_{-}^{+} e^{i α x} d x .

(2.5)

With this, we can express

A (α)

and

B (α)

as:

A (α) = \frac{1}{2} (l_{1} - \frac{l_{2}}{γ} + \frac{C_{K} i}{α + δ_{3} k}),

(2.6a)

B (α) = - \frac{1}{2} (l_{1} + \frac{l_{2}}{γ} + \frac{C_{K} i}{α + δ_{3} k}) .

(2.6b)

Fourier transforming conditions (2.1b) and (2.1c) gives us

\begin{matrix} S_{1} (α) A (α) & = \int_{0}^{\infty} \sum_{m = 0}^{M} (a_{m}^{(1)} \frac{\partial^{m} φ_{s}}{\partial x^{m}} + b_{m}^{(1)} \frac{\partial^{m + 1} φ_{s}}{\partial x^{m} \partial y}) e^{i α x} d x - \int_{- \infty}^{0} L_{1} {[φ_{I} (x, y)]}_{y = 0} e^{i x (α + δ_{1} k)} d x \\ = u_{1} + \frac{G_{1}}{α + δ_{1} k}, G_{1} \in C . \end{matrix}

(2.7)

and

\begin{matrix} S_{2} (α) B (α) & = \int_{0}^{\infty} \sum_{m = 0}^{M} (a_{m}^{(2)} \frac{\partial^{m} φ_{s}}{\partial x^{m}} + b_{m}^{(2)} \frac{\partial^{m + 1} φ_{s}}{\partial x^{m} \partial y}) e^{i α x} d x - \int_{- \infty}^{0} L_{2} {[φ_{I} (x, y)]}_{y = 0} e^{i x (α + δ_{1} k)} d x \\ = u_{2} + \frac{G_{2}}{α + δ_{1} k}, G_{2} \in C . \end{matrix}

(2.8)

From these equations, we can deduce

u_{1, 2}

are analytic in D_U while the second terms in each equation will be analytic in D_L. The functions

S_{1, 2} (α)

are related to the plate conditions via

S_{1} (α) = \sum_{m = 0}^{M} {(- i α)}^{m} (a_{m}^{(1)} - γ b_{m}^{(1)}),

(2.9a)

S_{2} (α) = \sum_{m = 0}^{M} {(- i α)}^{m} (a_{m}^{(2)} - γ b_{m}^{(2)}) .

(2.9b)

It is worth noting that the unknowns $u_{1}, u_{2}$ can be written in terms of the half-range transforms (written, for example, as $Φ_{\pm} (0^{\pm})$ in [32]) and also terms from the series expansion of $φ_{s}$ about 0, such as $φ_{s} (0), - i α \frac{\partial φ_{s}}{\partial x} (0)$ ⁠. In some instances, this could provide helpful information on our solution and its values at the origin or even help with ensuring the uniqueness of the solution. Indeed, it may relate closely to the aforementioned usage of junction conditions in specific examples.

Returning to our system of equations, (2.15) and (2.9) allow us to construct the matrix Wiener–Hopf equation

(\begin{matrix} S_{1} (α) & \frac{S_{1} (α)}{γ} \\ - S_{2} (α) & \frac{S_{2} (α)}{γ} \end{matrix}) [(\begin{matrix} l_{1} \\ - l_{2} \end{matrix}) + \frac{1}{α + δ_{3} k} (\begin{matrix} C_{K} i \\ 0 \end{matrix})] = \frac{2}{α + δ_{1} k} (\begin{matrix} G_{1} \\ G_{2} \end{matrix}) + 2 (\begin{matrix} u_{1} \\ u_{2} \end{matrix}) ​ .

(2.10)

It is convenient to write this in vector form and note the regions of analyticity for each term:

\frac{1}{2} \underline{\underline{K}} (α) (\underline{L} (α) + \underline{F} (α)) = \underline{G} (α) + \underline{U} (α),

(2.11)

where

\underline{L}, \underline{F}, \underline{G}

and

\underline{U}

are vectors containing the respective components of (2.11).

Assuming we can multiplicatively factorise the matrix kernel, we may rearrange the equation so that the left-hand side is analytic in D_U, and the right-hand side is analytic in D_L. This permits the analytic continuation of the equation to the union of these regions- the entire complex plane. Hence, each side of the equation must be equal to some entire function that, via the generalised Liouville theorem, must be a polynomial of some order determined by the asymptotic behaviour of the Wiener–Hopf equation. This polynomial can be determined using junction conditions or other restrictions on regularity, such as the Kutta–Joukowski condition. Good examples in which extra conditions are required to resolve this entire function occur commonly for problems with higher order boundary conditions such as poroelastic plates [33–35].

Thus, our Wiener–Hopf equation can be expressed as

{\underline{\underline{K}}}^{-} \underline{L} + {({\underline{\underline{K}}}^{-} {\underline{G}}^{+})}_{-} - 2 {({({\underline{\underline{K}}}^{+})}^{- 1} \underline{G})}_{-} = 2 {({\underline{\underline{K}}}^{+})}^{- 1} \underline{U} - {({\underline{\underline{K}}}^{-} \underline{F})}_{+} + 2 {({({\underline{\underline{K}}}^{+})}^{- 1} \underline{G})}_{+} = \underline{E} (α) .

(2.12)

We can now focus on the equation

{({\underline{\underline{K}}}^{+})}^{- 1} \underline{U} - {({\underline{\underline{K}}}^{-} \underline{F})}_{+} + 2 {({({\underline{\underline{K}}}^{+})}^{- 1} \underline{G})}_{+} = \underline{E} (α),

of which all components except

\underline{U}

can be calculated. After calculating

\underline{U},

we may return to (2.4) and find that

(\begin{matrix} A (α) \\ B (α) \end{matrix}) = (\begin{matrix} \frac{1}{S_{1} (α)} & 0 \\ 0 & \frac{1}{S_{2} (α)} \end{matrix}) (\underline{G} + \underline{U}) .

(2.13)

A similar result can be shown for $\underline{L}$ ⁠, but using $\underline{U}$ or $\underline{L}$ has no impact on the final result or the required analytical approach.

2.3 Additional analytical requirements

Although we calculated $A (α)$ and $B (α)$ in (2.13), several features of the Wiener–Hopf technique were omitted. We will review some neglected areas that may influence our solution and require extra care when modelling more complex real-world phenomena.

2.3.1 Choosing edge conditions

To solve the Wiener–Hopf equation uniquely, we require edge conditions for the solution such that we may apply the unsteady Kutta condition at the tip of the plate [36, 37], which should ensure the correct α-behaviour at infinity. For a traditional (one-sided) boundary condition problem, the edge conditions are usually of the form

\begin{matrix} φ_{s} (x, y) & \sim x^{a}, as x \to 0, \\ \frac{\partial φ_{s}}{\partial y} (x, y) & \sim x^{b}, as x \to 0. \end{matrix}

(2.14)

The values of a and b are determined case-by-case, depending on

L_{1, 2}

and the requirement of a non-singular solution at the edge. For example, the usual Sommerfeld problem with a Neumann boundary condition will have

a = 0, b = - \frac{1}{2}

⁠. For our method, edge conditions are applied to deduce the required large α behaviour of the components of

\underline{U}

and

\underline{L}

⁠, which can be found by looking at the small r behaviour of the quantities being transformed. More specifically, given a function.

f (x) = O (x^{ϖ}) a s x \to 0,

then the half transform² of f will satisfy

F (α) : = \int_{0}^{\infty} f (x) e^{i α x} d x = O (α^{- ϖ - 1}), as α \to \infty .

For clarity, we derive the large α scaling of each unknown term in (2.4), (2.7), (2.5) and (2.8) for a specific example at the end of this section.

By considering the Helmholtz equation in polar coordinates, one may write the general solution as a series

φ_{s} (r, θ) = \sum_{η} A_{η} r^{η} cos (η (θ - π)) + B_{η} r^{η} sin (η (θ - π)), η \geq 0.

(2.15)

The unknowns here are determined after inserting this general solution into the problem’s boundary conditions, from which a and b can be found and applied elsewhere.

Example: A ‘semi-rigid’ Ingard–Myers boundary condition

As a motivational example of this method, we define a ‘semi-rigid’ boundary condition as a boundary with one side considered rigid (having a Neumann boundary condition) while the other has some other condition. We choose the commonly used Ingard–Myers boundary condition [16, 38, 39] and revisit this in section 4.

For this boundary condition, our linear operators will be

\begin{matrix} L_{1} & = \frac{\partial}{\partial y} - \frac{i}{k Z} {(M \frac{\partial}{\partial x} + i k)}^{2}, \\ L_{2} & = - \frac{\partial}{\partial y} . \end{matrix}

(2.16)

We follow the approach of section 2.2.1 of [40] and find a solution to the Helmholtz equation in polar coordinates as

r \to 0

⁠. In this limit, we look for an expansion in r, noting that the Helmholtz equation will resemble the Laplace equation in the limit so that an appropriate asymptotic solution can be written in the form,

Ψ_{s} (r, θ) \sim \sum_{j = 1}^{\infty} r^{α_{j}} \sum_{m = 0}^{N_{j}} f_{m}^{(α_{j})} (θ) as r \to 0, 0 \leq α_{1} < α_{2} < \dots,

where each N_j is a finite number. These functions are found by inserting the approximation into the Helmholtz equation and then investigating the boundary conditions at increasing powers of r. We first look at

O (r^{α_{1} - 2})

and conclude what the lowest value for α₁ must be, before solving for the corresponding

f_{0}^{α_{1}}

⁠, for our example,

\frac{d^{2}}{d θ^{2}} f_{0}^{α_{1}} (θ) + α_{1}^{2} f_{0}^{(α_{1})} (θ) = 0,

(2.17a)

\frac{- i M^{2}}{K Z} α_{1} (α_{1} - 1) f_{0}^{α_{1}} (θ) = 0, θ = π,

(2.17b)

\frac{d}{d θ} f_{0}^{α_{1}} (θ) = 0, θ = - π .

(2.17c)

The first choice is

α_{1} = 0

⁠, for which

f_{0}^{0} = A_{0}

⁠. This will lead to a series

f_{n}^{0}

where each n is a positive integer. However, this is not the only possible solution. It can be observed that our boundary is acting as a perfect boundary (soft-hard), so we may look at

α_{2} = \frac{1}{4}

⁠. Our boundary conditions at

O (r^{- \frac{7}{4}})

become

\frac{- i M^{2}}{K Z} α_{2} (α_{2} - 1) f_{0}^{α_{2}} = 0, θ = π,

(2.18a)

\frac{d}{d θ} f_{0}^{α_{2}} = 0, θ = - π,

(2.18b)

for which we can choose

f_{0}^{\frac{1}{4}} = A_{\frac{1}{4}} cos (\frac{θ + π}{4}) + B_{\frac{1}{4}} sin (\frac{θ - π}{4})

and then construct a series

f_{n}^{\frac{1}{4}}

⁠. The final suitable choice would be

α_{3} = \frac{3}{4}

⁠, which will follow from the above and give

f_{0}^{\frac{3}{4}} = A_{\frac{3}{4}} cos (\frac{3 (θ + π)}{4}) + B_{\frac{3}{4}} sin (\frac{3 (θ - π)}{4}) .

