**2.1. The Scatterer**

We consider an infinitely long cylinder with uniform cross section. Without loss of generality we may assume that the axis of the cylinder is parallel to the *z*-axis. The cylinder is illuminated by an incident plane wave propagating with direction parallel to the *x*-*y* plane. We will assume that the cross-section *D* lying in the *x*-*y* plane has a closed boundary *∂D* that can be parameterised by

$$\mathbf{x}(t) = (\mathbf{x}\_1(t), \mathbf{x}\_2(t)), \qquad t \in \left[0, 2\pi\right]. \tag{1}$$

### **2.2. The incident and scattered fields**

[1] considered an approximate model relevant to the understanding of signal strength for phones in this environment. It studied the diffraction of an E-polarised wave by an absorbing rectangular cylinder, based upon Keller's method of GTD and its extensions to deal with multiple diffraction. It utilized the diffraction coefficient derived for the canonical problem of diffraction by an impedance corner to obtain relatively simple high frequency approximate expressions for the scattered far-field resulting from a plane wave obliquely incident on an

In order to validate the results of [1], Smith and Rawlins [2] undertook a numerical study of the scattering of an E-polarised plane wave by an infinite cylindrical structure in which an impedance boundary condition is enforced at all points on the cross-sectional boundary of the cylinder. It employed the integral equation formulation of Colton and Kress [3] for the unknown surface distribution comprising a single-layer potential and the adjoint of the double-layer potential. A Nyström method similar to that expounded by Colton and Kress [4] for the soft boundary condition was developed to obtain numerical solutions of this integral equation. The computed scattered far-fields were compared with the results of Rawlins [1] in order to validate his solutions over the range of impedances and wavenumbers examined. The study concluded that the approximations developed in [1] provide reasonably accurate patterns for rectangular structures for the range of wavenumbers and dimensions examined, but some divergences appear at smaller wavenumbers. There was a limitation to the study [2]: the method was applicable only to cylindrical cross-sections that are smooth (having a continuously varying normal vector at each point), and so the exactly rectangular structures investigated in [1] were treated by a replacing them by an appropriate "super-ellipse" that

In order to clarify the effect of corner rounding this paper examines the diffraction from cylindrical scatterers which possess corners, that is, points at which the normal changes discontinuously. Specifically we develop a numerical method for the scattering of an E-polarised plane wave by such cylindrical structures. The work in [5] is significantly extended. We examine three different boundary conditions: soft, hard and an impedance loaded boundary condition. In each case the boundary condition is enforced at all points on the cross-sectional boundary of the cylinder. We implement the Nyström method expounded by Colton and Kress [4] for the soft boundary condition to obtain numerical solutions of this integral equation. We then develop other Nyström methods similar to [4] for the hard and impedance boundary conditions to obtain numerical solutions of the respective integral

We use these numerical methods to examine the difference between a test structure with a corner and a rounded corner to assess the impact on near and far field scattering, as a function of the radius of curvature in the vicinity of the rounded corner point. We then extend the numerical methods developed thus far to examine a test structure with two corners. We conclude by examining the effect on the scattered field of rounding these corners

We consider an infinitely long cylinder with uniform cross section. Without loss of generality we may assume that the axis of the cylinder is parallel to the *z*-axis. The cylinder is

as a function of the radius of curvature in the vicinity of the rounded corner points.

imperfectly conducting rectangle.

2 Advanced Electromagnetic Waves

approximates the rectangle with rounded corners.

equations.

