#  Asymptotics of the meta-atom: plane wave scattering by a single Helmholtz
resonator

M. J. A. Smith  Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Wilberforce Road, CB3 0WA, UK P. A. Cotterill
Department of Mathematics, The University of Manchester, Oxford Road,
Manchester,
M13 9PL, UK  D. Nigro  Thales United Kingdom, 350 Longwater Avenue Green Park,
Reading, RG2 6GF, UK  W. J. Parnell  Department of Mathematics, The University
of Manchester, Oxford Road, Manchester,
M13 9PL, UK
I. D. Abrahams  Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Wilberforce Road, CB3 0WA, UK

###### Abstract

Using a combination of multipole methods and the method of matched
asymptotics, we present a solution procedure for acoustic plane wave
scattering by a single Helmholtz resonator in two dimensions. Closed-form
representations for the multipole scattering coefficients of the resonator are
derived, valid at low frequencies, with three fundamental configurations
examined in detail: the thin-walled, moderately thick-walled, and very thick-
walled limits. Additionally, we examine the impact of dissipation for very
thick-walled resonators, and also numerically evaluate the scattering,
absorption, and extinction cross sections (efficiencies) for representative
resonators in all three wall thickness regimes. In general, we observe strong
enhancement in both the scattered fields and cross sections at the Helmholtz
resonance frequencies. As expected, dissipation is shown to shift the
resonance frequency, reduce the amplitude of the field, and reduce the
extinction efficiency at the fundamental Helmholtz resonance. Finally, we
confirm results in the literature on Willis-like coupling effects for this
resonator design, and crucially, connect these findings to earlier works by
the authors on two-dimensional arrays of resonators, deducing that
depolarisability effects (off-diagonal terms) for a single resonator do not
ensure the existence of Willis coupling effects (bianisotropy) in bulk.

Submitted Manuscript

##  1 Introduction

Wave scattering is a process describing how an incident wave is modified by
the presence of an object, or ensemble of objects, in its path [1, 2]. Factors
such as the geometry of the scatterer or incident angle of the wave can induce
major changes in the observed response, and using rigorous mathematical
analysis, it is possible to provide insight on a diversity of observable
effects and real-world phenomena. For example, wave scattering can be used to
identify delaminations in composite media with ultrasound [3], and explains
how water droplets in the air can give rise to vibrant rainbows [1]. This
broad definition of scattering applies to waves of all types, from surface
waves on deep water through to electromagnetic, elastic and sound waves, and
in general, there are two canonical geometries whose scattering properties are
now generally well-understood: that of spheres and that of idealised
(infinitely tall) circular cylinders.

There has been extensive research attention on wave scattering by spheres
since at least the works of Mie [4], whose thorough outline of the separation
of variables solution for the vector Helmholtz equation continues to influence
scientific research well into the present day [1, 3]. This is due to the fact
that suspensions of spherical, or approximately spherical, objects arise in an
extensive range of applications, from the homogenisation of milk through to
air pollution modelling. Wave scattering by infinitely tall circular cylinders
has likewise received considerable attention, due to its extensive real-world
applications, from the design of fibrous materials such as kevlar through to
aerofoil design for aircraft [3]. It would appear that the first comprehensive
multipole solution for scattering by circular cylinders, in the setting of
Maxwell’s equations, was conducted by von Ignatowsky [5], although the first
scattering solution was outlined some decades earlier by Lord Rayleigh [6]. An
exhaustive early history of the scattering literature for spheres and
cylinders may be found in Kerker [2]. Interest in multipole solutions for
cylinders has grown significantly in recent decades with the advent of two-
dimensional photonic, phononic, and other crystals as well as with the advent
of metamaterials (discussed below) [7, 8]; however results are primarily
obtained using numerical methods, with few analytical treatments available for
non-circular cylindrical configurations.

In this work, we examine an important extension to scattering by an infinitely
tall cylinder: we consider two-dimensional acoustic plane-wave scattering by a
circular cylinder into which we have introduced a hollow circular cavity and
neck to form a Helmholtz resonator, where Neumann boundary conditions are
imposed on all walls. A representative resonator is presented in Figure 1 for
guidance. Using multipole methods and the method of matched asymptotics [9,
10] we construct a low-frequency, closed-form solution to describe the
scattering coefficients for the resonator and the resulting potential field in
the exterior domain. We are unaware of any other works in the literature which
present low-frequency analytical expressions for the scattering coefficients
of Helmholtz resonators in this manner. That said, scattering by a single
thin-walled elastic Helmholtz resonator shell in a fluid has been considered
previously [11] and their resonance condition (Eq. (C27)) in the rigid limit
is equivalent to ours in the thin-walled setting. Here, we consider three
resonator geometries explicitly: the thin-walled limit, moderately-thick
walled limit, and extremely thick-walled limits, providing closed-form
expressions and compact asymptotic forms in all instances.

A key objective of the present work is to obtain insight on wave propagation
through more complex arrangements of resonators, for example, when they are
tiled periodically to form finite clusters, gratings, or lattices [12, 13].
The scatterer considered here is a canonical geometry for two-dimensional
metamaterials, and following an established nomenclature in the literature
(i.e., Melnikov et al. [14]) our resonator is considered to be a “meta-atom”;
this terminology arises from the fact that a two-dimensional periodic lattice
of such resonators can give rise to a metamaterial, a type of composite
material that exhibits unusual and unexpected properties that are not readily
observed in conventional media [7, 8].

In addition to our multipole-matched asymptotic solution, we also compute
cross sections for a single cylindrical Helmholtz resonator. These are
measures of strength for different wave processes, for example, describing how
much incident power is scattered or absorbed by the resonator. We also
evaluate extinction cross sections, which refer to the power loss in the
downstream direction to the incident field (where a detector or observer would
be located), due to the presence of the object [1]. Formally, extinction is
defined as the sum of both absorption and scattering processes, and in
particular frequency regimes, or for certain geometries, either scattering or
absorption may dominate [1, Ch. 11]. In this work we consider cross sections
that are nondimensionalised by the geometric cross section of a closed
cylinder and we term the resulting quantities efficiency coefficients. Note
that the extinction cross section may be used to determine an estimate for the
absorbance (attenuance) of a large random ensemble of resonators; this is
obtained by multiplying the extinction cross section by the filling fraction
and the mean path length through the cluster (i.e., following the Beer-Lambert
law [15]). A substantial body of literature has been dedicated towards cross-
sections, particularly for spherical and cylindrical geometries, giving rise
to significant variations in definitions [16, 1, 17, 18, 19, 20]. In order to
avoid any possible confusion, and to facilitate comparison with the
literature, we present results for closed (ideal) Neumann cylinders where
possible.

To further complement the above, we examine the impact of dissipation by
considering a boundary layer in the neck region of a very thick-walled
resonator. A boundary layer refers to a wave-structure interaction effect
where fluid viscosity effects begin to emerge near the surface of an object,
ultimately dissipating wave energy. As discussed in the literature [21, 22],
there are a range of diffusive and dissipative mechanisms present in all
acoustic systems, such as bulk fluid thermal/viscous effects, radiation
damping losses, boundary layer effects, vortex shedding, and flow separation.
In this work we consider boundary layer effects (in the neck of very thick-
walled resonators only) as we consider this to be the dominant internal loss
mechanism. We are able to show numerically that dissipation generally reduces
the enhancement in the scattering and extinction cross sections (efficiencies)
to recover results for a closed Neumann cylinder, with the absorption
efficiency exhibiting a clear peak as the attenuation coefficient
$\bar{\alpha}$ increases. This absorption peak implicitly defines the
operating range of our model, and we do not advise considering $\bar{\alpha}$
values beyond this range, although in the limit
$\bar{\alpha}\rightarrow\infty$ results must return to those for a closed
Neumann cylinder, as the ever increasing resistance in the neck region
inhibits energy flow to the interior of the resonator.

We remark that our closed form (multipole) solution representation should
prove useful to those interested in the scattering behaviour of resonators, as
our expressions are rapidly evaluable with little computational overhead. Our
methodology avoids the need to use intensive numerical methods (where meshing
in the narrow neck region can become prohibitive), and permits the rapid
exploration of parameter spaces for optimisation. To this end, we present
closed-form representations for the dispersion equation corresponding to the
first Helmholtz resonance, in all three wall thickness regimes. In the very
thick-walled regime, the fundamental Helmholtz resonance can be pushed to very
low frequencies by tuning the wall thickness, outer radius, neck width, and
neck length carefully (see Smith and Abrahams [13]).