For completeness, we find the next term of each series: First, we find

f_{1}^{α_{j}}

by inspecting the boundary conditions at

O (r^{α_{j} - 1})

⁠. For α₁, the upper boundary condition will always be satisfied trivially, while the lower boundary condition remains the Neumann boundary condition. Therefore, we can set

f_{1}^{0} (θ) = A_{1} cos (θ + π) .

Then, for α₂, the boundary conditions will become

\frac{- i M^{2}}{K Z} \frac{- 3}{16} f_{1}^{\frac{1}{4}} + \frac{B_{\frac{1}{4}}}{4} = 0, θ = π,

(2.19a)

\frac{d}{d θ} f_{1}^{\frac{1}{4}} = 0, θ = - π .

(2.19b)

This will only have a solution if

B_{\frac{1}{4}} = 0

⁠, meaning we must eliminate this term from the expansion. We choose

f_{1}^{\frac{1}{4}} (θ) = A_{\frac{5}{4}} cos (\frac{5 (θ + π)}{4}) .

Similarly,

B_{\frac{3}{4}} = 0

and

f_{1}^{\frac{3}{4}} (θ) = A_{\frac{7}{4}} cos (\frac{7 (θ + π)}{4}) .

This trend continues onwards so that our small r expansion is of the form

Ψ_{s} (r, θ) \sim A_{0} + A_{\frac{1}{4}} r^{\frac{1}{4}} cos (\frac{θ + π}{4}) + A_{\frac{3}{4}} r^{\frac{3}{4}} cos (\frac{3 (θ + π)}{4}) + A_{1} r cos (θ + π) + O (r^{\frac{5}{4}}) .

(2.20)

For the traditional edge conditions such as (2.14), we find that

a = 0, b = - \frac{3}{4}

⁠. However, our method required the scaling of

Ψ_{s} (r, π) - Ψ_{s} (r, - π), as r \to 0,

(2.21a)

\frac{\partial Ψ_{s}}{\partial y} (r, π) - \frac{\partial Ψ_{s}}{\partial y} (r, - π), as r \to 0,

(2.21b)

L_{1} [Ψ_{s}] (r, 0), as r \to 0,

(2.21c)

L_{2} [Ψ_{s}] (r, 0), as r \to 0,

(2.21d)

for

l_{1}, l_{2}, u_{1}, u_{2}

respectively, after which we can find the large α scaling using known results on half transforms [32]. Inserting (2.20) into (2.21a), we see the constant term will cancel so that

Ψ_{s} (r, π) - Ψ_{s} (r, - π) \sim r^{\frac{1}{4}}

r \to 0

⁠. Similarly, inserting (2.20) into (2.21b) removes the constant term but keeps the next order term so that

Ψ_{s} (r, π) - Ψ_{s} (r, - π) \sim r^{- \frac{3}{4}}

r \to 0

⁠, this will also hold for (2.21d). Finally, for (2.21c), the lowest order will come from the second x derivative in

L_{1}

⁠, leaving the lowest order term scaling as

r^{- \frac{7}{4}}

⁠. Given these scalings,

l_{1} \sim α^{- \frac{5}{4}}, l_{2} \sim α^{- \frac{1}{4}}, u_{1} \sim α^{\frac{3}{4}}, u_{2} \sim α^{- \frac{1}{4}},

(2.22)

all as

α \to \infty

2.3.2 Regions of analyticity

With our general Wiener–Hopf equation constructed, we formally define initial regions D_U and D_L for our chosen governing equations. First, we can determine the regions of analyticity of the additional known terms from their half-range integrals once we have chosen ϵ such that $k = k + ϵ i$ ⁠. Omitting the analysis, we find that $C_{K} i {(α + δ_{3} k)}^{- 1}$ is analytic in the region ${z \in C | I (z) > - ϵ R (δ_{3}) - k I (δ_{3})},$ while $G_{1, 2} {(α + δ_{1} k)}^{- 1}$ are analytic in ${z \in C | I (z) < ϵ R (δ_{1}) + k I (δ_{1})}$ ⁠.

We will show in the next section that

\underline{\underline{K}}

can be written as

\underline{\underline{K}} = {\underline{\underline{K}}}^{+} {\underline{\underline{K}}}^{-}

such that

{\underline{\underline{K}}}^{+}

is analytic in some upper complex region D_U and

{\underline{\underline{K}}}^{-}

is analytic in some lower complex region D_L. For reasons that will be clearer in the next section, these regions will depend on the placement of any possible zeros of the functions

S_{1, 2} (α)

⁠. We write this set of zeros as

\begin{matrix} Z & = {α \in C | S_{1} (α) = 0 or S_{2} (α) = 0} \\ = Z_{U} \cup Z_{L}, \end{matrix}

(2.23)

Z_{U} = Z \cap {α \in C | I (α) > 0}, Z_{L} = Z \cap {α \in C | I (α) < 0} .

(2.24)

We assume all zeros lie off the real line. We may achieve this by introducing an imaginary part into the boundary conditions on the plate (in most aeroacoustics-related examples, this will follow from the ϵ imaginary part introduced into k). This step is for convenience; however, if computing factorisations numerically, it is possible to keep these zeros on the real line and deform the integration contour around them, as described in [6].

We may now define D_U to be the upper half region of the complex plane defined by

I (α) > c_{1}

and D_L to be the lower half region defined by

I (α) < c_{2}

with constants

c_{1, 2}

given by

c_{1} = \max [{- ϵ R (δ_{3}) - k I (δ_{3})} \cup {- ϵ} \cup {R (z) | z \in Z_{L}}] < 0,

(2.25a)

c_{2} = \min ({ϵ R (δ_{1}) + k I (δ_{1})} \cup {ϵ} \cup {I (k)} \cup {R (z) | z \in Z_{U}}) > 0.

(2.25b)

Figure 1 demonstrates the regions $D_{U, L}$ ⁠. As an arbitrary example, we assume there are four zeros in $Z$ that are defined with the subscript U or L denoting whether they are a member of $Z_{U}$ or $Z_{L}$ ⁠.

2.3.3 Acknowledging surface waves

To finish this section, it is important to mention the possibility that surface waves may arise depending on the chosen boundary conditions. In particular, one source of such waves may lie in the zeros of the kernels S₁ and S₂ or otherwise. Referring back to (2.13), these zeros would become poles in A, B that must be accounted for during the integration process. Surface waves are discussed throughout the literature, and a thorough parametric study of how they can arise for the Ingard–Myers boundary condition is tackled in [22]. For most practical examples, absorbing boundaries will ensure these zeros do not lie in the cut α plane, as shown in [16] for the Ingard–Myers boundary condition and [20] for the convective Robin boundary condition. One exception to this rule is [18], in which surface waves are explored in detail when purely imaginary impedance values are examined. In this case, the surface waves are equivalent to zeros of the kernel lying on the real axis. Therefore, their residues must be accounted for when integrating over the real line.

Extending this to more complicated boundary conditions, or even more generalised conditions, is a non-trivial but worthwhile study. However, since this article primarily focuses on adapting the Wiener–Hopf technique to more general scattering problems, we omit an in-depth study for brevity. Still, in Appendix 5, we outline a way to find the zeros of a scalar kernel $S (α)$ ⁠.

For many other applications, the far-field (large r) scattering is required. In this case, the steepest descent method is performed to ensure the integration occurs along the path of greatest decay. As we deform our integration contour from the real line to the path of steepest descent, we may well pick up poles that are equivalent to hydrodynamic or acoustic modes, as described in [22, 39, 41, 42].

3 Solving the Wiener–Hopf equation

Recall that we aim to solve the matrix Wiener–Hopf equation

{\underline{\underline{K}}}^{-} \underline{L} + {({\underline{\underline{K}}}^{-} \underline{F})}_{-} - 2 {({({\underline{\underline{K}}}^{+})}^{- 1} \underline{G})}_{-} = 2 {({\underline{\underline{K}}}^{+})}^{- 1} \underline{U} - {({\underline{\underline{K}}}^{-} \underline{F})}_{+} + 2 {({({\underline{\underline{K}}}^{+})}^{- 1} \underline{G})}_{+} = \underline{E} (α) .

(3.1)

Before rearranging for $\underline{U},$ one must investigate the asymptotic behaviour of each side of the equation to deduce $\underline{E}$ ⁠; this is the aforementioned imposition of the Kutta condition and edge conditions at the tip. To do so, the first task would be to factorise the matrix kernel multiplicatively. Then, we require additive factorisations for ${({\underline{\underline{K}}}^{+})}^{- 1} \underline{G}$ and ${({\underline{\underline{K}}}^{-})}^{- 1} \underline{F}$ ⁠. More general source terms can be computed directly using the integral equations (1.17) from [32] or use the method of pole decomposition under the assumption that these functions are meromorphic. These tasks will be performed for our specific wave problem later in this section.

Once these tasks are complete and all components of (2.11) are known, we find

\underline{U} = \frac{1}{2} {\underline{\underline{K}}}^{+} [{({\underline{\underline{K}}}^{-} \underline{F})}_{+} - 2 {({(\underline{\underline{K^{+}}})}^{- 1} \underline{G})}_{+} + \underline{E}],

(3.2)

from which we can calculate the unknowns A and B from (2.7) and (2.8).

To solve the Wiener–Hopf equation (2.11), we carefully generalise the method presented in [21] to split our arbitrary kernel $\underline{\underline{K}}$ and then briefly discuss how we can calculate the other components of (2.7) to obtain the unknown vector $\underline{U}$ ⁠.

The significant changes from the literature that we must consider are that $S_{1, 2} (α)$ are now of the form $S_{1, 2} (α) = U_{1, 2} (α) + γ V_{1, 2} (α)$ ⁠, where each $U_{j}, V_{j}$ are polynomials of α of any positive order that may or may not have zeros in the cut α-plane.

By adapting this process to two general linear conditions, we can investigate plates that are different above and below not only in impedance (via some impedance parameter) but are more fundamentally different in having two distinct boundary conditions where more than one parameter is altered.

3.1 Application of the Wiener–Hopf–Hilbert method

As in references [19–21], we perform a multiplicative splitting of our matrix by reducing the problem to solving a coupled pair of Hilbert equations, hence why this technique is sometimes referred to as the Wiener–Hopf–Hilbert method.

We will factor out our determinant to ensure the matrix we will split into two factors will remain non-singular throughout the cut plane. For our general case, this is equivalent to setting

\begin{matrix} \underline{\underline{K}} & = D (α) \underline{\underline{χ}}, \\ D^{2} (α) & = \det (\underline{\underline{K}}) = \frac{2 S_{1} S_{2}}{γ}, \\ \underline{\underline{χ}} & = (\begin{matrix} \sqrt{\frac{S_{1} γ}{2 S_{2}}} & \sqrt{\frac{S_{1}}{2 S_{2} γ}} \\ - \sqrt{\frac{S_{2} γ}{2 S_{1}}} & \sqrt{\frac{S_{2}}{2 S_{1} γ}} \end{matrix}) ​ . \end{matrix}

(3.3)

Factorising D can be performed using a Cauchy integral method described in [32]. Conversely, for simpler functions,

S_{1, 2}

can be factorised fully analytically by evaluating the integral around the branch cut, see [16, 18, 19, 21, 23]. Once this factorisation is complete, we need the factors

γ^{+}

and

γ^{-}

⁠. These can be written as

γ^{+} (α) = \sqrt{α + k} e^{- \frac{π i}{4}}

(3.4a)

γ^{-} (α) = \sqrt{α - k} e^{\frac{π i}{4}}

(3.4b)

to ensure that

γ^{+} (0) = γ^{-} (0)

holds given our chosen branch cuts. This condition is consistent with the usual assumption from the factorisation of

S_{1, 2}

[32]. Turning our attention to the matrix factorisation of

\underline{\underline{χ}}

seek a vector

\underline{u}

that is analytic in our region D_U and a vector

\underline{l}

that is analytic in D_L satisfying the homogeneous Wiener–Hopf equation

\underline{\underline{χ}} \underline{l} = \underline{u} .

