Plate arrays as a perfectly-transmitting negative-refraction metamaterial

doi:10.1016/j.wavemoti.2020.102673

Wave Motion

Volume 100, January 2021, 102673

https://doi.org/10.1016/j.wavemoti.2020.102673 Get rights and content

Highlights

•
A novel metamaterial is analysed using homogenisation and Bloch–Floquet methods.
•
The homogenisation method is a good approximation for plane waves and sources.
•
The plate array can act as a perfectly-transmitting negative refraction matematerial.

Abstract

A closely-spaced periodic array of identical thin rigid plates illuminated by incident waves is shown to exhibit properties of a negative refraction metamaterial under certain conditions. The close-spacing assumption is used as a basis for an approximation in which the region occupied by the plate array acts as an effective medium. Effective matching conditions on the plate array boundary are also derived. The approximation allows explicit expressions to be derived to wave scattering problems involving tilted plate arrays. This approximation is tested for its accuracy against an exact treatment of the problem based on Bloch–Floquet theory.

Both the exact and effective medium theory predict perfect wave transmission at all wave frequencies through the array when the tilt angle of plates in the array is the reverse of the incident wave direction: the array acts as an all-frequency perfectly-transmitting negative-refraction medium. For certain frequencies the array is also shown to act as an all-angle perfectly-transmitting negative-refraction material.

Introduction

In this paper we consider the two-dimensional problem of the scattering of waves by a staggered infinite periodic array of thin parallel plates. The problem has a long history in acoustics with application to blade rows in turbomachinery. More recent papers in this application area often relate to circular duct geometries and include effects such as swirling flow in addition to basic mean flow; for example [1]. In earlier papers the simpler two-dimensional linear blade row was considered by [2], [3] and [4]. These three papers all consider periodic arrays of thin plates with stagger in the presence of mean flow (equations for the pressure field include a Mach number, $M$ ). The work of [3] and [4] employ the Wiener–Hopf method to derive solutions, extending earlier work of [5], [6] and [7] for arrays without stagger. They sought to improve upon the simple approximate method of [2] who matched solutions outside the array with a continuum model for the field inside the array based on infinitesimal separation between adjacent blades.

In [2] it was highlighted that zeros of transmission at frequencies dependent on $M$ could be found at incident angles opposed to the inclination of elements in array. In the three papers cited above involving stagger much of the focus of results concerns the influence of mean flow as this is pertinent to the application area. However, in [2] it is also pointed out that without mean flow the transmitted and reflected wave amplitudes are symmetric with respect of the incident wave heading, regardless of stagger. This is certainly not an intuitive result. In this paper we confirm and extend this result with a different application area in mind.

The last 20 years has seen a vast expansion of interest in the science of so-called metamaterials. These are manufactured materials having properties not usually found in nature. A metamaterial typically possesses a microstructure whose effect upon the macroscopic field variables allows it to exhibit complex and counter-intuitive phenomena. The design and realisation of metamaterials have provided researchers with a range of new problems that can be considered. One of the key demands of metamaterials in wave engineering problems such as invisibility cloaking or perfect lensing is the ability to be able to redirect the path of waves without reflection and with a negative refractive index. For example, see [8], [9], [10] and [11] for examples related closely to the current work.

In this paper we bring together the ideas of closely-spaced plate arrays and metamaterials to illustrate two principal effects: (i) all-frequency perfectly-transmitting negative refraction based wave shifting; and (ii) all-angle perfectly-transmitting negative refraction wave shifting. We note that these features depend upon precise conditions being met and that plate arrays do not form a negative refractive index metamaterial in the sense presented in [12]. We shall also illustrate trapping of waves by long finite-width staggered plate arrays.

Two approaches are taken to the solving the staggered plate array problem. After defining the problem in Section 2 we focus in Section 3 on formulating an approximation to the solution based on a close-spacing assumption. This is, in essence, the approach adopted by [2]. The effect of the microstructure is captured by reducing (through a formal asymptotic procedure) the wave equation to allow wave motion only in directions aligned with plates in the array. This process might be referred to as homogenisation/continuum modelling or effective field theory depending on the context. With the addition of effective boundary conditions between the plate array and the exterior domain we derive explicit solutions to a variety of problems illustrating various interesting wave effects indicated in the paragraph above as well as considering guided waves trapped within the plate array and their excitation by point sources in the neighbourhood of the plate array.