The outline of this paper is as follows. In Section 2 we present a brief
solution outline for plane wave scattering by a closed (ideal) Neumann
cylinder. Next, we pose an ansatz for the total potential of a single
resonator satisfying Neumann conditions on the walls (which is known except
for an undetermined coefficient) in Section 3. In Section 3.2 we use matched
asymptotic methods to determine this amplitude for resonators in the thin-
walled, moderately thick-walled, and very thick-walled regimes. Closed-form
expressions for determining the Helmholtz resonance frequency are then derived
and presented for these three settings in Section 3.3. In Section 4,
asymptotic representations for the scattering coefficients and definitions for
the cross sections are given, with dissipation in very thick-walled cylinders
considered in Section 5. Numerical results are presented in Section 6 and are
followed by concluding remarks in Section 7. Finally, in Appendix A we discuss
and clarify Willis-like coupling effects that have been reported in the
literature for plane-wave scattering by resonators of this or similar design.

##  2 Scattering by a single cylinder

As a means of reference, we begin by briefly outlining the multipole solution
for time-harmonic plane wave scattering by a single circular cylinder
$\Omega_{\mathrm{c}}$ immersed in an infinitely extending fluid medium. We
consider the solution in the domain exterior to the cylinder where the field
satisfies the acoustic wave equation

$(\partial_{{x}}^{2}+\partial_{{y}}^{2}+1)\phi_{\mathrm{ext}}=0,\quad\mbox{for}(x,y)\in\mathbb{R}_{2}\backslash\Omega_{\mathrm{c}},$
(1)

where $x=k\bar{x}$ and $y=k\bar{y}$ denote nondimensional Cartesian
coordinates, $\phi_{\mathrm{ext}}$ is the fluid velocity potential in the
exterior domain, $k=\omega\sqrt{\rho/B}$ denotes the wavenumber, $\omega$ is
the angular frequency of the forcing and scattered fields, and $\rho$ and $B$
are the density and bulk modulus of the background medium, respectively. Here
the observed time-dependent field is given by
$\mathrm{Re}\left\\{\phi_{\mathrm{ext}}\exp(-\mathrm{i}\omega t)\right\\}$,
and the general solution to (1) takes the form

$\phi_{\mathrm{ext}}=\phi_{\mathrm{inc}}+\phi_{\mathrm{sc}}=\sum_{n=-\infty}^{\infty}\left\\{a_{n}\mathrm{J}_{n}(r)+b_{n}\mathrm{H}_{n}^{(1)}(r)\right\\}\mathrm{e}^{\mathrm{i}n\theta},$
(2)

where $\phi_{\mathrm{inc}}$ and $\phi_{\mathrm{sc}}$ are the incident and
scattered potentials, respectively, $a_{n}$ and $b_{n}$ refer to the as yet
unknown incoming and outgoing field coefficients, respectively,
$\mathrm{J}_{n}(z)$ is a Bessel functions of the first kind,
$\mathrm{H}_{n}^{(1)}(z)$ is a Hankel function of the first kind, $(r,\theta)$
is the polar form of $(x,y)$, and we specify Neumann conditions on the walls
$\partial\phi_{\mathrm{ext}}/\partial r\big{|}_{r=b}=0,$ with $\bar{b}$
denoting the dimensional cylinder radius (and $b=k\bar{b}$).

If we consider incident plane waves of the form
$\phi_{\mathrm{inc}}=\exp(\mathrm{i}x\cos\theta_{\mathrm{inc}}+\mathrm{i}y\sin\theta_{\mathrm{inc}})$,
where $\theta_{\mathrm{inc}}$ denotes the incident angle, then the solution is
obtained straightforwardly by using a Jacobi–Anger expansion for the incident
field [23, Eq. (8.511-4)]

$\mathrm{e}^{\mathrm{i}r\cos\theta}=\sum_{n=-\infty}^{\infty}\mathrm{i}^{n}\mathrm{J}_{n}(r)\mathrm{e}^{\mathrm{i}n\theta},$
(3)

and imposing the Neumann boundary condition, which gives the form

$\phi_{\mathrm{ext}}=\mathrm{e}^{\mathrm{i}r\cos(\theta-\theta_{\mathrm{inc}})}-\sum_{n=-\infty}^{\infty}\left\\{\frac{\mathrm{i}^{n}\mathrm{J}_{n}^{\prime}(b)}{\mathrm{H}_{n}^{(1)\prime}(b)}\mathrm{e}^{-\mathrm{i}n\theta_{\mathrm{inc}}}\right\\}\mathrm{H}_{n}^{(1)}(r)\,\mathrm{e}^{\mathrm{i}n\theta},$
(4)

where prime notation denotes the derivative with respect to argument, i.e.,
$\mathrm{J}_{n}^{\prime}(b)=\partial_{z}\mathrm{J}_{n}(z)|_{z=b}$. With this
canonical solution for plane wave scattering by a cylinder in mind, we now
proceed to our solution outline for a cylindrical resonator.

##  3 Scattering by a single Helmholtz resonator

In order to construct a multipole solution for a single resonator at low
frequencies, we combine the multipole expansion techniques outlined in Section
2 above, with the method of matched asymptotic expansions [9, 10]. This
procedure is outlined in extensive detail for two-dimensional homogeneous
arrays of resonators in earlier works by the authors [12, 13]; as the inner
solutions and matching procedure are unchanged from these earlier works, we
outline only the updated (leading-order) outer solutions here, and simply
state key results where appropriate. For reference, the acoustic wave equation
(1) governs wave propagation in the exterior domain for all outer solutions.

###  3.1 Leading-order outer solution for all wall thicknesses

We begin by posing an ansatz for the exterior domain ($r\geq b$), comprising
the plane wave, a monopole source at the (small) resonator mouth, and a
complete cylindrical harmonic basis, in the form

$\phi_{\mathrm{ext}}=\mathrm{e}^{\mathrm{i}r\cos(\theta-\theta_{\mathrm{inc}})}+A\mathrm{H}_{0}^{(1)}(\tilde{r})+\sum_{n=-\infty}^{\infty}c_{n}\mathrm{H}_{n}^{(1)}(r)\mathrm{e}^{\mathrm{i}n\theta},$
(5)

where $A$ and $c_{n}$ are as yet unknown coefficients,
$\tilde{r}=\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}$, and
$(x_{0},y_{0})=(b\cos\theta_{0},b\sin\theta_{0})$ represents the central
location of the resonator mouth. Without loss of generality we now fix the
location of the aperture by taking $\theta_{0}=0$ and consider only varying
incident angle $\theta_{\mathrm{inc}}$ in the present work. Figure 1 indicates
the geometry of the scatterer.

(a) Subfigure 6 list of figures text

(b) Subfigure 1 list of figures text

Figure 1:  LABEL:sub@fig:figxx1 Three-dimensional segment and
LABEL:sub@fig:figxx2 two-dimensional cross section of our infinitely tall
Helmholtz resonator in dimensionless coordinates.

For our leading-order outer problem [9, 12, 13], the resonator is an almost-
closed cylinder, and so we treat the aperture as a point source via the
boundary condition [12]

$\frac{\partial\phi_{\mathrm{ext}}}{\partial
r}\bigg{|}_{r=b}=\frac{\mathrm{i}A}{\pi
b}\sum_{n=-\infty}^{\infty}\mathrm{e}^{\mathrm{i}n\theta},$ (6)

which is compatible with the point source term in (5). After applying Graf’s
addition theorem [23, Eq. (8.530)]

$\mathrm{H}_{0}^{(1)}(\widetilde{r})=\sum\limits_{n=-\infty}^{\infty}\mathrm{J}_{n}(b)\mathrm{H}_{n}^{(1)}(r)\mathrm{e}^{\mathrm{i}n(\theta-\theta_{0})},\quad\mbox{for}\quad
r>b,$ (7)

on the initial monopole term in (5), and the Jacobi–Anger expansion (3) on the
plane wave term, we impose the Neumann condition (6) above to obtain

$c_{n}=-\mathrm{i}^{n}\frac{\mathrm{J}^{\prime}_{n}(b)}{\mathrm{H}_{n}^{(1)\prime}(b)}\mathrm{e}^{-\mathrm{i}n\theta_{\mathrm{inc}}}-\frac{AQ_{n}}{2\mathrm{H}_{n}^{(1)\prime}(b)},$
(8)

where
$Q_{n}=\mathrm{J}_{n}(b)\mathrm{H}_{n}^{(1)\prime}(b)+\mathrm{J}_{n}^{\prime}(b)\mathrm{H}_{n}^{(1)}(b)$.
Accordingly, only the amplitude $A$ remains unknown at this stage, and in the
limit as we approach the exterior mouth we find that

$\lim_{r\rightarrow b}\lim_{\theta\rightarrow
0}\phi_{\mathrm{ext}}\sim\mathrm{e}^{\mathrm{i}b\cos\theta_{\mathrm{inc}}}+\frac{2\mathrm{i}A}{\pi}\left(\gamma_{\mathrm{e}}-\frac{\mathrm{i}\pi}{2}+\log\left(\frac{\tilde{r}}{2}\right)\right)\\\
-\sum_{n=-\infty}^{\infty}\left[\mathrm{i}^{n}\mathrm{J}_{n}^{\prime}(b)\mathrm{e}^{-\mathrm{i}n\theta_{\mathrm{inc}}}+\frac{AQ_{n}}{2}\right]\frac{\mathrm{H}_{n}^{(1)}(b)}{\mathrm{H}_{n}^{(1)\prime}(b)},$
(9)

where $\gamma_{\mathrm{e}}\approx 0.577216\ldots$ denotes the Euler–Mascheroni
constant. Having derived an appropriate asymptotic form for the outer solution
in the exterior domain (9), as we approach the aperture, we now consider
asymptotic matching to determine $A$ at low frequencies.