(3.5)

This equation provides an analytic continuation of $\underline{u}$ from D_U to $D_{U} \cup D_{L} ∖ Γ^{- k} .$ In Fig. 2, we demonstrate our branch cut notation and the jumps in γ we expect to see on each side. This is crucial to the analysis that follows in this section.

Demonstration of branch cuts Γk,−k and the values γ(α) takes on each side. We also include the notation for the evaluation of matrix K__ above and below each branch cut.

Fig. 2

Demonstration of branch cuts $Γ^{k, - k}$ and the values $γ (α)$ takes on each side. We also include the notation for the evaluation of matrix $\underline{\underline{K}}$ above and below each branch cut.

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Across the lower branch cut,

\underline{l} |_{- k, U} = \underline{l} |_{- k, L}

⁠, thus we can split (3.5) into two equations on each side of this branch cut:

\underline{\underline{χ}} |_{- k, U} \underline{l} |_{- k, U} = \underline{u} |_{- k, U},

(3.6a)

\underline{\underline{χ}} |_{- k, L} \underline{l} |_{- k, U} = \underline{u} |_{- k, L} .

(3.6b)

Rearranging each equation for

l |_{- k, U}

and then equating both to each other gives us the equation

\underline{\underline{χ}} |_{- k, U} {\underline{\underline{χ}}}^{- 1} |_{- k, L} \underline{u} |_{- k, L} = \underline{u} |_{- k, U} .

(3.7)

By explicitly investigating the form of the matrix kernel

\underline{\underline{χ}}

and it’s inverse on each side of the branch cut (where γ has two different values), we can set

\underline{\underline{H}} = \underline{\underline{χ}} |_{- k, U} {\underline{\underline{χ}}}^{- 1} |_{- k, L} = - i (\begin{matrix} 0 & \sqrt{\frac{S_{1} |_{- k, U} S_{1} |_{- k, L}}{S_{2} |_{- k, U} S_{2} |_{- k, L}}} \\ \sqrt{\frac{S_{2} |_{- k, U} S_{2} |_{- k, L}}{S_{1} |_{- k, U} S_{1} |_{- k, L}}} & 0 \end{matrix}) ​ .

(3.8)

This reduces our system to a simpler form

\underline{\underline{H}} \underline{u} |_{- k, L} = \underline{u} |_{- k, U} .

(3.9)

Considering u_j to be the jth entry of the vector

\underline{u}

⁠, we define

V (α) = u_{1} (α) u_{2} (α), W (α) = \frac{u_{1} (α)}{u_{2} (α)},

(3.10)

and use (3.9) to obtain the two Hilbert problems

V |_{- k, U} = - V |_{- k, L},

(3.11a)

W |_{- k, U} W |_{- k, L} = \frac{S_{1} |_{- k, U} S_{1} |_{- k, L}}{S_{2} |_{- k, U} S_{2} |_{- k, L}} .

(3.11b)

These Hilbert problems can be solved using techniques in [12, 19, 21]; however, it is clear that to approach this more generally, care must be taken to avoid zeros and poles in our factors which can be handled on a case-by-case basis. Furthermore, we pay more attention to the imaginary part of k, causing the branch cuts to be at an angle and affecting the calculations needed for W. This is included since we will use a wavenumber with a small imaginary part in our example since this aids the numerical integration.

Solving 3.11a is straightforward:

V (α) = {(γ^{+})}^{2 n - 1}, n \in Z .

(3.12)

We can determine the value of n by ensuring (3.9) is satisfied and our matrices are non-singular at the branch point, just as in [21]. We postpone this until after we have solved (3.11b).

For W, we focus on the method of [19]. Due to the different values for

γ^{+}

across the lower branch cut, we write

S_{i} (α)

in the more concise form

U_{i} (α) + γ V_{i} (α)

⁠, where

U_{i}

and

V_{i}

are simple polynomials in α, for each

j = 1, 2,

⁠. Then,

\begin{matrix} S_{j} |_{- k, U} S_{j} |_{- k, L} & = (U_{j} + | γ | e^{i ϑ} V_{j}) (U_{j} - | γ | e^{i ϑ} | V_{j}) \\ = U_{j}^{2} - | (α^{2} - k^{2}) | e^{2 i ϑ} V_{j}^{2} \\ = U_{j}^{2} - (| α | e^{i ϑ} - k) (| α | e^{i ϑ} + k) V_{j}^{2} \\ = {\tilde{S}}_{j} (| α | e^{i ϑ}), \end{matrix}

where

{\tilde{S}}_{j}

is a polynomial of positive order. We may obtain a solution for

W (α)

as an extension of Appendices A and B from [19] by considering the respective roots of these polynomials.

\begin{matrix} W |_{- k, U} W |_{- k, L} & = \frac{{\tilde{S}}_{1}}{{\tilde{S}}_{2}} \\ = \frac{C_{1} \prod_{j = 1}^{N} (| α | e^{i ϑ} - s_{j})}{C_{2} \prod_{j = 1}^{M} (| α | e^{i ϑ} - t_{j})} \end{matrix}

(3.13)

where s_j and t_j are the N (⁠

M)

roots of

{\tilde{S}}_{1}

and

{\tilde{S}}_{2}

respectively and

C_{1, 2}

are the respective leading coefficients. We can take logs on each side of (3.13) and divide each side by a suitable scalar multiple of

| γ^{+} |

⁠. This is to take advantage of the jump in

γ^{+}

across

Γ^{- k}

so that we may rewrite (3.13) as

{[\frac{log W}{γ^{+}}]}_{- k, U} - {[\frac{log W}{γ^{+}}]}_{- k, L} = \frac{1}{| γ^{+} | e^{\frac{i}{2} (\frac{π}{2} + θ)}} log [\frac{C_{1} \prod_{j = 1}^{N} (| α | e^{i ϑ} - s_{j})}{C_{2} \prod_{j = 1}^{M} (| α | e^{i ϑ} - t_{j})}] .

(3.14)

C_{1, 2}

denotes the coefficient of the highest power for the polynomials

{\tilde{S}}_{1}

and

{\tilde{S}}_{2}

⁠. This can be solved using the usual methods for Riemann–Hilbert problems as described in [18]

W (α) = exp {\frac{- γ^{+} e^{\frac{i}{2} (\frac{π}{2} - ϑ)}}{2 π} \int_{- k \infty}^{- k} \frac{1}{| t + k |^{1 / 2} (t - α)} log [\frac{C_{1} \prod_{j = 1}^{N} (| t | e^{i ϑ} - s_{j})}{C_{2} \prod_{j = 1}^{M} (| t | e^{i ϑ} - t_{j})}] d t} .

(3.15)

Writing this as

W (α) = e^{Ψ (α)},

we must calculate

Ψ (α) = \frac{- γ^{+} e^{\frac{i}{2} (\frac{π}{2} - ϑ)}}{2 π} \int_{- k \infty}^{- k} \frac{1}{| t + k |^{1 / 2} (t - α)} (log (\frac{C_{1}}{C_{2}}) + \sum_{j = 1}^{N} log (| t | e^{i ϑ} - s_{j}) - \sum_{j = 1}^{M} log (| t | e^{i ϑ} - t_{j})) d t .

(3.16)

To solve this, we change the integral contour to the positive real line using the change of coordinates

t = - e^{i ϑ} (\tilde{t} + | k |)

and use integrals

I (α), J (α)

derived in Appendix B of [19]. Omitting the algebra, we obtain the particular solution

W (α) = e^{\frac{π i (N - M)}{4}} \sqrt{\frac{C_{1}}{C_{2}}} \frac{\prod_{j = 1}^{N} (γ^{+} (α) + γ^{+} (s_{j}))}{\prod_{j = 1}^{M} (γ^{+} (α) + γ^{+} (t_{j}))} .

(3.17)

A key issue omitted in the literature is how the presence of zeros (and consequently poles) in the kernel can influence this process. For our more general setup, although we can choose to factorise a kernel with finite determinant as in [19], this matrix will have poles and zeros. Therefore, our splitting must reflect this but with poles and zeros placed in the correct domain, D_L for ${\underline{\underline{K}}}^{+}$ and D_U for ${\underline{\underline{K}}}^{-}$ ⁠. We must check that this will not affect our solution at the end of the procedure.

Another interesting consequence of (3.18b) is that if the branch point $α = - k$ is a root of either $\tilde{S_{1}}$ or $\tilde{S_{2}},$ we will see that W can have poles or zeros at the branch point. This problem will likely arise if Neumann is the upper or lower boundary condition. However, we will show later that the structure of the solution will ensure this does not pose a problem to the analyticity of our factors. This additional jump will be reflected in $\underline{\underline{H}}$ from (3.9), so we can still choose the same n value for V to ensure the factors are non-singular.

3.2 Finding the particular solution for $u_{1, 2}$

With V and W calculated, we know that

u_{1}^{(n)} = {(γ^{+})}^{n - 1 / 2} \sqrt{W},

(3.18a)

u_{2}^{(n)} = {(- 1)}^{n} \frac{{(γ^{+})}^{n - 1 / 2}}{\sqrt{W}} .

(3.18b)

After substituting these

u_{1, 2}^{(n)}

into (3.9), we deduce that n must be an even integer. We aim to choose some specific fixed

n^{*}

such that

\begin{matrix} {\underline{\underline{χ}}}^{+} & = (\begin{matrix} u_{1}^{(n^{*} + 1)} & u_{1}^{(n^{*})} \\ u_{2}^{(n^{*} + 1)} & u_{2}^{(n^{*})} \end{matrix}), \\ {({\underline{\underline{χ}}}^{-})}^{- 1} & = (\begin{matrix} \sqrt{\frac{S_{2}}{2 S_{1} γ}} & - \sqrt{\frac{S_{1}}{2 S_{2} γ}} \\ \sqrt{\frac{S_{2} γ}{2 S_{1}}} & \sqrt{\frac{S_{1} γ}{2 S_{2}}} . \end{matrix}) (\begin{matrix} u_{1}^{(n^{*} + 1)} & u_{1}^{(n^{*})} \\ u_{2}^{(n^{*} + 1)} & u_{2}^{(n^{*})} \end{matrix}), \\ = (\begin{matrix} l_{1}^{(n^{*} + 1)} & l_{1}^{(n^{*})} \\ l_{2}^{(n^{*} + 1)} & l_{2}^{(n^{*})} \end{matrix}), \end{matrix}

(3.19)

will satisfy all necessary analytical conditions. Referring to our Wiener–Hopf equation (3.1), we need to ensure the following conditions are satisfied to apply an analytic continuation argument and then the generalised Liouville’s Theorem that dictates the form of

\underline{E}

⁠; the asymptotic growth of each side of the equation will dictate the order of each polynomial component within the vector

\underline{E}

⁠:

${\underline{\underline{K}}}^{+} {\underline{\underline{K}}}^{-} = \underline{\underline{K}}$ ⁠.
${\underline{\underline{χ}}}^{+}$ and ${\underline{\underline{χ}}}^{-}$ are non-singular in the entire cut plane.
${({\underline{\underline{K}}}^{+})}^{- 1}$ is analytic in D_U.
${\underline{\underline{K}}}^{-}$ is analytic in D_L.