**2. Formulation**

**2.1. The Scatterer**

The incident field illuminating the scatterer induces a scattered field. We assume that the incident and scattered fields are time harmonic with a temporal factor *e*−*iω<sup>t</sup>* . The spatial component *uinc*(*x*, *y*) of the incident wave travelling in the direction of the unit vector d = (cos *θ*0, sin *θ*0) takes the form

$$u^{inc}(\mathbf{x}, \mathbf{y}) = e^{ik\mathbf{x} \cdot \mathbf{d}},\tag{2}$$

and satisfies the Helmholtz equation

$$
\Delta u^{\rm inc}(\mathbf{x}, y) + k^2 u^{\rm inc}(\mathbf{x}, y) = 0, \qquad (\mathbf{x}, y) \in \mathbb{R}^2. \tag{3}
$$

The spatial component *usc*(*x*, *y*) of the scattered field obeys the Helmholtz equation

$$
\Delta u^{\rm sc}(\mathbf{x}, y) + k^2 u^{\rm sc}(\mathbf{x}, y) = 0, \qquad (\mathbf{x}, y) \in \mathbb{R}^2,\tag{4}
$$

at all points (*x*, *y*) exterior to the body, where *k* = *ω*/*c* is the wavenumber and *c* the speed of light in free space; moreover it obeys the two-dimensional form of the Sommerfeld radiation condition [4]

$$\lim\_{|x|\to\infty} \sqrt{|x|} \left( \frac{\partial u^{\rm sc}(x)}{\partial x} - iku^{\rm sc}(x) \right) = 0, \qquad x \in \mathbb{R}^2 \backslash D. \tag{5}$$

### **2.3. The boundary conditions**

The nature of the scatterer imposes certain conditions that must be satisfied by the total field

$$
\mu^{\rm tot} = \mathfrak{u}^{\rm inc} + \mathfrak{u}^{\rm sc},\tag{6}
$$

on the boundary of the scatterer *∂D*.

This work considers sound soft scatterers, sound hard scatterers, and impedance loaded scatterers. All the scatterers induce a scattered acoustic potential. We define the boundary conditions for the different scatterers below.

### *2.3.1. Sound soft scatterers*

The total field *utot* vanishes on the boundary of a sound soft scatterer *∂D*. Thus

$$u^{tot}(x) = 0, \qquad x \in \partial D,\tag{7}$$

and from (6) we determine

$$u^{\rm sc}(x) = -u^{\rm inc}(x), \qquad x \in \partial D. \tag{8}$$

This sound soft boundary condition is a Dirichlet boundary condition.

### *2.3.2. Sound hard scatterers*

The normal derivative of the total field with respect to the unit outward normal n to *∂D*, vanishes on the boundary of a sound hard scatterer *∂D*. Thus

$$\frac{\partial u^{tot}}{\partial n}(x) = 0, \qquad x \in \partial D,\tag{9}$$

and from (6) we determine

$$\frac{\partial u^{\rm sc}}{\partial n}(x) = -\frac{\partial u^{\rm inc}}{\partial n}(x), \qquad x \in \partial D. \tag{10}$$

This sound hard boundary condition is a Neumann boundary condition.

### *2.3.3. Impedance loaded scatterers*

The impedance boundary value problem is prescribed by the boundary condition

$$\frac{\partial u^{tot}}{\partial n}(x) + ik\lambda u^{tot}(x) = 0, \qquad x \in \partial D,\tag{11}$$

where n(*x*) is the unit outward normal to the boundary at the point x and *λ* = *λ*(x) is a continuous function of position. From (6) we determine

$$\frac{\partial \mu^{sc}}{\partial n}(x) + ik\lambda \mu^{sc}(x) = -\frac{\partial \mu^{inc}}{\partial n}(x) - ik\lambda \mu^{inc}(x), \qquad x \in \partial D. \tag{12}$$

The scattered field is uniquely determined by the boundary and radiation conditions, provided Re(*λ*) is positive on the boundary *∂D*. In this work, *λ* will be restricted to be a (complex) constant.

### **2.4. Green's function**

*2.3.1. Sound soft scatterers*

4 Advanced Electromagnetic Waves

and from (6) we determine

*2.3.2. Sound hard scatterers*

and from (6) we determine

*2.3.3. Impedance loaded scatterers*

*∂usc*

(complex) constant.