###  3.2 Results from asymptotic matching

By employing the asymptotic matching procedure outlined in [12, 13], where
expressions for all other leading-order outer solutions (i.e., for the
resonator interior $\phi_{\mathrm{int}}$ and the resonator neck
$\phi_{\mathrm{neck}}$) and all inner solutions (i.e., $\Phi$ and $\Psi$) are
given explicitly, we now use the representation for $\phi_{\mathrm{ext}}$ in
(9) of the present work, and find after considerable algebraic manipulation
that the amplitude is given by

$A=-\frac{2\mathrm{i}}{\pi
bh_{\varepsilon}}\sum_{p=-\infty}^{\infty}\frac{\mathrm{i}^{p}}{\mathrm{H}_{p}^{(1)\prime}(b)}\mathrm{e}^{-\mathrm{i}p\theta_{\mathrm{inc}}},$
(10)

where

$h_{\varepsilon}=\begin{cases}\dfrac{4\mathrm{i}}{\pi}\left(\gamma_{\mathrm{e}}-\dfrac{\mathrm{i}\pi}{2}+\log\left(\dfrac{\varepsilon}{4}\right)\right)-\dfrac{1}{2}\sum\limits_{m=-\infty}^{\infty}\dfrac{Q_{m}^{2}}{\mathrm{J}_{m}^{\prime}(b)\mathrm{H}_{m}^{(1)\prime}(b)},\quad\mbox{when
thin-walled, }&\\\\[12.0pt]
\dfrac{4\mathrm{i}}{\pi}\left(\gamma_{\mathrm{e}}-\dfrac{\mathrm{i}\pi}{2}+\log\left(\dfrac{\sqrt{q}\,\mathcal{C}(q)\varepsilon}{2}\right)\right)-\dfrac{1}{2}\sum\limits_{m=-\infty}^{\infty}\left\\{\dfrac{\check{Q}_{m}\mathrm{J}_{m}(a)}{\mathrm{J}_{m}^{\prime}(a)}+\dfrac{Q_{m}\mathrm{H}_{m}^{(1)}(b)}{\mathrm{H}_{m}^{(1)\prime}(b)}\right\\},\qquad&\\\
\hskip 184.9429pt\mbox{when moderately thick-walled},&\\\\[12.0pt]
\dfrac{2\mathrm{i}}{\pi}\left(\gamma_{\mathrm{e}}-\dfrac{\mathrm{i}\pi}{2}-\log\left(\dfrac{\pi}{\varepsilon}\right)-\left[\dfrac{2\mathrm{i}\tau_{3}}{\pi}+\tau_{4}\tau_{5}\right]\left[\dfrac{2\mathrm{i}\tau_{1}}{\pi}+\tau_{2}\tau_{5}\right]^{-1}\right)-\dfrac{1}{2}\sum\limits_{m=-\infty}^{\infty}\dfrac{Q_{m}\mathrm{H}_{m}^{(1)}(b)}{\mathrm{H}_{m}^{(1)\prime}(b)},\qquad&\\\\[12.0pt]
\hskip 213.39566pt\mbox{when very thick-walled. }\end{cases}$ (11)

In these expressions, we introduce the small dimensionless parameter
$\varepsilon=k\bar{\ell}$, where $2\bar{\ell}$ denotes the total aperture
width, $2\bar{\mathbcal{m}}$ is the total aperture length (where
$\mathbcal{m}=k\bar{\mathbcal{m}}$), $a=b-2\mathbcal{m}$ denotes the inner
radius (where $a=k\bar{a}$),
$\check{Q}_{m}=\mathrm{J}_{m}(a)\mathrm{H}_{m}^{(1)\prime}(a)+\mathrm{J}_{m}^{\prime}(a)\mathrm{H}_{m}^{(1)}(a)$,
and the quantity $q$ is obtained by solving the relation [12]

$\frac{\mathbcal{m}}{\ell}=\frac{1}{2}\left[2E(q^{2})+(q^{2}-1)K(q^{2})\right]^{-1}\left[-2E(1-q^{2})+(1+q^{2})K(1-q^{2})\right],$
(12)

with $\mathcal{C}(q)=1/\\{2E(q^{2})+(q^{2}-1)K(q^{2})\\},$ where $E(z)$ and
$K(z)$ are complete Elliptic integrals of the first and second kind,
respectively. Furthermore, as in Smith and Abrahams [13] we define

$\displaystyle\tau_{1}$ $\displaystyle=\frac{2\varepsilon}{\pi}(1-\log
2)\sin(2\mathbcal{m})-\cos(2\mathbcal{m}),\qquad\tau_{4}=-\frac{2\varepsilon}{\pi}(1-\log
2)\sin(2\mathbcal{m})+\cos(2\mathbcal{m}),$ (13a) $\displaystyle\tau_{2}$
$\displaystyle=-\frac{2\varepsilon}{\pi}\sin(2\mathbcal{m}),\hskip
56.9055pt\tau_{5}=\dfrac{2\mathrm{i}}{\pi}\left[\gamma_{\mathrm{e}}-\dfrac{\mathrm{i}\pi}{2}-\log\left(\frac{\pi}{\varepsilon}\right)\right]-\dfrac{1}{2}\sum\limits_{m=-\infty}^{\infty}\dfrac{\check{Q}_{m}\mathrm{J}_{m}(a)}{\mathrm{J}_{m}^{\prime}(a)},$
(13b) $\displaystyle\tau_{3}$
$\displaystyle=\left[\frac{2\varepsilon}{\pi}(1-\log
2)^{2}-\frac{\pi}{2\varepsilon}\right]\sin(2\mathbcal{m})-2(1-\log
2)\cos(2\mathbcal{m}).$ (13c)

Thus, with the form of $A$ in (10) and the representation for
$h_{\varepsilon}$ in (11), in addition to the expression for $c_{n}$ in (8),
we have now fully prescribed $\phi_{\mathrm{ext}}$ in (5) in the low-frequency
asymptotic limit for all wall thickness configurations. For reference, we also
define the channel aspect ratio $h=2\bar{\mathbcal{m}}/2\bar{\ell}$ and note
that in the thin-walled limit $h\rightarrow 0$, we have $q\rightarrow 1$ and
$\mathcal{C}(q)\rightarrow 1/2$, which returns consistent expressions for
$h_{\varepsilon}$ in (11).

Next, we examine the explicit form of the Helmholtz resonance condition for a
single resonator.