The first point follows as a consequence of (3.5) and (3.19) where we assume that the matrix

{\underline{\underline{χ}}}^{-}

can be written as the product

{({\underline{\underline{χ}}}^{+})}^{- 1} \underline{\underline{χ}}

⁠, (the splitting assumes analyticity in required regions; we have yet to show this). Since

\underline{\underline{χ}}

is orthogonal, we only need to ensure that

{\underline{\underline{χ}}}^{+}

has non-zero determinant. Since

\begin{matrix} \det ({\underline{\underline{χ}}}^{+}) & = u_{1}^{(n^{*} + 1)} u_{2}^{(n^{*})} - u_{2}^{(n^{*} + 1)} u_{1}^{(n^{*})} \\ = {(- 1)}^{n^{*}} {(γ^{+})}^{2 n^{*}} - {(- 1)}^{n^{*} + 1} {(γ^{+})}^{2 n^{*}} \\ = 2 {(- 1)}^{n^{*}} {(- (α + k))}^{n^{*}} . \end{matrix}

We can choose $n^{*} = 0$ and ensure our matrices are non-singular.

Now that we know

n^{*}

⁠, we can define our matrices

{({\underline{\underline{K}}}^{+})}^{- 1}

and

{\underline{\underline{K}}}^{-}

and check for analyticity in the required regions.

\begin{matrix} {({\underline{\underline{K}}}^{+})}^{- 1} & = \frac{1}{D^{+} (α)} {({\underline{\underline{χ}}}^{+})}^{- 1} \\ = \sqrt{\frac{1}{2 S_{1}^{+} S_{2}^{+}}} (\begin{matrix} γ^{+} \sqrt{W} & \sqrt{W} \\ - γ^{+} \frac{1}{\sqrt{W}} & \frac{1}{\sqrt{W}} \end{matrix}), \end{matrix}

(3.20)

\begin{matrix} {\underline{\underline{K}}}^{-} & = D^{-} (α) {({\underline{\underline{χ}}}^{+})}^{- 1} \underline{\underline{χ}} \\ = \sqrt{\frac{2 S_{1}^{-} S_{2}^{-}}{γ^{-}}} \frac{1}{2} (\begin{matrix} \sqrt{\frac{1}{γ^{+} W}} & - \sqrt{\frac{W}{γ^{+}}} \\ \sqrt{\frac{γ^{+}}{W}} & \sqrt{W γ^{+}} \end{matrix}) (\begin{matrix} \sqrt{\frac{S_{1} γ}{2 S_{2}}} & \sqrt{\frac{S_{1}}{2 S_{2} γ}} \\ - \sqrt{\frac{S_{2} γ}{2 S_{1}}} & \sqrt{\frac{S_{2}}{2 S_{1} γ}} \end{matrix}) \\ = \frac{\sqrt{S_{1}^{-} S_{2}^{-}}}{2} (\begin{matrix} (\sqrt{\frac{S_{1}}{S_{2} W}} + \sqrt{\frac{S_{2} W}{S_{1}}}) & \frac{1}{γ} (\sqrt{\frac{S_{1}}{S_{2} W}} - \sqrt{\frac{S_{2} W}{S_{1}}}) \\ γ^{+} (\sqrt{\frac{S_{1}}{S_{2} W}} - \sqrt{\frac{S_{2} W}{S_{1}}}) & \frac{1}{γ^{-}} (\sqrt{\frac{S_{1}}{S_{2} W}} + \sqrt{\frac{S_{2} W}{S_{1}}}) \end{matrix}) ​ . \end{matrix}

(3.21)

Regarding ${({\underline{\underline{K}}}^{+})}^{- 1}$ ⁠, our matrix consists of ‘plus functions’ and W. It can be observed that W is fully analytic in the complex plane, excluding the branch cut $Γ^{- k}$ ⁠, so it is analytic in D_U.

Ensuring

{\underline{\underline{K}}}^{-}

is analytic in D_L requires more care. We must check that it has no poles or zeros in D_L and no discontinuity across the branch cut

Γ^{- k}

⁠. To check there is no branch cut discontinuity, it is sufficient to solely check the matrix

{({\underline{\underline{χ}}}^{-})}^{(- 1)}

⁠.³ We can use the following derivation from [21]:

\begin{matrix} [{\underline{\underline{χ}}}^{- 1} {\underline{\underline{χ}}}^{+}] |_{- k, U} & = {\underline{\underline{χ}}}^{- 1} |_{- k, U} \underline{\underline{H}} {\underline{\underline{χ}}}^{+} |_{- k, L} \\ = {\underline{\underline{χ}}}^{- 1} |_{- k, U} \underline{\underline{χ}} |_{- k, U} {\underline{\underline{χ}}}^{- 1} |_{- k, L} {\underline{\underline{χ}}}^{+} |_{- k, L} \\ = [{\underline{\underline{χ}}}^{- 1} {\underline{\underline{χ}}}^{+}] |_{- k, L} . \end{matrix}

(3.22)

This demonstrates there is no discontinuity in ${\underline{\underline{χ}}}^{-}$ across the lower branch cut.

By inspection, the only possible poles or zeros lie at the branch point—k or any potential zeros of S₁ and S₂. The former issue can be checked thoroughly, as described in [21]:

\lim_{α \to - k} W (α) = \frac{S_{1} (- k)}{S_{2} (- k)} + O (γ^{+}),

from which we can conclude every entry of

{\underline{\underline{χ}}}^{-}

will be finite and non-zero at the branch point.

For two-sided wave scattering problems in the literature, the boundaries are assumed to be absorbing (or supporting surface waves), which ensures either $S_{1, 2}$ have either no zeros in the cut plane or manageable zeros on the real line. The presence of zeros can be analysed using other matrix factorisation methods, as described in section 1. Still, for this study, we are more interested in a specific subclass of matrices relating to our application. Therefore, we suggest an amendment to the factorisation that can account for potential zeros in D_L arising from the $S_{1, 2}$ terms in ${\underline{\underline{χ}}}^{-}$ ⁠.

In [23], it is noted that zeros in the kernels can be accounted for with the introduction of eigensolutions to the governing equation. This effectively ensures that a zero in one of the kernels in D_L (without loss of generality) can be transferred to $S_{1}^{+}$ and vice versa. This ensures the definitions of the analytic splitting can be adhered to while also satisfying the governing equations. Our goal is to account for zeros similarly within our matrix kernel.

Returning to

3.11 a

⁠, we amend the usual requirement for the Riemann–Hilbert problem: the solutions must be holomorphic in the entire complex plane (excluding the branch cut) and replace this with meromorphic solutions. Since our issue lies solely in zeros of

S_{1, 2}

in D_L, we suppose that all these zeros lie at

z_{1}, z_{2}, \dots, z_{N}

and define

Ξ (α) = \prod_{n = 1}^{N} (z - z_{n}) .

We replace our solution of V in 3.11a by $V (α) / Ξ (α)$ ⁠, which is analytic in D_U and meromorphic in D_L. Inserting this new solution into ${\underline{\underline{χ}}}^{+}$ will not affect the analyticity in D_U. In contrast, when we then calculate ${\underline{\underline{K}}}^{-}$ ⁠, a factor of $\sqrt{Ξ (α)}$ will ensure there are no longer any poles or zeros in D_L. We note that ${\underline{\underline{χ}}}^{+}$ will also be singular at the zeros in D_L, but this will not affect the formation or rearrangement of our Wiener–Hopf equation.

Our matrix factorisation can be written as

{\underline{\underline{K}}}^{+} = 2^{\frac{1}{4}} \sqrt{\frac{S_{1}^{+} S_{2}^{+}}{Ξ (α) W (α)}} (\begin{matrix} W (α) & \frac{W (α)}{γ^{+}} \\ - 1 & \frac{1}{γ^{+}} \end{matrix}),

(3.23a)

{\underline{\underline{K}}}^{-} = \frac{\sqrt{Ξ (α) S_{1}^{-} S_{2}^{-}}}{2^{\frac{5}{4}}} (\begin{matrix} (\sqrt{\frac{S_{1}}{S_{2} W}} + \sqrt{\frac{S_{2} W}{S_{1}}}) & \frac{1}{γ} (\sqrt{\frac{S_{1}}{S_{2} W}} - \sqrt{\frac{S_{2} W}{S_{1}}}) \\ γ^{+} (\sqrt{\frac{S_{1}}{S_{2} W}} - \sqrt{\frac{S_{2} W}{S_{1}}}) & \frac{1}{γ^{-}} (\sqrt{\frac{S_{1}}{S_{2} W}} + \sqrt{\frac{S_{2} W}{S_{1}}}) \end{matrix}) ​ .

(3.23b)

In the case that each side has the same boundary condition,⁴S₁ = S₂, W = 1 and

{\underline{\underline{K}}}^{+} = 2^{\frac{1}{4}} \sqrt{\frac{1}{Ξ (α)}} (\begin{matrix} S_{1}^{+} (α) & \frac{S_{1}^{+} (α)}{γ^{+}} \\ - S_{1}^{+} (α) & \frac{S_{1}^{+} (α)}{γ^{+}} \end{matrix}),

(3.24a)

{\underline{\underline{K}}}^{-} = \frac{\sqrt{Ξ (α)}}{2^{\frac{1}{4}}} (\begin{matrix} S_{1}^{-} (α) & 0 \\ 0 & \frac{S_{1}^{-} (α)}{γ^{-}} \end{matrix}) ​ .

(3.24b)

Finally, it is worth noting that if we had our plate on the positive x-axis, such as in the leading-edge scattering problem, the roles of

\underline{L}

and

\underline{U}

would swap in our Wiener–Hopf equation, which would now be of the form

\underline{\underline{K}} (α) (\underline{U} (α) + {\underline{F}}^{-} (α)) = 2 {\underline{G}}^{+} (α) + 2 \underline{L} (α),

(3.25)

so that we would instead need to factorise the matrix kernel into the form

\underline{\underline{K}} = {\underline{\underline{K}}}^{-} {\underline{\underline{K}}}^{+}

⁠. For this, the whole procedure must be repeated. We instead derive

V (α)

and

W (α)

by looking at the upper branch cut discontinuity of γ. The derivations after this follow similarly.

3.3 Solving for $φ_{s}$

Having factorised the matrix, all that remains is to find every component of (2.7). For this, we may complete two more straightforward tasks on a case-by-case basis. First, additive factorisations of the functions ${\underline{\underline{K}}}^{-} \underline{F}$ and ${(\underline{\underline{K^{+}}})}^{- 1} \underline{G}$ can be found using the method of pole decomposition [32].