The total field *utot* vanishes on the boundary of a sound soft scatterer *∂D*. Thus

This sound soft boundary condition is a Dirichlet boundary condition.

*∂utot*

*<sup>∂</sup>*<sup>n</sup> (x) = <sup>−</sup>*∂uinc*

The impedance boundary value problem is prescribed by the boundary condition

where n(*x*) is the unit outward normal to the boundary at the point x and *λ* = *λ*(x) is a

The scattered field is uniquely determined by the boundary and radiation conditions, provided Re(*λ*) is positive on the boundary *∂D*. In this work, *λ* will be restricted to be a

This sound hard boundary condition is a Neumann boundary condition.

vanishes on the boundary of a sound hard scatterer *∂D*. Thus

*∂usc*

*∂utot*

continuous function of position. From (6) we determine

*<sup>∂</sup>*<sup>n</sup> (x) + *ikλusc*(x) = <sup>−</sup>*∂uinc*

The normal derivative of the total field with respect to the unit outward normal n to *∂D*,

*<sup>u</sup>tot*(x) = 0, <sup>x</sup> <sup>∈</sup> *<sup>∂</sup>D*, (7)

*<sup>u</sup>sc*(x) = <sup>−</sup>*uinc*(x), <sup>x</sup> <sup>∈</sup> *<sup>∂</sup>D*. (8)

*<sup>∂</sup>*<sup>n</sup> (x) = 0, <sup>x</sup> <sup>∈</sup> *<sup>∂</sup>D*, (9)

*<sup>∂</sup>*<sup>n</sup> (x) + *ikλutot*(x) = 0, <sup>x</sup> <sup>∈</sup> *<sup>∂</sup>D*, (11)

*<sup>∂</sup>*<sup>n</sup> (x), <sup>x</sup> <sup>∈</sup> *<sup>∂</sup>D*. (10)

*<sup>∂</sup>*<sup>n</sup> (x) <sup>−</sup> *ikλuinc*(x), <sup>x</sup> <sup>∈</sup> *<sup>∂</sup>D*. (12)

As shown in [3], the problem of determining the scattered field may be solved by employing the single- and double-layer potentials associated with the two dimensional free-space Green's function

$$G(x,y) = \frac{i}{4}H\_0^{(1)}k(|x-y|),\tag{13}$$

where *H*(1) <sup>0</sup> denotes the Hankel function of first kind and order zero. The Green's function satisfies the Helmholtz equation

$$
\Delta\_x G(x, y) + k^2 G(x, y) = 0,\tag{14}
$$

everywhere except at x = y, and satisfies the Sommerfeld radiation condition (5).

For a fixed point y ∈ *∂D*, the normal derivative of the Green's function with respect to the outward unit normal at y is

$$\frac{\partial G(x,y)}{\partial n(y)} = \nabla\_y G(x,y) \cdot n(y). \tag{15}$$

It satisfies the Helmholtz equation (14) except at x = y, and satisfies the Sommerfeld radiation condition (5).

### **2.5. Integral operators**

We define two operators associated with the single- and double-layer potentials of a continuous density *φ*(y) defined on the boundary *∂D*,

$$\mathcal{G}(\mathcal{S}\phi)(x) = 2 \int\_{\partial D} G(x, y) \phi(y) ds(y),\tag{16}$$

$$\delta(\mathcal{K}\phi)(x) = 2 \int\_{\partial D} \frac{\partial G(x, y)}{\partial n(y)} \phi(y) ds(y);\tag{17}$$

their normal derivatives are, respectively

$$(\mathcal{K}'\phi)(x) = 2\int\_{\partial D} \frac{\partial G(x,y)}{\partial n(x)} \phi(y) ds(y),\tag{18}$$

$$\delta(\mathcal{T}\phi)(x) = 2\frac{\partial}{\partial n(x)} \int\_{\partial D} \frac{\partial G(x,y)}{\partial n(y)} \phi(y) ds(y). \tag{19}$$

The integral operators (16), (17), (18) and (19) are compact [3].