###  3.3 Helmholtz resonance frequency

For our plane wave scattering problem, the Helmholtz resonance frequency is
obtained straightforwardly by searching for the minimum value of the
denominator of $A$ in (10). This corresponds to the vanishing of the imaginary
part of $h_{\varepsilon}$. In the limit as the wave becomes long relative to
all geometric parameters, we find that

$\lim_{\left\\{a,b,\mathbcal{m}\right\\}\rightarrow
0}h_{\varepsilon}\approx\begin{cases}\dfrac{\mathrm{i}}{\pi}\left[\dfrac{2}{b^{2}}-\dfrac{1}{4}+\gamma_{\mathrm{e}}-\dfrac{\mathrm{i}\pi}{2}+\log\left(\dfrac{\varepsilon^{4}}{2^{5}b^{3}}\right)\right],\quad\mbox{when
thin-walled, }&\\\\[12.0pt]
\dfrac{\mathrm{i}}{\pi}\left[\dfrac{2}{a^{2}}-\dfrac{1}{4}+\gamma_{\mathrm{e}}-\dfrac{\mathrm{i}\pi}{2}+\log\left(\dfrac{q^{2}\mathcal{C}(q)^{4}\varepsilon^{4}}{2a^{2}b}\right)\right],\quad\mbox{when
moderately thick-walled},&\\\\[12.0pt]
\dfrac{\mathrm{i}}{\pi}\left\\{\gamma_{\mathrm{e}}-\dfrac{\mathrm{i}\pi}{2}-\log\left(\dfrac{\pi^{2}b}{2\varepsilon^{2}}\right)+2\left[\dfrac{1}{a^{2}}-\dfrac{\mathbcal{m}\pi}{\varepsilon}-\dfrac{17}{8}-\log\left(\dfrac{\pi
a}{2^{3}\varepsilon}\right)\right]\right.\qquad&\\\\[12.0pt] \hskip
28.45274pt\cdot\left.\left[1+\dfrac{4\varepsilon\mathbcal{m}}{\pi}\left(\dfrac{1}{a^{2}}-\dfrac{1}{8}-\log\left(\dfrac{\pi
a}{2\varepsilon}\right)\right)\right]^{-1}\right\\},\quad\mbox{when very
thick-walled. }\end{cases}$ (14)

If we then prescribe $\mathbcal{m}=\kappa_{\mathbcal}{m}\varepsilon^{\mu}$ and
$a=\kappa_{a}\varepsilon^{\gamma}$ where $\kappa_{\mathbcal}{m}$ and
$\kappa_{a}$ are real constants and $0<\mu,\gamma<1$, and examine the narrow
aperture limit $\varepsilon\rightarrow 0$, it follows that under the dominant
balance scaling $\mu+1-2\gamma>0$ [13] we have

$h_{\varepsilon}\approx\dfrac{\mathrm{i}}{\pi}\left\\{\dfrac{2}{a^{2}}-\dfrac{2\mathbcal{m}\pi}{\varepsilon}-\dfrac{17}{4}+\gamma_{\mathrm{e}}-\dfrac{\mathrm{i}\pi}{2}+\log\left(\dfrac{2^{7}\varepsilon^{4}}{\pi^{4}a^{2}b}\right)\right\\},\quad\mbox{when
very thick-walled. }$ (15)

Accordingly we write the Helmholtz resonance condition in our three canonical
limits as

$\displaystyle\dfrac{2}{b^{2}}-\dfrac{1}{4}+\gamma_{\mathrm{e}}+\log\left(\dfrac{\varepsilon^{4}}{2^{5}b^{3}}\right)$
$\displaystyle\approx 0,\quad\mbox{when thin-walled, }$ (16a)
$\displaystyle\dfrac{2}{a^{2}}-\dfrac{1}{4}+\gamma_{\mathrm{e}}+\log\left(\dfrac{q^{2}\mathcal{C}(q)^{4}\varepsilon^{4}}{2a^{2}b}\right)$
$\displaystyle\approx 0,\quad\mbox{when moderately thick-walled},$ (16b)
$\displaystyle\dfrac{2}{a^{2}}-\dfrac{2\mathbcal{m}\pi}{\varepsilon}-\dfrac{17}{4}+\gamma_{\mathrm{e}}+\log\left(\dfrac{2^{7}\varepsilon^{4}}{\pi^{4}a^{2}b}\right)$
$\displaystyle\approx 0,\quad\mbox{when very thick-walled, }$ (16c)

and note that in all cases $h_{\varepsilon}$ takes the value $1/2$ at
resonance. These expressions must be solved numerically due to the presence of
$\log(k)$ terms, in contrast to the corresponding expressions for a two-
dimensional array of resonators [12, 13].

The expressions in (16) are useful for rapidly determining the location of the
first Helmholtz resonance. For example, should we specify a total aperture
width of $2\bar{\ell}\approx 1\,$mm, which generally corresponds to an
acoustic length scale before viscous effects start to emerge, as well as a
desired resonance frequency $\nu$ (i.e., $\nu=1$ kHz, where $\omega=2\pi\nu$
or $k=18.3074$ m-1 in air) then in the thin-walled limit we solve (16a) to
find that an outer radius of $\bar{b}\approx 17.8$ mm is required; such values
correspond to a resonator design that is readily fabricated using contemporary
methods (i.e., using additive manufacturing techniques).

##  4 Asymptotic representations of scattering coefficients

In this section, we return to the leading-order exterior field representation
from Section 3.1 and examine the multipole coefficients in closer detail. We
wish to obtain simple explicit expressions valid at low frequencies, i.e., for
$b=k\bar{b}\ll 1$, ensuring that we preserve the partitioning $\varepsilon\ll
b$. To this end, we write the ansatz (5) as

$\phi_{\mathrm{ext}}=\phi_{\mathrm{inc}}+\phi_{\mathrm{sc}}=\mathrm{e}^{\mathrm{i}r\cos(\theta-\theta_{\mathrm{inc}})}+\sum_{n=-\infty}^{\infty}d_{n}\mathrm{H}_{n}^{(1)}(r)\mathrm{e}^{\mathrm{i}n\theta},$
(17)

which arises after an application of Graf’s addition theorem (7), where

$d_{n}=A\mathrm{J}_{n}(b)+c_{n}=\frac{\mathrm{i}A}{\pi
b\mathrm{H}_{n}^{(1)\prime}(b)}-\mathrm{i}^{n}\frac{\mathrm{J}^{\prime}_{n}(b)}{\mathrm{H}_{n}^{(1)\prime}(b)}\mathrm{e}^{-\mathrm{i}n\theta_{\mathrm{inc}}}.$
(18)

On inspection, we see that the first term in (18) is due to the resonator and
the second term corresponds to the scattering coefficient for an ideal Neumann
cylinder (4). Next, we truncate the sum in (17) so that all terms of
$O(b^{4})$ are captured (i.e., we consider the orders $n=-2,\ldots,2$) and
presume that such a truncation is sufficient for describing the response of
the resonator at low frequencies. Consequently, in the vanishing $b$ limit we
find that

$\displaystyle\lim_{b\rightarrow 0}d_{0}$
$\displaystyle\approx\frac{\pi\mathrm{i}b^{2}}{8f_{\varepsilon}}\left[1-2f_{\varepsilon}\vphantom{\frac{1}{2}}\right]-\frac{\pi
b^{3}\cos(\theta_{\mathrm{inc}})}{4f_{\varepsilon}}$ (19a)
$\displaystyle\hskip
28.45274pt+\frac{\pi\mathrm{i}b^{4}}{8}\left\\{\left(1-\frac{1}{f_{\varepsilon}}\right)\left[\frac{1}{2}-\gamma_{\mathrm{e}}+\frac{\mathrm{i}\pi}{2}-\log\left(\frac{b}{2}\right)\right]+\frac{1}{4}-\frac{\cos(2\theta_{\mathrm{inc}})}{2f_{\varepsilon}}\right\\},$
(19b) $\displaystyle\lim_{b\rightarrow 0}d_{\pm 1}$
$\displaystyle\approx\mp\frac{\pi
b^{2}}{4}\mathrm{e}^{\mp\mathrm{i}\theta_{\mathrm{inc}}}\pm\frac{\pi\mathrm{i}b^{3}}{8f_{\varepsilon}}+\frac{\pi
b^{4}}{4}\left\\{\pm\frac{\mathrm{e}^{\mp\mathrm{i}\theta_{\mathrm{inc}}}}{2}\left[\frac{5}{4}+\gamma_{\mathrm{e}}-\frac{\mathrm{i}\pi}{2}+\log\left(\frac{b}{2}\right)\right]\mp\frac{\cos(\theta_{\mathrm{inc}})}{f_{\varepsilon}}\right\\},$
(19c) $\displaystyle\lim_{b\rightarrow 0}d_{\pm 2}$
$\displaystyle\approx\frac{\pi\mathrm{i}b^{4}}{32}\left[\frac{1}{f_{\varepsilon}}-\mathrm{e}^{\mp
2\mathrm{i}\theta_{\mathrm{inc}}}\right],$

where we have used the scaling
$h_{\varepsilon}=4\mathrm{i}f_{\varepsilon}/(\pi b^{2})$ and the result

$\lim_{b\rightarrow
0}A\approx\frac{\pi\mathrm{i}b^{2}}{4f_{\varepsilon}}\left(1+2\mathrm{i}b\cos(\theta_{\mathrm{inc}})\right)+\frac{\mathrm{i}\pi
b^{4}}{8f_{\varepsilon}}\left[-\frac{1}{2}+\gamma_{\mathrm{e}}-\frac{\mathrm{i}\pi}{2}-\cos(2\theta_{\mathrm{inc}})+\log\left(\frac{b}{2}\right)\right].$
(20)

With these representations in mind, we now consider cross sections for our
Helmholtz resonator.