Although we have denoted

\underline{F}

as an upper analytic function, it only has one pole in the complex plane at

- δ_{3} k

⁠, therefore we can write

{\underline{\underline{K}}}^{-} \underline{F} = \underset{Analytic in Lower Region}{\underset{︸}{({\underline{\underline{K}}}^{-} (α) \underline{F} - {\underline{\underline{K}}}^{-} (- δ_{3} k) \underline{F})}} + \underset{Analytic in Upper Region}{\underset{︸}{{\underline{\underline{K}}}^{-} (- δ_{3} k) \underline{F}}} .

(3.26)

Similarly, for the other term, we can write

{(\underline{\underline{K^{+}}})}^{- 1} \underline{G} = \underset{Analytic in Upper Region}{\underset{︸}{({(\underline{\underline{K^{+}}})}^{- 1} (α) \underline{G} - {(\underline{\underline{K^{+}}})}^{- 1} (- δ_{1}) \underline{G})}} + \underset{Analytic in Lower Region}{\underset{︸}{{(\underline{\underline{K^{+}}})}^{- 1} (- δ_{1}) \underline{G}}} .

(3.27)

Second, we must deduce the entire function $\underline{E} (α)$ using asymptotic arguments. This can be done case-by-case using the correct edge conditions and the solution’s small r expansion.

We now can calculate the unknowns

A (α), B (α)

that are needed for the scattered solution

φ_{s} (x, y)

⁠:

\begin{matrix} (\begin{matrix} A (α) \\ B (α) \end{matrix}) & = (\begin{matrix} \frac{1}{S_{1}} & 0 \\ 0 & \frac{1}{S_{2}} \end{matrix}) {\underline{\underline{K}}}^{+} (α) \frac{C_{K} i}{2 (α + δ_{3} k)} (\begin{matrix} {[{\underline{\underline{K}}}^{-} (- δ_{3} k)]}_{11} \\ {[{\underline{\underline{K}}}^{-} (- δ_{3} k)]}_{21} \end{matrix}) \\ + (\begin{matrix} \frac{1}{S_{1}} & 0 \\ 0 & \frac{1}{S_{2}} \end{matrix}) {\underline{\underline{K}}}^{+} (α) \frac{1}{(α + δ_{1} k)} (\begin{matrix} G_{1} {[{({\underline{\underline{K}}}^{+})}^{- 1} (- δ_{1} k)]}_{11} + G_{2} {[{({\underline{\underline{K}}}^{+})}^{- 1} (- δ_{1} k)]}_{12} \\ G_{1} {[{({\underline{\underline{K}}}^{+})}^{- 1} (- δ_{1} k)]}_{21} + G_{2} {[{({\underline{\underline{K}}}^{+})}^{- 1} (- δ_{1} k)]}_{22} \end{matrix}) \end{matrix}

(3.28)

4 Example: scattering a sound wave off a two-sided edge in flow

To demonstrate how we may use this general solution and methodology, we shall find the solution to a specific problem in which a sound wave scatters off the edge of a semi-infinite plate with an upper layer added made from some hypothetical material used to reduce noise. For simplicity, we model this plate by considering some two-sided boundary, one rigid and the other satisfies the Ingard–Myers impedance boundary condition for this pre-determined impedance $Z \in C$ [28].

To emphasise how our method generalises the results from existing literature such as [19–21], we change the fundamental mathematical structure of the boundary conditions above and below (not just the impedance parameter) and extend the method to account for a more mathematically intricate impedance boundary condition. The Ingard–Myers condition is known to be a better physical representation of the behaviour of a plate in flow; therefore, it is vital to create a more general method to handle more realistic conditions.

4.1 Governing equations

We define

\tilde{ψ}

as the total field that includes the incident wave and the scattered contribution we would like to find. It represents the velocity potential

\nabla \tilde{ψ} = \underline{u}

⁠. The total field satisfies the convective wave equation

{(\frac{\partial}{\partial t} + U_{\infty} \frac{\partial}{\partial x})}^{2} \tilde{ψ} - c_{0}^{2} \nabla^{2} \tilde{ψ} = 0,

(4.1)

where

U_{\infty}

is the constant mean flow speed, and we define

M = U_{\infty} / c_{0}

to be the mean flow Mach number. We consider time harmonic perturbations

\tilde{ψ} = ψ (x, y) e^{i ω t}

with

ω = \frac{k}{c_{0}}

⁠, and from now on suppress the

e^{i ω t}

term in all functions. We introduce an important convective constant for our derivations,

β = \sqrt{1 - M^{2}}

and give the simplified governing equations in full:

β^{2} \frac{\partial^{2} ψ}{\partial x^{2}} + \frac{\partial^{2} ψ}{\partial y^{2}} - 2 i k M \frac{\partial ψ}{\partial x} + k^{2} ψ = 0,

(4.2a)

\frac{\partial ψ}{\partial y} - \frac{i}{k Z} {(M \frac{\partial}{\partial x} + i k)}^{2} ψ = 0, x < 0, y = 0^{+},

(4.2b)

\frac{\partial ψ}{\partial y} = 0, x < 0, y = 0^{-},

(4.2c)

{[ψ]}_{-}^{+} = C_{K} e^{ikx / M}, x > 0, y = 0,

(4.2d)

{[\frac{\partial ψ}{\partial y}]}_{-}^{+} = 0, x > 0, y = 0.

(4.2e)

The first two conditions are our plate boundary condition (Ingard–Myers on

0^{+}

and Neumann on

0^{-}

⁠). The latter two are continuity conditions in the wake as described in [19],

C_{K}

being a constant that will be fixed during the implementation of the Kutta condition at the tip of the plate. We utilise the same change of variables as [19]:

x = β X, y = Y, Z = β^{- 3} Z, k = β K,

(4.3a)

ψ (x, y) = Ψ (X, Y) e^{i K M X} .

(4.3b)

With this, we decompose our total field into a scattered component and an incident wave:

\begin{matrix} Ψ (X, Y) & = Ψ_{s} (X, Y) + Ψ_{I} (X, Y), \\ Ψ_{s} & = \frac{1}{2 π} {\begin{matrix} \int_{- \infty}^{\infty} A (α) e^{- i α X - γ Y}, d α Y > 0, \\ \int_{- \infty}^{\infty} B (α) e^{- i α X + γ Y} d α, Y < 0. \end{matrix} \\ Ψ_{I} & = e^{i K (X cos (θ_{0}) - Y sin (θ_{0}))}, \end{matrix}

(4.4)

for some angle

θ_{0} \in (0, \frac{π}{2})

that we measure clockwise from the negative x-axis as per Fig. 3. Our final governing equations become

\frac{\partial^{2} Ψ_{s}}{\partial X^{2}} + \frac{\partial^{2} Ψ_{s}}{\partial Y^{2}} + K^{2} Ψ_{s} = 0,

(4.5a)

\frac{\partial Ψ_{s}}{\partial Y} - \frac{i}{K Z} {(M \frac{\partial}{\partial X} + i K)}^{2} Ψ_{s} = i K C_{0} e^{i K_{0} X} X < 0, Y = 0^{+},

(4.5b)

\frac{\partial Ψ_{s}}{\partial Y} = i K sin (θ_{0}) e^{i K_{0} X} X < 0, Y = 0^{-},

(4.5c)

{[Ψ_{s}]}_{-}^{+} = C_{K} e^{i K_{M} X}, X > 0, Y = 0,

(4.5d)

{[\frac{\partial Ψ_{s}}{\partial Y}]}_{-}^{+} = 0, X > 0, Y = 0,

(4.5e)

where we set

C_{0} = (sin (θ_{0}) - \frac{1}{Z} {(M cos (θ_{0}) + 1)}^{2}),

(4.6a)

K_{0} = K cos (θ_{0}),

(4.6b)

K_{M} = \frac{K}{M} .

(4.6c)

Fig. 3

Picking D_U and D_L for a general system with $| Z | = 4$ and some positive real $δ_{1, 2, 3}$ that satisfy $δ_{1} < 1$ and $δ_{2, 3} > 1$ ⁠.

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Equation (2.2) is of the form (2.1), hence the analysis of section 2 may be used. Compared with our generalised solution, the constants will be

δ_{1} = - cos (θ_{0}), δ_{2} = sin (θ_{0}), δ_{3} = \frac{1}{M} .

(4.7a)

G_{1} = K C_{0}, G_{2} = K sin (θ_{0}) .

(4.7b)

The complex α plane, including the poles, branch cuts, and regions of analyticity, is represented in Fig. 4, to be compared with Fig. 1. As shown in Fig. 4, D_U is the region $I (α) > - ϵ$ while D_L is the region $I (α) < ϵ cos θ_{0}$ ⁠. As desired, the real line will lie in the region of overlap $D_{U} \cap D_{L}$ ⁠.

$Geometrical setup for our diffraction problem. The two-dimensional model ignores the spanwise direction and focuses on z = 0.$

Fig. 4

Geometrical setup for our diffraction problem. The two-dimensional model ignores the spanwise direction and focuses on z = 0.

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Fig. 5

α-plane representation of regions of analyticity for the wave scattering example.

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Fig. 6

Contour plots of $R (ψ_{s} (x, y))$ for the rigid boundary problem alongside some semi-rigid examples inspired by real materials. For this example, we choose frequency f₁, Mach number M₁ and incident angle θ₁

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Fig. 7

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Fig. 8

Contour plots of $R (ψ_{s} (x, y))$ for the rigid boundary problem alongside some semi-rigid examples inspired by real materials. For this example, we choose frequency f₂, Mach number M₁ and incident angle θ₁

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After performing the Fourier transforms as outlined in section 2, we will obtain a Wiener–Hopf equation of the form

(\begin{matrix} S_{1} (α) & \frac{S_{1} (α)}{γ} \\ γ & - 1 \end{matrix}) [(\begin{matrix} l_{1} \\ - l_{2} \end{matrix}) + \frac{i}{α - K_{m}} (\begin{matrix} C_{K} \\ 0 \end{matrix})] = 2 (\begin{matrix} u_{1} \\ u_{2} \end{matrix}) - \frac{2}{α + K_{0}} (\begin{matrix} G_{1} \\ G_{2} \end{matrix}),

(4.8)

where the scalar kernels are of the form

\begin{matrix} S_{1} (α) & = \frac{i M^{2}}{K Z} α^{2} - \frac{2 i M}{Z} α + \frac{i K}{Z} - γ, \\ S_{2} (α) & = - γ . \end{matrix}

(4.9)

4.2 Solving the Wiener–Hopf equation

Following section 3, we must first split the matrix $\underline{\underline{K}},$ before performing additive splittings and deducing the form of the entire function $\underline{E}$ ⁠.

To find W, we need to calculate the roots $s_{j}, t_{j}$ of $S_{1} |_{- k, U} S_{1} |_{- k, L}$ and $S_{2} |_{- k, U} S_{2} |_{- k, L}$ respectively.