The acoustic single-layer potential *u* with integrable density *φ* is

$$
\mu \left( x \right) = \frac{1}{2} \mathcal{S} \phi(x),
\tag{20}
$$

and is continuous and bounded throughout **<sup>R</sup>**2\*∂<sup>D</sup>* and at all points on the boundary *<sup>∂</sup><sup>D</sup>* [4]. The double-layer potential *v* with integrable density *φ* is

$$v\left(x\right) = \frac{1}{2}\mathcal{K}\phi(x),\tag{21}$$

and is continuous and bounded throughout **<sup>R</sup>**2\*∂D*. It is discontinuous at all points on the boundary *<sup>∂</sup>D*, but can be continuously extended form *<sup>D</sup>* to *<sup>D</sup>*¯ and from **<sup>R</sup>**2\*∂D*¯ to **<sup>R</sup>**2\*∂<sup>D</sup>* with limiting values [4]

$$v\_{\pm}(x) = \int\_{\partial D} \frac{\partial G(x, y)}{\partial n(y)} \phi(y) ds(y) \pm \frac{\phi(x)}{2}, \qquad x \in \partial D,\tag{22}$$

where

$$v\_{\pm}(x) = \lim\_{h \to +0} v(x \pm h n(x)). \tag{23}$$

### **2.6. Integral representations**

The solution to the exterior Dirichlet problem for all <sup>x</sup> <sup>∈</sup> **<sup>R</sup>**2\*D*¯ , is based on representing the scattered field as a combination of the single (20) and double-layer (21) potentials

$$u^{\pounds}(x) = \int\_{\partial D} \left\{ \frac{\partial G(x, y)}{\partial n(y)} - i\eta G(x, y) \right\} \phi(y) ds(y), \qquad x \in \mathbb{R}^2 \backslash \bar{D}, \tag{24}$$

where *η* is a coupling parameter, provided the continuous density *φ*(x) is a solution to the following integral equation on *∂D*:

$$
\hbar \, I \phi + \mathcal{K} \phi - i \eta \mathcal{S} \phi = \mathcal{Q}\_{\prime} \tag{25}
$$

where *<sup>g</sup>* <sup>=</sup> <sup>−</sup>2*uinc*. This integral equation is uniquely solvable for all wave numbers satisfying Im *k* ≥ 0 [3].

What Effect does Rounding the Corners have on Diffraction from Structures with Corners? http://dx.doi.org/10.5772/61152 7

The single-layer potential (20)

$$u^{\rm sc}(x) = \int\_{\partial D} G(x, y)\phi(y)ds(y), \qquad x \in \mathbb{R}^2 \backslash \bar{D}, \tag{26}$$

is a solution to the exterior Neumann problem for all <sup>x</sup> <sup>∈</sup> **<sup>R</sup>**2\*D*¯ , provided that the continuous density *φ*(x) is a solution of the following integral equation on *∂D* [6]:

$$
\phi - \mathcal{K}' \phi = -2\hbar,\tag{27}
$$

where

The integral operators (16), (17), (18) and (19) are compact [3].

The acoustic single-layer potential *u* with integrable density *φ* is

The double-layer potential *v* with integrable density *φ* is

*v*±(x) =

*∂G*(x, y)

*v*±(x) = lim

scattered field as a combination of the single (20) and double-layer (21) potentials

*<sup>∂</sup>*n(y) <sup>−</sup> *<sup>i</sup>ηG*(x, <sup>y</sup>)

*∂D*

with limiting values [4]

6 Advanced Electromagnetic Waves

**2.6. Integral representations**

*usc*(x) =

following integral equation on *∂D*:

Im *k* ≥ 0 [3].