###  4.1 Scattering, absorption, and extinction cross sections

In order to compactly describe how our resonator influences the incident plane
wave, we evaluate cross sections for the scattering, absorption and extinction
strength of a given resonator, which are all functions of frequency. We
characterise the scattering strength using the (dimensional) scattering cross
section for a cylinder or resonator [18, Sec. E.1]

$\bar{\sigma}_{\mathrm{sc}}=\frac{4}{k}\sum_{m=-\infty}^{\infty}|d_{m}|^{2},$
(21)

which we scale by the geometric cross section of the scatterer to define the
nondimensional coefficient
$Q_{\mathrm{sc}}=\bar{\sigma}_{\mathrm{sc}}/(2\bar{b}),$ which is termed the
scattering efficiency [1]. Additionally we have the (dimensional) extinction
cross section

$\bar{\sigma}_{\mathrm{ext}}=-\frac{4}{k}\,\sum_{m=-\infty}^{\infty}\mathrm{Re}\left\\{d_{m}\mathrm{e}^{-\mathrm{i}m(\pi/2-\theta_{\mathrm{inc}})}\right\\},$
(22)

with the corresponding extinction efficiency
$Q_{\mathrm{ext}}=\bar{\sigma}_{\mathrm{ext}}/(2\bar{b})$. Furthermore, we
note that the extinction efficiency is given by the sum [1]:

$Q_{\mathrm{ext}}=Q_{\mathrm{sc}}+Q_{\mathrm{abs}},$ (23)

where $Q_{\mathrm{abs}}=\bar{\sigma}_{\mathrm{abs}}/(2\bar{b})$ denotes the
absorption efficiency. Therefore, in the absence of viscosity (or other
dissipative processes) we have that $Q_{\mathrm{ext}}=Q_{\mathrm{sc}}$. Having
obtained cross section expressions in preparation for numerical investigations
in Section 6, we now examine the impact of dissipation in thick-walled
resonators.

##  5 Dissipation in very thick-walled resonators

In this section, we discuss the impact of incorporating viscosity within the
neck region of a very thick-walled resonator, where a boundary layer could be
expected to appear in the fluid, giving rise to viscous losses [22]. For thin-
and moderately thick-walled resonators, boundary layer effects can be expected
to be minimal, due to the smaller neck length, and so we do not consider these
configurations here. To incorporate dissipative loss in a simple fashion, we
specify a complex-valued wavenumber within the neck region via the replacement
$k\mapsto k+\mathrm{i}\bar{\alpha}$ where $\bar{\alpha}$ is the dimensional
attenuation coefficient (units $1/\mathrm{m}$). This gives rise to the outer
solution in the neck (cf., [13, Eq. (2.6)])

$\lim_{\varepsilon\rightarrow 0}\phi_{\mathrm{neck}}\sim
p_{0}\mathrm{e}^{\mathrm{i}(1+\mathrm{i}\alpha)\tilde{y}}+q_{0}\mathrm{e}^{-\mathrm{i}(1+\mathrm{i}\alpha)\tilde{y}},$
(24)

where $\alpha=\bar{\alpha}/k$ is the nondimensional attenuation constant,
$p_{0}$ and $q_{0}$ are unknown constants, and $(\tilde{x},\tilde{y})$ denote
a local rotated coordinate frame whose origin is centered at the exterior
mouth of the resonator [13]. Using the outer solution in the exterior domain
$\phi_{\mathrm{ext}}$ (9), the outer solution in the neck region
$\phi_{\mathrm{neck}}$ (24), and the outer solution in the interior [12, Eq.
(6.11)], in tandem with the inner solutions $\Phi$ and $\Psi$ given in Smith
and Abrahams [13], we obtain a result identical to (10) after asymptotic
matching, but with the replacement $h_{\varepsilon}\mapsto
h_{\varepsilon}^{\mathrm{d}}$ where

$h_{\varepsilon}^{\mathrm{d}}=\dfrac{2\mathrm{i}}{\pi}\left(\gamma_{\mathrm{e}}-\dfrac{\mathrm{i}\pi}{2}-\log\left(\dfrac{\pi}{\varepsilon}\right)-\left[\dfrac{2\mathrm{i}\tau_{3}^{\mathrm{d}}}{\pi}+\tau_{4}^{\mathrm{d}}\tau_{5}\right]\left[\dfrac{2\mathrm{i}\tau_{1}^{\mathrm{d}}}{\pi}+\tau_{2}^{\mathrm{d}}\tau_{5}\right]^{-1}\right)-\dfrac{1}{2}\sum\limits_{m=-\infty}^{\infty}\dfrac{Q_{m}\mathrm{H}_{m}^{(1)}(b)}{\mathrm{H}_{m}^{(1)\prime}(b)},$
(25)

along with the dissipative forms

$\displaystyle\tau_{1}^{\mathrm{d}}$
$\displaystyle=\frac{2\varepsilon\gamma_{\mathrm{d}}}{\pi}(1-\log
2)\sin(2\mathbcal{m}\gamma_{\mathrm{d}})-\cos(2\mathbcal{m}\gamma_{\mathrm{d}}),$
(26a) $\displaystyle\tau_{2}^{\mathrm{d}}$
$\displaystyle=-\frac{2\varepsilon\gamma_{\mathrm{d}}}{\pi}\sin(2\mathbcal{m}\gamma_{\mathrm{d}}),$
(26b) $\displaystyle\tau_{3}^{\mathrm{d}}$
$\displaystyle=\left[\frac{2\varepsilon\gamma_{\mathrm{d}}}{\pi}(1-\log
2)^{2}-\frac{\pi}{2\varepsilon\gamma_{\mathrm{d}}}\right]\sin(2\mathbcal{m}\gamma_{\mathrm{d}})-2(1-\log
2)\cos(2\mathbcal{m}\gamma_{\mathrm{d}}),$ (26c)
$\displaystyle\tau_{4}^{\mathrm{d}}$
$\displaystyle=-\frac{2\varepsilon\gamma_{\mathrm{d}}}{\pi}(1-\log
2)\sin(2\mathbcal{m}\gamma_{\mathrm{d}})+\cos(2\mathbcal{m}\gamma_{\mathrm{d}}),$
(26d)

in which $\gamma_{\mathrm{d}}=1+\mathrm{i}\alpha$. Presuming that $\alpha$ is
small (i.e., $\alpha\sim O(1)$) we have that

$\lim_{\left\\{a,b,\mathbcal{m}\right\\}\rightarrow
0}h_{\varepsilon}^{\mathrm{d}}\approx\frac{2\mathrm{i}}{\pi}\left\\{\frac{\gamma_{\mathrm{e}}}{2}-\frac{\mathrm{i}\pi}{4}-\frac{1}{2}\log\left(\frac{\pi^{2}b}{2\varepsilon^{2}}\right)\right.\\\
\left.+\left[\frac{1}{a^{2}}-\frac{\mathbcal{m}\pi}{\varepsilon}-\frac{2\pi\mathbcal{m}^{3}\gamma_{\mathrm{d}}^{2}}{3\varepsilon}-\frac{2\mathbcal{m}^{2}\gamma_{\mathrm{d}}^{2}}{a^{2}}-\frac{17}{8}-\log\left(\frac{\pi
a}{8\varepsilon}\right)\right]\cdot\left[1-2\mathbcal{m}^{2}\gamma_{\mathrm{d}}^{2}+\frac{4\varepsilon\mathbcal{m}\gamma_{\mathrm{d}}^{2}}{\pi
a^{2}}\right]^{-1}\right\\},$ (27)

which, under the $\mu+1-2\gamma>0$ dominant balance limit from Section 3.3,
takes the form

$h_{\varepsilon}^{\mathrm{d}}\approx\frac{\mathrm{i}}{\pi}\left\\{\gamma_{\mathrm{e}}-\frac{\mathrm{i}\pi}{2}-\frac{17}{4}+\frac{2}{a^{2}}-\frac{2\mathbcal{m}\pi}{\varepsilon}+\log\left(\frac{2^{7}\varepsilon^{4}}{\pi^{4}a^{2}b}\right)-4\mathbcal{m}^{2}\gamma_{\mathrm{d}}^{2}\left[\frac{1}{a^{2}}+\frac{\mathbcal{m}\pi}{3\varepsilon}\right]\right\\},$
(28)

and subsequently, the Helmholtz resonance condition is given by

$\frac{2}{a^{2}}-\frac{2\mathbcal{m}\pi}{\varepsilon}-\frac{17}{4}+\gamma_{\mathrm{e}}+\log\left(\frac{2^{7}\varepsilon^{4}}{\pi^{4}a^{2}b}\right)-4\mathbcal{m}^{2}(1-\alpha^{2})\left[\frac{1}{a^{2}}+\frac{\mathbcal{m}\pi}{3\varepsilon}\right]=0.$
(29)