As we allude to in section 3, since the lower surface is rigid,

S_{2} |_{- k, U} S_{2} |_{- k, L} = γ^{2}

⁠, therefore

t_{1} = K, t_{2} = - K

⁠. Conversely,

S_{1} |_{- k, U} S_{1} |_{- k, L} = - \frac{M^{4}}{K^{2} Z^{2}} {(α - \frac{K}{M})}^{4} - α^{2} + K^{2}

(4.10)

will have four roots, of which none will be the branch points

\pm k

due to the restrictions on the model parameters.

Putting this together,

W (α) = \frac{i M^{2}}{K Z} \frac{\prod_{j = 1}^{4} (γ^{+} (α) + γ^{+} (s_{j}))}{γ^{+} (α) (γ^{+} (α) + \sqrt{2 K})}

(4.11)

and we may also calculate

S_{1}^{+}

numerically so that all components of (2.12) and (2.13) can be computed.

We solve (2.7) for

\underline{E}

in the Appendix 5, finding

\underline{E} (α) = (\begin{matrix} E \\ 0 \end{matrix})

for some complex constant

E

⁠.

Appendix 5 outlines the procedure for applying the edge conditions derived in section 2. Moreover, we also apply the Kutta condition at the tip of the plate to enforce finite velocity and pressure. With this, we deduce $C_{K}$ and also show $E = 0$ ⁠.

Now we have all components of the Wiener–Hopf equation (3.1) and we can find both

A (α)

and

B (α)

from (2.7) and (2.8).

\begin{matrix} (\begin{matrix} A (α) \\ B (α) \end{matrix}) & = (\begin{matrix} \frac{1}{S_{1}} & 0 \\ 0 & - \frac{1}{γ} \end{matrix}) {\underline{\underline{K}}}^{+} (α) \frac{C_{K} i}{2 (α + \frac{K}{M})} (\begin{matrix} {[{\underline{\underline{K}}}^{-} (- \frac{K}{M})]}_{11} \\ {[{\underline{\underline{K}}}^{-} (- \frac{K}{M})]}_{21} \end{matrix}) \\ + (\begin{matrix} \frac{1}{S_{1}} & 0 \\ 0 & - \frac{1}{γ} \end{matrix}) {\underline{\underline{K}}}^{+} (α) \frac{1}{(α - K cos (θ_{0}))} (\begin{matrix} G_{1} {[{({\underline{\underline{K}}}^{+})}^{- 1} (K cos (θ_{0}))]}_{11} + G_{2} {[{({\underline{\underline{K}}}^{+})}^{- 1} (K cos (θ_{0}))]}_{12} \\ G_{1} {[{({\underline{\underline{K}}}^{+})}^{- 1} (K cos (θ_{0}))]}_{21} + G_{2} {[{({\underline{\underline{K}}}^{+})}^{- 1} (K cos (θ_{0}))]}_{22} \end{matrix}) \end{matrix}

(4.12)

\begin{matrix} C_{K} & = 2 i \frac{G_{1} {[{({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1}]}_{2, 1} + G_{2} {[{({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1}]}_{2, 2}}{{[{\underline{\underline{K}}}^{-} (K_{M})]}_{2, 1}} \\ = 2 i K \frac{(sin (θ_{0}) - \frac{{(M cos (θ_{0}) + 1)}^{2}}{Z}) {[{({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1}]}_{2, 1} + sin (θ_{0}) {[{({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1}]}_{2, 2}}{{[{\underline{\underline{K}}}^{-} (K_{M})]}_{2, 1}} \end{matrix}

(4.13)

4.3 Results

We finish this section with a demonstration of the effects of our boundary condition; we plot near-field results of the fully rigid plate alongside the semi-rigid plate with various parameter values for $f, M, θ_{0}$ and $Z$ ⁠. We keep $R (Z) > 0$ ⁠, which ensures the boundary absorbs energy [19] and allows us to disregard potential zeros in the kernel, as shown in [16].

To calculate our integral, we use the reliable and efficient rational 4-to-1 mapping [43]

α = \frac{ζ + ζ^{3}}{{(1 - ζ^{2})}^{2}}

so that, for any point in the upper half (x, y) plane our calculation will now be

ψ (x, y) = \frac{e^{\frac{ikMx}{1 - M^{2}}}}{2 π} \int_{- 1}^{1} A (\frac{ζ + ζ^{3}}{{(1 - ζ^{2})}^{2}}) \frac{ζ^{4} + 6 ζ^{2} + 1}{{(1 - ζ^{2})}^{3}} e^{- i \frac{ζ + ζ^{3}}{{(1 - ζ^{2})}^{2}} \frac{x}{\sqrt{1 - M^{2}}} - γ (\frac{ζ + ζ^{3}}{{(1 - ζ^{2})}^{2}}) y} d ζ,

(4.14)

with an analogous result for a point in the lower half (x, y) plane. This mapping was chosen due to the expected small r behaviour of both the rigid and semi-rigid scattered solutions.

For our plots, we fix two frequencies and investigate two combinations of Mach number and incident angle for each chosen frequency. We add a small imaginary part

k = \frac{2 π f}{c_{0}} + 0.05 i

to the wavenumber for computational purposes. Our chosen frequencies, Mach numbers and incident angles are

f_{1} = 400 Hz, f_{2} = 800 Hz, M_{1} = 0.3, M_{2} = 0.6, θ_{1} = \frac{π}{4}, θ_{2} = \frac{π}{6} .

The first impedance values we choose to test are taken from [20], $Z_{F} = \frac{1}{2} + 1 i$ ⁠, resembling an absorbing fibrous sheet, $Z_{S} = \frac{3}{2}$ ⁠, resembling a perforated steel sheet. We expect similar reductions in ψ_s due to these coatings, but the presence of the rigid surface at $y = 0^{-}$ and the effects of $W (α)$ are harder to predict.

We observe many interesting phenomena when comparing our scattered field of the semi-rigid sheets with the fully rigid boundary. First, we notice expected reductions in ψ_s both above and below the plate for 6b and 6c. More specifically, it appears that in 6b the reflected wave from the upper surface is nominal. Conversely in 7b the reflected wave below the plate seems to have been either cancelled due to destructive interference or otherwise, while the reflected wave above seems to be more significant. We believe this is due to the interact dependence of the parameter W on our model parameters. This phenomena may have interesting consequences when choosing a material with a specific impedance $Z$ to reduce sound.

The best performing boundary is the perforated fibre with impedance $Z_{F}$ ⁠, particularly in the second quadrant. In 6c, we see there are regions where the wave field is reduced, potentially from the destructive interference of reflected or diffracted components. This is most prevalent at the higher Mach number M₂ for $Z_{S}$ ⁠. A final note for these plots is that we find some impedance values break the symmetry about the negative x axis found in cases 6a and 7a.

At the higher frequency, f₂, we notice similar features and trends regarding the symmetry about $x < 0, y = 0$ and the effect of each impedance on the reflected waves above and below. There seem to be larger reduction regions, particularly for 8b and 8c. It can be observed that the angles where destructive interference occurs remain the same.

We follow this by testing three more impedance values that are within a similar range to

Z_{F}

and

Z_{S}

⁠, however for brevity, this will only be for one frequency (f₁), but for both cases:⁵

Z_{1} = \frac{1}{2} i, Z_{2} = \frac{1}{2} - i, Z_{3} = 1 + \frac{1}{2} i .

The varied natures of impedance values $Z_{1}, Z_{2}, Z_{3}$ demonstrate interesting and varied effects on the scattered field ψ_s. For every case, the upper surface of the plate has absorbed some energy and demonstrates a weakened reflected wave. The only significant increase tends to lie in the first quadrant at the wake. This can be attributed to the choice of the wake constant C_K varying with our parameters (particularly $Z$ ⁠) due to implementing the Kutta condition. One final interesting comparison is that by changing the sign of the imaginary part of $Z$ from positive in 6c to negative in 8c we notice there is less prominent reduction in the second quadrant but more generally less impact upon the reflected field below the plate. Figure 11 test $Z_{1, 2, 3}$ at f₁, M₂ and θ₂. For these parameters, only $Z_{3}$ retains the symmetry in the third and fourth quadrants, unlike Fig. 10, suggesting a non-trivial dependence for this feature on $Z$ ⁠. We see that $Z_{1}$ has significant reflected waves above the plate, suggesting that at higher Mach numbers, a purely imaginary impedance may not perform as well as others, perhaps due to convective effects. A feature repeated from Fig. 10 is the varying strengths in the jump across the wake. A final note is that we have chosen constant values for each impedance Z. In reality, Z is actually a function, $Z (ω)$ ⁠. This can be straightforwardly incorporated into the model, which may prove very useful for some applications of the effects of metamaterials in aeroacoustics and beyond.

Fig. 9

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Fig. 10

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Fig. 11

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5 Conclusions

In this article, we presented a generalised framework for solving scattering problems with a semi-infinite boundary: our boundary is a semi-infinite plate on which two arbitrary linear boundary conditions are applied on the upper and lower sides of the plate, while we may consider the problem with any incident wave field by accounting for forcing terms in our boundary conditions on the negative x-axis. At the same time, we introduce a jump condition on the positive x-axis to represent the possible presence of a wake.

The primary goal was to extend the approach in papers such as [19–21] to solve similar scattering problems to apply them to popular industrial problems in aeroacoustics, such as leading- and trailing-edge noise. Although [12] provides an outline for solving the matrix factorisation that arises in similar problems, we have instead focused on the generalised scattering problem for which the matrix factorisation is an important component but not the sole issue that requires attention.

We reduced the matrix factorisation problem to Hilbert problems by considering jumps across square root branch cuts. The extension from a simple first-order boundary condition to a general linear boundary condition follows through manageable extensions to the functions V and W, which must be carefully constructed by paying attention to scalar kernel functions, effectively reducing the matrix problem to a scalar one. The two functions V and W that arise from the governing Hilbert equation are solved by investigating a second-order scalar kernel that can be dealt with numerically at little extra cost than a simpler impedance boundary condition (such as the standard convective Robin boundary condition). During this generalised process, attention was focused on important analytical features of the Wiener–Hopf–Hilbert method that must be carefully accounted for. These include the possible zeros of the scalar kernel that can lead to the presence of surface waves in the solution and singularities in the kernel, the importance of choosing the correct edge condition, the use of the generalised Liouville theorem, and the implementation of a Kutta condition.

Finally, an industrially focused example is presented and solved. A coated rigid plate is modelled with a Neumann boundary condition below and the Ingard–Myers boundary condition. Previous examples in the literature consider a first-order boundary condition, and the impedance parameter varies above and below the plate. The benefit of our method is that we can easily adapt the approach from a simpler mathematical boundary condition to a more complicated yet physically realistic boundary condition. Impedance values from [20] that represent realistic absorbing sheets of different materials were tested for this model. Results demonstrated that both the real and imaginary parts of the impedance $Z$ can reduce the scattered near-field in both magnitude and direction.

Footnotes

Our boundary conditions have no source term for the jump in $\frac{\partial φ_{s}}{\partial y}$ ⁠, however, the analysis would allow for a suitable term here that would give rise to an upper analytic $F_{2} (α)$ function.