To confirm the validity of the approximation of Section 3, the problem of plane wave scattering, without approximation, by an infinite periodic array of thin staggered plates is considered in Section 4. Standard arguments, based on periodicity, are adopted to reduce the boundary-value problem to one that lies within a fundamental periodic strip of the domain and the application of Fourier transforms lead to integral equations which can be solved accurately and efficiently using well-established approximation methods. An analysis of the structure of this solution shows that all of the key symmetry and transparency relations present in the approximation are shared by this exact treatment of the problem. This approach appears to be much simpler than the ones used by [3] and [4].

Results are produced throughout the paper to demonstrate various aspects of the theory and the paper is concluded in Section 5.

Section snippets

Description of the problem

A periodic array is comprised of thin plates each of length $2 L$ , separated from their neighbours by a perpendicular distance $d$ , with centres lying along $y = 0$ and tilted at an angle $δ$ to the positive $y$ -axis (see Fig. 1). The array occupies the strip $- b < y < b$ , $- \infty < x < \infty$ such that $b = L cos δ$ and along the edges $y = \pm b$ the edges of adjacent plates are separated by a distance $l = d ∕ cos δ$ . The overlap (or stagger) between two adjacent plates in the array is $a = d tan δ$ .

The periodic array of plates is embedded in an

Approximation for closely-spaced plates

An approximation is developed under the assumption that the spacing between the plates is small in relation to the plate length: $ϵ = d ∕ L ≪ 1$ . It is also assumed that $k L = O (1)$ which implies that $k d = O (ϵ) ≪ 1$ . This implies the wavelength is much larger than the perpendicular distance between adjacent plates.

We start by considering the solution in $| y | < b$ in the domain occupied by the plate array. Every point in this domain can be written as $X = X_{n} + d X^{'}$ and $Y = X_{n} tan δ + L Y^{'}$ where $X^{'} \in (0, 1)$ , $- 1 + ϵ tan δ < Y^{'} < 1$ and $- \infty < n < \infty$

Exact treatment of the scattering problem

In this section we set about solving the problem of plane wave scattering by a periodic array of tilted plates without making a close-spacing approximation.

On account of the periodicity of the geometry and the form of the incident wave it must be that $ϕ (x + l, y) = e^{i α_{0} l} ϕ (x, y) .$ The idea is to exploit the periodicity and to solve the problem in a single fundamental strip, aligned with the array, allowing the solution outside this strip to be determined by (4.1). Thus, we are required to consider the

Conclusion

Scattering of waves by an infinite periodic array of inclined (or staggered) thin parallel plates occupying a region of finite width and infinite length has been considered. An approximation has been developed based on close spacing between plates, relative to their length, within the array allowing the region occupied by the plate array to be replaced by an effective medium and matching conditions on the boundary of the plate array replaced by effective matching conditions. This reduction in

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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View full text

Plate arrays as a perfectly-transmitting negative-refraction metamaterial

Highlights

Abstract

Introduction

Section snippets

Description of the problem

Approximation for closely-spaced plates

Exact treatment of the scattering problem

Conclusion

Declaration of Competing Interest

J. Sound Vib.

J. Sound Vib.

Appl. Ocean Res.

Upstream-radiated rotor-stator interaction noise in mean swirling flow

J. Fluid Mech.

Propagation of sound waves through a blade row

J. Sound Vib.

The reflection of an electromagnetic plane wave by an infinite set of plates, II

Quart. Appl. Math.

Reflection of an electromagnetic plane wave by an infinite set of plates, II

Quart. Appl. Math.

Reflection of an electromagnetic plane wave by an infinite set of plates, III

Quart. Appl. Math.

Zero-reflection acoustic metamaterial with a negative refractive index

Sci. Rep.