Accordingly, by substituting (29) into (28) we find that at the Helmholtz
resonance we have

$h_{\varepsilon}^{\mathrm{d}}\approx\frac{1}{2}+\frac{8}{\pi}\alpha\mathbcal{m}^{2}\left[\frac{1}{a^{2}}+\frac{\mathbcal{m}\pi}{3\varepsilon}\right],$
(30)

and so from the definition of $A$ in (10) (with the replacement
$h_{\varepsilon}\mapsto h_{\varepsilon}^{\mathrm{d}}$) we see that the
presence of dissipation acts to lower the amplitude of the monopole source
when on-resonance. That is, $h_{\varepsilon}^{\mathrm{d}}$ increases from the
value of $1/2$ observed in the lossless case to (30). For reference, we use
$h_{\varepsilon}^{\mathrm{d}}=4\mathrm{i}f_{\varepsilon}^{\mathrm{d}}/(\pi
b^{2})$ for the replacement $f_{\varepsilon}\mapsto
f_{\varepsilon}^{\mathrm{d}}$. Furthermore, the above forms for
$h_{\varepsilon}^{\mathrm{d}}$ (25) and $\tau_{j}^{\mathrm{d}}$ (26) correctly
limit to the earlier nondissipative results for $h_{\varepsilon}$ (11) and
$\tau_{j}$ (13) in the limit of vanishing loss $\alpha\rightarrow 0$.

##  6 Numerical Results

In this section, we compute a selection of scattered field profiles
$\phi_{\mathrm{sc}}$ for representative resonators from all three wall
thickness configurations, numerically evaluate a selection of scattering,
absorption, and extinction efficiencies as a function of nondimensionalised
frequency (wavenumber), and examine how well the asymptotic representations
for $d_{n}$ in (19) perform relative to the full multipole forms (18).
Additionally, we consider the impact of dissipative loss (from a boundary
layer in the neck) on all cross sections for a representative resonator in the
very thick-walled limit. In all examples we consider air as the background
medium possessing a bulk modulus $B=141.83$ KPa and density $\rho=1.2041$
kg/m3, although this is done without loss of generality.

(a) Subfigure 6 list of figures text

(b) Subfigure 1 list of figures text

(c) Subfigure 2 list of figures text

(d) Subfigure 3 list of figures text

(e) Subfigure 2 list of figures text

(f) Subfigure 3 list of figures text

Figure 2: Scattered field profiles $\phi_{\mathrm{sc}}$ (5) for
LABEL:sub@fig:figa1,LABEL:sub@fig:figa2 a thin-walled resonator at
$k=k_{\mathrm{H}}/2$ and $k=k_{\mathrm{H}}\approx 16.2136$ m-1 respectively,
where $k_{\mathrm{H}}$ satisfies the Helmholtz resonance condition (16a);
LABEL:sub@fig:figa3,LABEL:sub@fig:figa4 a moderately thick-walled resonator
($h=2\bar{\mathbcal{m}}/2\bar{\ell}=4$) at $k_{\mathrm{H}}/2$ and
$k_{\mathrm{H}}\approx 13.3001$ m-1 respectively, where $k_{\mathrm{H}}$
satisfies (16b); LABEL:sub@fig:figa5,LABEL:sub@fig:figa6 a very thick-walled
resonator ($h=15$) at $k_{\mathrm{H}}/2$ and $k_{\mathrm{H}}\approx 26.9376$
m-1 respectively, where $k_{\mathrm{H}}$ satisfies (16c). In all figures we
use $\bar{b}=20$ mm, $2\bar{\ell}=1$ mm, and $\theta_{\mathrm{inc}}=\pi/6$.

In Figure 2 we present the scattered field $\phi_{\mathrm{sc}}$ for a
representative thin-walled, moderately thick-walled, and very thick-walled
resonator, where all resonators possess the same outer radius $\bar{b}$ and
aperture width $2\bar{\ell}$. In each setting, we examine the scattered
response at the first Helmholtz resonance frequency, $k_{\mathrm{H}}$,
satisfying the relevant condition in (16), and at the low frequency
$k_{\mathrm{H}}/2$ (i.e., away from the resonance frequency), to consider how
the field profile is modified as we approach the resonance. In the thin-walled
case we observe a field enhancement of over two orders of magnitude in the
transition from $k=k_{\mathrm{H}}/2$ to $k=k_{\mathrm{H}}$, with
$\phi_{\mathrm{sc}}$ taking a maximum value at the resonator mouth. A similar
behaviour is observed for the moderately thick-walled case ($h=4$), although
we find that the field enhancement is approximately halved. This reduction
continues for the very thick-walled resonator where the field enhancement is
now slightly over one order of magnitude, and is likely due to the fact that
the volume inside the resonator decreases, since we increase the aperture neck
length $2\bar{\mathbcal{m}}$ whilst keeping the outer radius $\bar{b}$
constant. Accordingly, to achieve the strongest field enhancements we advise
that the resonator wall thickness be taken as thin as possible. Additionally,
the strongest backscattering is observed in the thin and very thick-walled
representative configurations. For reference, a plane wave of incidence angle
$\theta_{\mathrm{inc}}=\pi/6$ is considered in all cases.

(a) Subfigure 6 list of figures text

(b) Subfigure 1 list of figures text

(c) Subfigure 2 list of figures text

(d) Subfigure 3 list of figures text

Figure 3: Scattering efficiency $Q_{\mathrm{sc}}$ (from (21)) versus (scaled)
frequency for a LABEL:sub@fig:figb1 thin-walled resonator, LABEL:sub@fig:figb2
moderately thick-walled resonator ($h=4$), and LABEL:sub@fig:figb3 very thick-
walled resonator ($h=15$); LABEL:sub@fig:figb4 gives the maximum value for
$Q_{\mathrm{sc}}$ at the first Helmholtz resonance versus the channel aspect
ratio $h=2\bar{\mathbcal{m}}/2\bar{\ell}$ (with results obtained using the
thick-walled formulation). In all figures we use $\bar{b}=20$ mm,
$2\bar{\ell}=1$ mm, and $\theta_{\mathrm{inc}}=\pi/6$, and results for a
single closed Neumann cylinder of the same radius are superposed for reference
(dashed red). Inset figures: resonator geometry (not to scale).

In Figure 3 we compute the scattering efficiency $Q_{\mathrm{sc}}$ (following
the definition for the scattering cross section $\bar{\sigma}_{\mathrm{sc}}$
in (21)) for the three resonator configurations considered in Figure 2. For
all $Q_{\mathrm{sc}}$ figures we superpose the result for a closed Neumann
cylinder as a means of reference (dashed red curves). We find that for all
resonator configurations, a considerable enhancement is observed in the
scattering efficiency at the first Helmholtz resonance frequency
$k_{\mathrm{H}}$ (obtained by solving (16) for each geometry), and also
observe that the $Q_{\mathrm{sc}}$ curves tend to that of a closed Neumann
cylinder away from the higher-frequency Helmholtz resonance peaks. As the wall
thickness increases, we observe that the spacing between Helmholtz resonances
increases (as the enclosed internal resonator area becomes smaller), however
we also find that the peak scattering efficiency $\max(Q_{\mathrm{sc}})$ does
not exhibit an obvious trend. To investigate this behaviour, we plot
$\max(Q_{\mathrm{sc}})$ against the channel aspect ratio $h$ in Figure 3d,
where we find that a maximum scattering efficiency, for our chosen outer
radius and aperture width, occurs at $h\approx 4.11$ and $h\approx 5.01$. The
oscillations in the $\max(Q_{\mathrm{sc}})$ curve are due to phase
cancellation effects within the neck (see $\phi_{\mathrm{neck}}$ in (24) with
$\alpha=0$), where the outward and inward propagating wave components of the
solution either destructively or constructively interfere with each other.