The result is independent of which x-axis we integrate along

It can be shown that if there is a discontinuity in ${({\underline{\underline{χ}}}^{-})}^{(- 1)}$ across a branch cut then it will have to hold for ${\underline{\underline{χ}}}^{-}$ too. Furthermore, by construction, $D^{-} (α)$ is continuous in the region D_L that includes the branch cut

This is assuming all y derivatives in the boundary condition are derivatives concerning the normal pointing out of the plate

Each of these impedance values lie within the range of sensible values for the impedance of these steel sheet in [20]

Acknowledgements

A. D. G. Hales would like to thank I. D. Abrahams for his guidance with the formulation of the article and the excellent feedback from anonymous reviewers. A. D. G. Hales acknowledges support from EPSRC studentship EP/T517847/1, L. J. Ayton acknowledges support from EPSRC Early Career Fellowship EP/P015980/1.

References

[1]

Hopf

Über Eine Klasse Singulärer Integralgleichungen; Bemerkungen zur Methode von Hardy und Titchmarsh

J. Lond. Math. Soc.

s1-4

(

1929

)

–

Google Scholar

Crossref

WorldCat

[2]

Kisil

A. V.

Abrahams

I. D.

Mishuris

Rogosin

S. V.

The Wiener–Hopf technique, its generalizations and applications: constructive and approximate methods

Proc. R. Soc. A

477

(

2021

)

20210533

Google Scholar

Crossref

WorldCat

[3]

Buck

Oerlemans

Palo

Experimental validation of a wind turbine turbulent inflow noise prediction code

AIAA J.

(

2018

)

1495

–

1506

Google Scholar

Crossref

WorldCat

[4]

Daniele

On the factorization of Wiener-Hopf matrices in problems solvable with Hurd’s method

IEEE Trans. Antennas Propag

(

1978

)

614

–

616

Google Scholar

Crossref

WorldCat

[5]

Khrapkov

A. A.

Certain cases of the elastic equilibrium of an infinite wedge with a nonsymmetric notch at the vertex, subjected to concentrated forces

J. Appl. Math. Mech

(

1971

)

625

–

637

Google Scholar

Crossref

WorldCat

[6]

Veitch

B. H.

Abrahams

I. D.

On the commutative factorization of n×n matrix Wiener–Hopf kernels with distinct eigenvalues

Proc. R. Soc. A

463

(

2007

)

613

–

639

Google Scholar

Crossref

WorldCat

[7]

Daniele

V. G.

On the solution of two coupled Wiener–Hopf equations

SIAM J. Appl. Math.

(

1984

)

667

–

680

Google Scholar

Crossref

WorldCat

[8]

Jones

D. S.

Commutative Wiener–Hopf factorization of a matrix

Proc. R. Soc. Lond. Ser. A

393

(

1984

)

185

–

192

Google Scholar

OpenURL Placeholder Text

WorldCat

[9]

Jones

D. S.

Factorization of a Wiener—Hopf matrix

IMA J. Appl. Math

(

1984

)

211

–

220

Google Scholar

Crossref

WorldCat

[10]

Moiseev

N. G.

On factorization of matrix-valued functions of special form

Sov. Maths Dokl

(

1989

)

264

–

267

Google Scholar

OpenURL Placeholder Text

WorldCat

[11]

Moiseev

N. G.

Partial factorization indices of a special class of matrix-valued functions

Dokl. Akad. Nauk

(

1994

)

640

–

644

Google Scholar

OpenURL Placeholder Text

WorldCat

[12]

Rawlins

A. D.

Williams

W. E.

Matrix Wiener–Hopf factorisation

Q. J. Mech. Appl. Math

(

1981

)

–

Google Scholar

Crossref

WorldCat

[13]

Williams

W. E.

Recognition of some readily “Wiener—Hopf” factorizable matrices

IMA J. Appl. Math

(

1984

)

367

–

378

Google Scholar

Crossref

WorldCat

[14]

Hurd

R. A.

The Wiener–Hopf–Hilbert method for diffraction problems

Can. J. Phys.

(

1976

)

775

–

780

Google Scholar

Crossref

WorldCat

[15]

Rawlins

A. D.

The explicit Wiener-Hopf factorisation of a special matrix

ZAMM—J. Appl. Math. Mech

(

1981

)

527

–

528

Google Scholar

Crossref

WorldCat

[16]

Ahmad

An improved model for noise barriers in a moving fluid

J. Math. Anal. Appl

321

(

2006

)

609

–

620

Google Scholar

Crossref

WorldCat

[17]

Teixeira

F. S.

dos Santos

A. F.

Lebre

A. B.

The diffraction problem for a half-plane with different face impedances revisited

J. Math. Anal. Appl

140

(

1989

)

485

–

509

Google Scholar

OpenURL Placeholder Text

WorldCat

[18]

Barton

P. G.

Rawlins

Acoustic diffraction by a semi-infinite plane with different face impedances

Q. J. Mech. Appl. Math

(

1999

)

469

–

487

Google Scholar

Crossref

WorldCat

[19]

Barton

P. G.

Rawlins

A. D.

Diffraction by a half-plane in a moving fluid

Q. J. Mech. Appl. Math

(

2005

)

459

–

479

Google Scholar

Crossref

WorldCat

[20]

Rawlins

A. D.

30.—Acoustic diffraction by an absorbing semi-infinite half plane in a moving fluid

Proc. R. Soc. Edinburgh A

(

1975

)

337

–

357

Google Scholar

Crossref

WorldCat

[21]

Hurd

R. A.

Przeździecki

Diffraction by a half-plane with different face impedances—a re-examination

Can. J. Phys.

(

1981

)

1337

–

1347

Google Scholar

Crossref

WorldCat

[22]

Rienstra

S. W.

Hydrodynamic instabilities and surface waves in a flow over an impedance wall, Aero- and Hydro-Acoustics

(ed.

Comte-Bellot

Ffowcs-Williams

J. E.

;

Springer

Berlin Heidelberg

1986

)

483

–

490

Google Scholar

Google Preview

OpenURL Placeholder Text

WorldCat

[23]

Abrahams

I. D.

Lawrie

J. B.

On the factorization of a class of Wiener–Hopf kernels

IMA J. Appl. Math

(

1995

)

–

Google Scholar

Crossref

WorldCat

[24]

Zhou

Zhong

Zhang

On the effect of velvet structures on trailing edge noise: experimental investigation and theoretical analysis

J. Fluid Mech.

919

(

2021

)

A11

Google Scholar

Crossref

WorldCat

[25]

Antipov

Y. A.

Exact solution to the problem of acoustic wave propagation in a semi-infinite waveguide with flexible walls

Wave Motion

(

2019

Google Scholar

OpenURL Placeholder Text

WorldCat

[26]

Simões

A wave diffraction problem with higher order impedance boundary conditions

Int. J. Math. Models Methods Appl. Sci

(

2022

)

–

Google Scholar

OpenURL Placeholder Text

WorldCat

[27]

Brambley

, A well-posed modified Myers boundary condition, 16th AIAA/CEAS Aeroacoustics Conference (

2010

[28]

Ingard

Influence of fluid motion past a plane boundary on sound reflection, absorption, and transmission

J. Acoust. Soc. Am

(

1959

)

1035

–

1036

Google Scholar

Crossref

WorldCat

[29]

Ul-Hassan

Rawlins

A. D.

Two problems of waveguides carrying mean fluid flow

J. Sound Vibr

216

(

1998

)

713

–

738

Google Scholar

Crossref

WorldCat

[30]

Rawlins

A. D.

7.—Acoustic diffraction by an absorbing semi-infinite plane in a moving fluid, II

Proc. R. Soc. Edinburgh A

216

(

1976

)

–

Google Scholar

OpenURL Placeholder Text

WorldCat

[31]

Kisil

, Approximate Wiener–Hopf factorisation with stability analysis. Ph.D. Thesis, University of Cambridge (

2015

[32]

Noble

Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations

(

Chelsea Publishing, second (unaltered)

. edition

1988

Google Scholar

Google Preview

OpenURL Placeholder Text

WorldCat

[33]

Cannell

P. A.

Edge scattering of aerodynamic sound by a lightly loaded elastic half-plane

Proc. R. Soc. Lond. A

347

(

1975

)

213

–

238

Google Scholar

OpenURL Placeholder Text

WorldCat

[34]

Jaworski

J. W.

Peake

Aerodynamic noise from a poroelastic edge with implications for the silent flight of owls

J. Fluid Mech.

723

(

2013

)

456

–

479

Google Scholar

Crossref

WorldCat

[35]

Scott

J. F. M.

Acoustic scattering by a finite elastic strip

Philos. Trans. R. Soc. Lond. A

338

(

1992

)

145

–

167

Google Scholar

OpenURL Placeholder Text

WorldCat

[36]

Crighton

D. G.

The Kutta condition in unsteady flow

Annu. Rev. Fluid Mech.

(

1985

)

411

–

445

Google Scholar

Crossref

WorldCat

[37]

Ayton

L. J.

Gill

J. R.

Peake

The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction

J. Sound Vibr

378

(

2016

)

–

Google Scholar

Crossref

WorldCat

[38]

Hales

A. D. G

Ayton

L. J.

, Adapting a trailing-edge noise model to an impedance boundary condition. AIAA (

2023

[39]

Rienstra

S. W

Hirschberg

, An Introduction to Acoustics: IWDE Report 92-06. Instituut Wiskundige Dienstverlening (

1992

[40]

Babich

V. M.

Lyalinov

M. A.

Grikurov

V. E.

, Diffraction theory: the Sommerfeld-Malyuzhinets technique.

Alpha Science series on wave phenomena

(

Alpha Science International Ltd

Oxford

2008

Google Scholar

Google Preview

OpenURL Placeholder Text

WorldCat

[41]

Rienstra

S. W.

Darau

Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall

J. Fluid Mech.

671

(

2011

)

559

–

573

Google Scholar

Crossref

WorldCat

[42]

Tsai

C. T.

, Effect of airfoil thickness on high-frequency gust interaction noise. Ph.D. Thesis, University of Arizona (

1992

[43]

Llewellyn Smith

S. G.

Luca

Numerical solution of scattering problems using a Riemann–Hilbert formulation

Proc. R. Soc. A

475

(

2019

)

20190105

Google Scholar

Crossref

WorldCat

Appendix A: Finding the zeros of a scalar kernel

It can be shown that polynomials in α and γ are uniquely determined by their zeros. This is important for factorising scalar kernels [23]. If we introduce the change of variables

α = - k sin (t), γ = i k cos (t), t \in [- 3 π / 2, - π / 2] \times i R

⁠, then a kernel of the form

S (α) = \sum_{m = 0}^{M} a_{m} {(- i α)}^{m} - γ \sum_{m = 0}^{M} b_{m} {(- i α)}^{m},

(A.1)

can be re-written into the form

S (t) = \sum_{m = 0}^{M} a_{m} {(i k sin (t))}^{m} - i k cos (t) \sum_{m = 0}^{M} b_{m} {(i k sin (t))}^{m} .

(A.2)

One may then define

T = e^{i t}

and re-write this trigonometric polynomial as

S (T) = \sum_{m = 0}^{M} a_{m} {((T - T^{- 1}) / 2)}^{m} - i k (T + T^{- 1}) / 2 \sum_{m = 0}^{M} b_{m} {((T - T^{- 1}) / 2)}^{m} .