*∂G*(x, y)

*∂D*

where

*<sup>u</sup>* (x) <sup>=</sup> <sup>1</sup>

*<sup>v</sup>* (x) <sup>=</sup> <sup>1</sup>

2

and is continuous and bounded throughout **<sup>R</sup>**2\*∂<sup>D</sup>* and at all points on the boundary *<sup>∂</sup><sup>D</sup>* [4].

2

and is continuous and bounded throughout **<sup>R</sup>**2\*∂D*. It is discontinuous at all points on the boundary *<sup>∂</sup>D*, but can be continuously extended form *<sup>D</sup>* to *<sup>D</sup>*¯ and from **<sup>R</sup>**2\*∂D*¯ to **<sup>R</sup>**2\*∂<sup>D</sup>*

*<sup>∂</sup>*n(y) *<sup>φ</sup>*(y)*ds*(y) <sup>±</sup> *<sup>φ</sup>*(x)

*h*→+0

The solution to the exterior Dirichlet problem for all <sup>x</sup> <sup>∈</sup> **<sup>R</sup>**2\*D*¯ , is based on representing the

where *η* is a coupling parameter, provided the continuous density *φ*(x) is a solution to the

where *<sup>g</sup>* <sup>=</sup> <sup>−</sup>2*uinc*. This integral equation is uniquely solvable for all wave numbers satisfying

S*φ*(x), (20)

K*φ*(x), (21)

<sup>2</sup> , <sup>x</sup> <sup>∈</sup> <sup>∂</sup>*D*, (22)

*v*(x ± *h*n(x)). (23)

*<sup>φ</sup>*(y)*ds*(y), <sup>x</sup> <sup>∈</sup> **<sup>R</sup>**2\*D*¯ , (24)

*Iφ* + K*φ* − *iη*S*φ* = 2*g*, (25)

$$h(x) = -\frac{\partial u^{inc}}{\partial n}(x), \qquad x \in \partial D,\tag{28}$$

and *φ*(x) satisfies

$$\int\_{\partial D} \phi ds = 0.\tag{29}$$

Further, in **R**2, the exterior Neumann problem is uniquely solvable if and only if

$$\int\_{\partial D} h ds = 0,\tag{30}$$

is satisfied [6].

The solution to the exterior impedance problem for all <sup>x</sup> <sup>∈</sup> **<sup>R</sup>**2\*D*¯ , is

$$u^{\mathcal{K}}(x) = \int\_{\partial D} G(x, y)\phi(y)ds(y), \qquad x \in \mathbb{R}^2 \backslash \bar{D}, \tag{31}$$

provided *φ*(x) is a solution to

$$
\phi - \mathcal{K}'\phi - ik\lambda\mathcal{S}\phi = -2m,\tag{32}
$$

where

$$m(x) = -\frac{\partial u^{inc}}{\partial n}(x) - ik\lambda u^{inc}(x), \qquad x \in \partial D. \tag{33}$$

This solution is unique provided that *k* is not an interior Dirichlet eigenvalue [3]. Uniqueness is guaranteed by considering a suitable combination of single- and double-layer potentials, ie the combined potential

$$u^{\circledast \mathbf{c}}(x) = \int\_{\partial D} \left\{ \frac{\partial G(x, y)}{\partial n(y)} - i\eta G(x, y) \right\} \phi(y) ds(y), \qquad x \in \mathbb{R}^2 \backslash \bar{D}, \tag{34}$$

where *η* = 0 such that *η* Re *k* ≥ 0, solves the exterior impedance problem uniquely provided that the density *φ*(x) ∈ *∂D* is a solution of the integral equation [3]

$$\left(\left(I-\mathrm{i}\eta\lambda\right)\phi-\left(\mathcal{K}'+\mathrm{i}\eta\mathcal{T}+\mathrm{i}\eta\lambda\mathcal{K}+\lambda\mathcal{S}\right)\phi=-2m.\tag{35}$$