(a) Subfigure 6 list of figures text

(b) Subfigure 1 list of figures text

(c) Subfigure 2 list of figures text

(d) Subfigure 3 list of figures text

Figure 4: The impact of introducing dissipation in the neck region of a very
thick-walled resonator, via the attenuation coefficient $\bar{\alpha}$:
LABEL:sub@fig:figc1 scattering efficiency $Q_{\mathrm{sc}}$ (defined from
(21)), LABEL:sub@fig:figc2 absorption efficiency $Q_{\mathrm{abs}}$ (defined
from (23)), and LABEL:sub@fig:figc3 extinction efficiency $Q_{\mathrm{ext}}$
(defined from (22)), versus the (scaled) frequency; LABEL:sub@fig:figc4 gives
the maximum value for $Q_{\mathrm{abs}}$ at the first Helmholtz resonance
versus the (scaled) attenuation coefficient $\bar{\alpha}$. In all figures we
use $\bar{b}=20$ mm, $2\bar{\ell}=1$ mm, $h=15$, and
$\theta_{\mathrm{inc}}=\pi/6$. Inset figures: resonator geometry (not to
scale).

In Figure 4 we examine the impact of introducing a boundary layer in the neck
of our representative very thick-walled resonator ($h=15$) from Figure 2 near
the first Helmholtz resonance. We compute the scattering, absorption, and
extinction efficiencies for a range of attenuation values $\bar{\alpha}$,
observing that in the limit as $\bar{\alpha}\rightarrow\infty$, results for
the scattering and extinction efficiency coefficients tend to the results for
(lossless) Neumann cylinders straightforwardly (see the dashed red curve in
Figure 3c for reference). Such behaviour is expected due to the increasing
resistance in the neck. However, the absorption efficiency coefficient is
found to first rise with increasing loss, and then to decrease monotonically
towards zero; the presence of this maximum $Q_{\mathrm{abs}}$ therefore
suggests a range of validity for $\bar{\alpha}$ in our treatment (and
resonator geometry), i.e. $0<\bar{\alpha}\bar{b}<0.08$ when considering
dissipative loss. In general, we advise that only $\alpha$ values that lie
below this peak should be considered as physically reliable. For all
attenuation values we observe that scattering is the dominant process in the
extinction of the incident power, with the reduction in $Q_{\mathrm{ext}}$
being driven by the reduction in $Q_{\mathrm{sc}}$.

(a) Subfigure 6 list of figures text

(b) Subfigure 1 list of figures text

(c) Subfigure 6 list of figures text

(d) Subfigure 1 list of figures text

Figure 5: The impact of incident angle $\theta_{\mathrm{inc}}$ on the maximum
(lossless) scattering efficiency $\max(Q_{\mathrm{sc}})$ (defined from (21)),
for a LABEL:sub@fig:figtheta1 thin-walled and LABEL:sub@fig:figtheta2
moderately thick-walled ($h=4$) resonator. Inset figures: resonator geometry
(not to scale). Corresponding scattered field profiles $\phi_{\mathrm{sc}}$
(5) for the thin-walled resonator configuration in Figure 5a at
$k=k_{\mathrm{H}}=16.2136$ m-1 for LABEL:sub@fig:figtheta3
$\theta_{\mathrm{inc}}=0$ and LABEL:sub@fig:figtheta4
$\theta_{\mathrm{inc}}=\pi$. Here we use $\bar{b}=20$ mm and $2\bar{\ell}=1$
mm where applicable.

In Figure 5 we examine how the scattering efficiency $Q_{\mathrm{sc}}$ at the
first Helmholtz resonance is impacted by the incident angle of the incoming
plane wave. We present results for a thin- and moderately thick-walled
resonator ($h=4$) showing that an absolute maximum scattering efficiency is
observed at incidence angles along the mirror plane for the resonator (recall
that the aperture is located at $\theta_{0}=0$), with a minimum observed in
the orthogonal direction at $\theta_{\mathrm{inc}}=\pi/2$. That is, the
incident wave does not have to be directed into the resonator
$\theta_{\mathrm{inc}}=\pi$ in order to achieve maximal scattering efficiency,
as the same result is obtained for $\theta_{\mathrm{inc}}=0$. An identical
behaviour is observed for very thick resonator configurations and so we do not
present the corresponding figure here. Figures 5c and 5d present the scattered
field $\phi_{\mathrm{sc}}$ for the thin-walled resonator at the Helmholtz
resonance frequency $k=k_{\mathrm{H}}=16.2136$ m-1 for the incidence angles
$\theta_{\mathrm{inc}}=0$ and $\theta_{\mathrm{inc}}=\pi$, respectively,
demonstrating how very different field profiles can still return identical
scattering cross sections (i.e., an identical amount of incident power
scattered).

(a) Subfigure 6 list of figures text

(b) Subfigure 1 list of figures text

Figure 6: Comparing the scattering efficiency $Q_{\mathrm{sc}}$ (defined from
(21)), evaluated using the full (18) (dotted black line) and asymptotic forms
(19) (blue line) for $d_{n}$ corresponding to LABEL:sub@fig:figw1 a thin-
walled resonator, with the asymptotic form for $h_{\varepsilon}$ (14) taken to
$O(b^{4})$, and LABEL:sub@fig:figw2 moderately thick-walled ($h=4$) resonator.
In these figures we use $\bar{b}=20$ mm, $2\bar{\ell}=1$ mm, and
$\theta_{\mathrm{inc}}=\pi$ where applicable; the result for a single closed
Neumann cylinder of the same radius is superposed for reference (dashed red).
Inset figure: resonator geometry (not to scale).

In Figure 6 we consider the full multipolar (18) and asymptotic (19) forms for
$d_{n}$ when evaluating the (lossless) scattering efficiency
$Q_{\mathrm{sc}}$. Results are given for the thin-walled and moderately thick-
walled ($h=4$) resonator geometries considered in Figure 2. In general, we
find that the asymptotic estimates (19) work well for frequencies below the
first Helmholtz resonance $k_{\mathrm{H}}$. However, for the thin-walled
geometry, asymptotic forms for $h_{\varepsilon}$ (14) must be taken to
$O(b^{4})$ in order to recover the $Q_{\mathrm{sc}}$ peak as shown; this
representation of $h_{\varepsilon}$ is not presented here for compactness.
Additionally, results for the very thick-walled ($h=15$) configuration are not
presented here as we require asymptotic forms for $h_{\varepsilon}$ and
$d_{0,\pm 1,\pm 2}$ to a very high order in $b$ for accuracy, although for
$k\ll k_{\mathrm{H}}$ we find that the asymptotic forms (19) for $d_{n}$ work
well for all resonator configurations.

For reference, numerical investigations examining the impact of varying the
channel aspect ratio $h$ and the aperture half-angle $\theta_{\mathrm{ap}}$ on
the first Helmholtz resonance frequency $k_{\mathrm{H}}$ for a very thick-
walled resonator ($h=15$) return a similar qualitative behaviour to that seen
in Smith and Abrahams [13] for two-dimensional arrays. Additionally, the cross
section results for lossless cylinders match those presented in Rubinow [24]
after a factor 2 correction in their definition is taken into account.

##  7 Discussion

In this paper we have presented an analytic solution method for deducing plane
wave scattering by a single two-dimensional Helmholtz resonator. Our solution
procedure, valid at low frequencies, combines multipole methods with the
method of matched asymptotic expansions, extending results from earlier works
by the authors on homogeneous two-dimensional arrays of resonators [12, 13].
In addition to describing the scattered field, we determine the extinction,
absorption, and scattering cross sections for a selection of resonator designs
and compare these against results for an isolated Neumann cylinder.

Numerical investigations demonstrate considerable field enhancement near the
resonator mouth, at the first Helmholtz resonance frequency, with a strong
dependence on the wall thickness. We have found that optimal wall thicknesses
exist to achieve maximal cross sections (efficiencies) for a prescribed outer
radius and incidence angle. Furthermore, we consider the impact of a boundary
layer emerging in the resonator neck, giving rise to viscous dissipative
losses. Although a lossy neck gives rise to moderate values for the absorption
efficiency, the corresponding reduction in the scattering efficiency has the
net effect of diminishing the extinction efficiency at the first Helmholtz
resonance and beyond, where in the limit of large loss, we find that all cross
sections return to the results for a Neumann cylinder, as expected. In general
the maximal scattering efficiency $Q_{\mathrm{sc}}$ is found for incidence
wave angles that lie along the mirror symmetry plane for the resonator,
despite the fact that very different scattered field profiles are observed for
$\theta_{\mathrm{inc}}=0$ and $\theta_{\mathrm{inc}}=\pi$. The formulation
presented here should prove useful for ongoing theoretical and experimental
work by other groups [25, 14, 26].