(A.3)

This now belongs to the polynomial ring

C [T, T^{- 1}]

which is a unique factorisation domain and therefore can be uniquely written as a polynomial of the form

S (T) = \frac{C}{T^{N_{1}}} \prod_{m = 0}^{N_{2}} (T - T_{m}),

(A.4)

for some positive integers

N_{1}, N_{2}

⁠. Each T_m is a zero of the original kernel and can be written in α-space as

α_{m} = - i log T_{m}

(A.5)

Appendix B: Additional analysis for the semi-rigid wave scattering example

I: Finding the entire function $\underline{E}$

To find

\underline{E} (α),

we must investigate the large α behaviour of every component of the Wiener–Hopf equation

\begin{matrix} \underset{Lower Analytic}{\underset{︸}{{\underline{\underline{K}}}^{-} \underline{L} + {({\underline{\underline{K}}}^{-} \underline{F})}_{-} - 2 {({({\underline{\underline{K}}}^{+})}^{- 1} \underline{G})}_{-}}} \\ = \underset{Upper Analytic}{\underset{︸}{2 {({\underline{\underline{K}}}^{+})}^{- 1} \underline{U} + 2 {({({\underline{\underline{K}}}^{+})}^{- 1} \underline{G})}_{+} - {({\underline{\underline{K}}}^{-} \underline{F})}_{+}}} \end{matrix}

(B.1)

We first rewrite this equation to account for each additive factorisation

\begin{matrix} {\underline{\underline{K}}}^{-} (α) \underline{L} (α) + ({\underline{\underline{K}}}^{-} (α) - {\underline{\underline{K}}}^{-} (K_{M})) \underline{F} (α) \\ - 2 {({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1} \underline{G} (α) \\ = 2 {({\underline{\underline{K}}}^{+} (α))}^{- 1} \underline{U} (α) - {\underline{\underline{K}}}^{-} (K_{M}) \underline{F} (α) \\ + 2 ({({\underline{\underline{K}}}^{+} (α))}^{- 1} - {({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1}) \underline{G} (α) . \end{matrix}

(B.2)

We note that $\underline{F} (α) \sim α^{- 1}$ and $\underline{G} (α) \sim α^{- 1}$ as $α \to \infty,$ allowing us to ignore two terms from each side of the equation when finding $\underline{E}$ ⁠.

It can be shown that

S_{1}^{+} \sim α,

S_{1}^{-} \sim α

and

W \sim α,

so that

\begin{matrix} {({\underline{\underline{K}}}^{+})}^{- 1} (α) & = (\begin{matrix} O (α^{- \frac{3}{4}}) & O (α^{\frac{1}{4}}) \\ O (α^{- \frac{5}{4}}) & O (α^{- \frac{1}{4}}) \end{matrix}), \\ {\underline{\underline{K}}}^{-} (α) & \sim (\begin{matrix} O (α^{\frac{5}{4}}) & O (α^{\frac{1}{4}}) \\ O (α^{\frac{3}{4}}) & O (α^{- \frac{1}{4}}) \end{matrix}) ​ . \end{matrix}

(B.3)

Omitting the four matrices in (B.2) that scale like

O (α^{- 1})

⁠, we use Landau notation to represent the asymptotics of matrix entries when performing the necessary multiplication:

\begin{matrix} (\begin{matrix} O (α^{\frac{5}{4}}) & O (α^{\frac{1}{4}}) \\ O (α^{\frac{3}{4}}) & O (α^{- \frac{1}{4}}) \end{matrix}) (\begin{matrix} O (α^{- \frac{5}{4}}) \\ O (α^{- \frac{1}{4}}) \end{matrix}) + (\begin{matrix} O (α^{\frac{5}{4}}) & O (α^{\frac{1}{4}}) \\ O (α^{\frac{3}{4}}) & O (α^{- \frac{1}{4}}) \end{matrix}) (\begin{matrix} O (α^{- 1}) \\ O (α^{- 1}) \end{matrix}) \\ = (\begin{matrix} O (α^{- \frac{3}{4}}) & O (α^{\frac{1}{4}}) \\ O (α^{- \frac{5}{4}}) & O (α^{- \frac{1}{4}}) \end{matrix}) (\begin{matrix} O (α^{\frac{3}{4}}) \\ O (α^{- \frac{1}{4}}) \end{matrix}) + (\begin{matrix} O (α^{- \frac{3}{4}}) & O (α^{\frac{1}{4}}) \\ O (α^{- \frac{5}{4}}) & O (α^{- \frac{1}{4}}) \end{matrix}) (\begin{matrix} O (α^{- 1}) \\ O (α^{- 1}) \end{matrix}) \\ = \underline{E} (α) . \end{matrix}

(B.4)

After simplifying, we see that

\underset{LHS}{\underset{︸}{(\begin{matrix} O (1) \\ O (α^{- 1 / 2}) \end{matrix}) + (\begin{matrix} O (α^{\frac{1}{4}}) \\ O (α^{- \frac{1}{4}}) \end{matrix})}} = \underset{RHS}{\underset{︸}{(\begin{matrix} O (1) \\ O (α^{- 1 / 2}) \end{matrix}) + (\begin{matrix} O (α^{- \frac{3}{4}}) \\ O (α^{- \frac{5}{4}}) \end{matrix})}} = \underset{Entire}{\underset{︸}{(\begin{matrix} E_{1} (α) \\ E_{2} (α) \end{matrix})}} .

(B.5)

Applying Liouville’s theorem, we see that our entire functions must be of the form

\underline{E} (α) = (\begin{matrix} E \\ 0 \end{matrix})

(B.6)

For some $E \in C$ ⁠.

II: Application of the Kutta condition

We will choose these constants to ensure a finite jump in velocity and pressure at the tip of the trailing edge occurs. That is the terms for which

u_{1} \sim 1

and

u_{2} \sim α^{- 1}

must be set to zero via these free constants. Since

\underline{U} = \frac{1}{2} {\underline{\underline{K}}}^{+} (α) \underline{E} - (\underline{\underline{I}} - {\underline{\underline{K}}}^{+} (α) {({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1}) \underline{G} + \frac{1}{2} {\underline{\underline{K}}}^{+} (α) {\underline{\underline{K}}}^{-} (K_{M}) \underline{F},

(B.7)

\begin{matrix} (\begin{matrix} u_{1} (α) \\ u_{2} (α) \end{matrix}) & \sim (\begin{matrix} O (α^{\frac{3}{4}}) & O (α^{\frac{5}{4}}) \\ O (α^{- \frac{1}{4}}) & O (α^{\frac{1}{4}}) \end{matrix}) [(\begin{matrix} \frac{E}{2} \\ 0 \end{matrix}) + {({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1} (\begin{matrix} G_{1} α^{- 1} \\ G_{2} α^{- 1} \end{matrix}) + {\underline{\underline{K}}}^{-} (K_{M}) (\begin{matrix} \frac{C_{K} i}{2} α^{- 1} \\ 0 \end{matrix})] \\ + (\begin{matrix} G_{1} α^{- 1} \\ G_{2} α^{- 1} \end{matrix}) . \end{matrix}

(B.8)

The Kutta condition states that we must have finite pressure at the tip. Since pressure can be expressed in terms of

Ψ_{s}

via the convective derivative:

p_{s} = - ρ_{0} (U \frac{\partial Ψ_{s}}{\partial x} + i ω Ψ_{s}),

(B.9)

considering the previously calculated expansion eliminates the possibility of the

r^{\frac{1}{4}}

and

r^{\frac{3}{4}}

terms. In α-space, this is equivalent to any terms of

A (α)

that are

O (α^{- \frac{5}{4}})

or worse (equivalently, terms in u₁ that are

O (α^{\frac{3}{4}})

or worse). Since

u_{1} (α) = \frac{A (α)}{S_{1} (α)}

⁠, we have to conclude that

E = 0

⁠.

Similar to [19], we choose our constant C_K to eliminate the

α^{\frac{1}{4}}

term that is equivalent to the

r^{\frac{3}{4}}

term in the

Ψ_{s}

expansion. Using the matrix notation

{[\underline{\underline{M}}]}_{i, j}

to denote the

(i, j) -

th entry of

\underline{\underline{M}}

⁠. Expanding the matrix equation inside the brackets, we need to ensure that

{[{({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1}]}_{2, 1} G_{1} + {[{({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1}]}_{2, 2} G_{2} + {[{\underline{\underline{K}}}^{-} (K_{M})]}_{2, 1} \frac{i C_{K}}{2} = 0,

(B.10)

from which we set

C_{K} = 2 i \frac{G_{1} {[{({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1}]}_{2, 1} + G_{2} {[{({\underline{\underline{K}}}^{+} (- K_{0}))}^{- 1}]}_{2, 2}}{{[{\underline{\underline{K}}}^{-} (K_{M})]}_{2, 1}} .

(B.11)

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

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Article Contents

Analytical insights into semi-infinite plate scattering: the wiener–hopf technique and two-sided linear boundary conditions

Summary

1 Introduction

2 Constructing Wiener–Hopf equations for scattering problems in flow

2.1 Governing equations and initial assumptions

2.2 Constructing the Wiener–Hopf equation

Notation and definitions

2.3 Additional analytical requirements

2.3.1 Choosing edge conditions

Example: A ‘semi-rigid’ Ingard–Myers boundary condition

2.3.2 Regions of analyticity

2.3.3 Acknowledging surface waves

3 Solving the Wiener–Hopf equation

3.1 Application of the Wiener–Hopf–Hilbert method

3.2 Finding the particular solution for $u_{1, 2}$

3.3 Solving for $φ_{s}$

4 Example: scattering a sound wave off a two-sided edge in flow

4.1 Governing equations

4.2 Solving the Wiener–Hopf equation

4.3 Results

5 Conclusions

Footnotes

Acknowledgements

References

Appendix A: Finding the zeros of a scalar kernel

Appendix B: Additional analysis for the semi-rigid wave scattering example

I: Finding the entire function $\underline{E}$

II: Application of the Kutta condition

Citations

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Article Contents

Analytical insights into semi-infinite plate scattering: the wiener–hopf technique and two-sided linear boundary conditions

Summary

1 Introduction

2 Constructing Wiener–Hopf equations for scattering problems in flow

2.1 Governing equations and initial assumptions

2.2 Constructing the Wiener–Hopf equation

Notation and definitions

2.3 Additional analytical requirements

2.3.1 Choosing edge conditions

Example: A ‘semi-rigid’ Ingard–Myers boundary condition

2.3.2 Regions of analyticity

2.3.3 Acknowledging surface waves

3 Solving the Wiener–Hopf equation

3.1 Application of the Wiener–Hopf–Hilbert method

3.2 Finding the particular solution for u1,2

3.3 Solving for φs

4 Example: scattering a sound wave off a two-sided edge in flow

4.1 Governing equations

4.2 Solving the Wiener–Hopf equation

4.3 Results

5 Conclusions

Footnotes

Acknowledgements

References

Appendix A: Finding the zeros of a scalar kernel

Appendix B: Additional analysis for the semi-rigid wave scattering example

I: Finding the entire function E_

II: Application of the Kutta condition

Citations

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3.2 Finding the particular solution for $u_{1, 2}$

3.3 Solving for $φ_{s}$

I: Finding the entire function $\underline{E}$