Recently, Melnikov et al. [14] has conducted experimental and theoretical work
on plane wave scattering by a single Helmholtz resonator of the type
considered here. In their work they use a lumped-element model [22] to
describe the resonator response which takes the form of a third-order ordinary
differential equation (a non-linear spring model). Such phenomenological-type
modelling is not required to determine the plane wave scattering response of a
single resonator, as we have shown here; that is, the asymptotic matching
procedure recovers field representations that are theoretically
indistinguishable from the genuine field at low frequencies. Furthermore,
Melnikov et al. [14] constructs an acoustic analogue to the electric
polarisability tensor in electrostatics [3], with non-zero off-diagonal terms
that are referred to as elements of a Willis coupling tensor (see Appendix A
in the present work for an in-depth discussion). However, we stress that
Willis coupling in general refers to the tensors that emerge within
generalised constitutive relations for effective structured media, and
furthermore, as shown in earlier work on two-dimensional arrays of resonators
by the authors [13], that anisotropy and not bianisotropy is observed in bulk.
Accordingly, the presence of Willis-like behaviour (depolarisability effects)
in the acoustic polarisability tensor [14], as shown in our Appendix, does not
necessarily correlate with an effective Willis tensor effect in bulk. Although
Willis coupling tensors are vanishing at low frequencies (for centrosymmetric
unit cells), they are still nonetheless present at low frequencies, and given
their absence in the effective dispersion equation in bulk [12, 13], we
therefore see no evidence that two-dimensional arrays of Helmholtz resonators
exhibit bianisotropy (Willis coupling).

In addition to how this work may relate to Willis coupling, future work
includes careful parameter sweeps to examine the impact of aperture width on
all cross sections, as well as studies considering scattering by random
suspensions of resonators, finite clusters, and one-dimensional arrays (with
the latter currently under investigation by the authors). The impact of
different internal resonator geometries (i.e., square) may be of interest.
Although the results derived here are formally derived for small aperture
widths $2\bar{\ell}$, our treatment appears to hold for much wider apertures
than that considered numerically in the present work (see [12, 13]). Other
future work includes an extension of our model to incorporate nonlinear
effects in the fluid within the neck, which removes the need to assume the
existence of nonlinearity a priori, as is required for lumped element model
treatments.

##  Appendix A Acoustic polarisability and Willis-like coupling

When designing periodically structured media it is important to consider the
possibility that a metamaterial, comprised of purely linear phases, may behave
as a Willis (bi-anisotropic) medium. It is also possible that the metamaterial
may exhibit enhanced nonlinear constitutive behaviour [27], but we do not
consider that possibility here. Recently, a selection of works [25, 14, 26]
have claimed that an analogue to Willis coupling is exhibited in the
scattering response of a single Helmholtz resonator, which we now examine
using our present formalism.

Following the conventions outlined in Quan [26] we begin by writing the
(dimensional) scattered pressure for the resonator in terms of an ideal
(point-source) monopole and dipole at the origin

$\overline{P}_{\mathrm{sc}}=-\mathrm{i}k^{2}\frac{c_{\mathrm{p}}}{4}{M}\mathrm{H}_{0}^{(1)}(k\overline{r})-\mathrm{i}\frac{k^{3}c_{\mathrm{p}}^{2}}{4}\left({D}_{\overline{x}}\cos\theta+{D}_{\overline{y}}\sin\theta\right)\mathrm{H}_{1}^{(1)}(k\overline{r}),$
(31)

where $\overline{r}$ denotes the dimensional distance from the origin,
$c_{\mathrm{p}}=\sqrt{B/\rho}$ is the free-space phase velocity, ${M}$ is the
acoustic monopole amplitude, $({D}_{\overline{x}},{D}_{\overline{y}})$ are the
acoustic dipole amplitudes, and we consider $\exp(-\mathrm{i}\omega t)$ time-
harmonic dependence as in the main text. Through the use of orthogonality
relations it follows that [26, Eq.(2.51)]

$\displaystyle M$ $\displaystyle=\frac{2\mathrm{i}c_{\mathrm{p}}^{-2}}{\pi
k^{2}\mathrm{H}_{0}^{(1)}(k\overline{r})}\,\int_{0}^{2\pi}\overline{P}_{\mathrm{sc}}(\overline{r},\theta)\,\mathrm{d}\theta,$
(32a) $\displaystyle D_{\overline{x}}$
$\displaystyle=\frac{4\mathrm{i}c_{\mathrm{p}}^{-2}}{\pi
k^{3}\mathrm{H}_{1}^{(1)}(k\overline{r})}\,\int_{0}^{2\pi}\overline{P}_{\mathrm{sc}}(\overline{r},\theta)\cos(\theta)\,\mathrm{d}\theta,$
(32b) $\displaystyle D_{\overline{y}}$
$\displaystyle=\frac{4\mathrm{i}c_{\mathrm{p}}^{-2}}{\pi
k^{3}\mathrm{H}_{1}^{(1)}(k\overline{r})}\,\int_{0}^{2\pi}\overline{P}_{\mathrm{sc}}(\overline{r},\theta)\sin(\theta)\,\mathrm{d}\theta,$
(32c)

where
$\overline{P}_{\mathrm{sc}}=\mathrm{i}\omega\rho\overline{\phi}_{\mathrm{sc}}$
in turn follows from the linearised form of Bernoulli’s law [22]. Thus, using
the dimensionless form of $\phi_{\mathrm{sc}}$ given in (17), we are able to
write (32) in the form

$\left[\begin{array}[]{c}M\\\\[5.0pt] D_{\overline{x}}\\\\[5.0pt]
D_{\overline{y}}\end{array}\right]=\left[\begin{array}[]{ccc}{\alpha}_{11}&{\alpha}_{12}&0\\\\[5.0pt]
{\alpha}_{21}&{\alpha}_{22}&0\\\\[5.0pt]
0&0&{\alpha}_{33}\end{array}\right]\left[\begin{array}[]{c}\overline{P}_{\mathrm{inc}}^{(0)}\\\\[5.0pt]
\bar{v}_{\mathrm{inc}}^{(0)}\\\\[5.0pt]
\bar{w}_{\mathrm{inc}}^{(0)}\end{array}\right],$ (33)

where

$\displaystyle{\alpha}_{11}$
$\displaystyle=\frac{4\mathrm{i}}{(c_{\mathrm{p}}k)^{2}}\left(\frac{2h_{\varepsilon}^{-1}}{\left[\pi
k\bar{b}H_{0}^{(1)\prime}(k\bar{b})\right]^{2}}-\frac{\mathrm{J}_{0}^{\prime}(k\bar{b})}{\mathrm{H}_{0}^{(1)\prime}(k\bar{b})}\right),\quad{\alpha}_{12}=-\frac{16\rho
h_{\varepsilon}^{-1}}{\pi^{2}c_{\mathrm{p}}k^{4}\bar{b}^{2}\mathrm{H}_{0}^{(1)\prime}(k\bar{b})\mathrm{H}_{1}^{(1)\prime}(k\bar{b})},$
(34a) $\displaystyle{\alpha}_{21}$
$\displaystyle=\frac{16\mathrm{i}h_{\varepsilon}^{-1}}{\pi^{2}k^{5}\bar{b}^{2}c_{\mathrm{p}}^{2}\mathrm{H}_{0}^{(1)\prime}(k\bar{b})\mathrm{H}_{1}^{(1)\prime}(k\bar{b})},\quad{\alpha}_{22}=\frac{8\rho}{k^{3}c_{\mathrm{p}}}\frac{\mathrm{J}_{1}^{\prime}(k\bar{b})}{\mathrm{H}_{1}^{(1)\prime}(k\bar{b})}-\frac{32\rho
h_{\varepsilon}^{-1}}{c_{\mathrm{p}}k^{3}\left[\pi
k\bar{b}\mathrm{H}_{1}^{(1)\prime}(k\bar{b})\right]^{2}},$ (34b)
$\displaystyle{\alpha}_{33}$
$\displaystyle=\frac{8\rho}{k^{3}c_{\mathrm{p}}}\frac{\mathrm{J}_{1}^{\prime}(k\bar{b})}{\mathrm{H}_{1}^{(1)\prime}(k\bar{b})},$
(34c)

with
$\overline{P}_{\mathrm{inc}}^{(0)}=\overline{P}_{\mathrm{inc}}(\overline{r}=0)=\mathrm{i}kc_{\mathrm{p}}\rho$
denoting the incident pressure, and the vector
$(\bar{v}_{\mathrm{inc}}^{(0)},\bar{w}_{\mathrm{inc}}^{(0)})=(\partial_{\overline{x}}\overline{\phi}_{\mathrm{inc}}|_{\overline{r}=0},\partial_{\overline{y}}\overline{\phi}_{\mathrm{inc}}|_{\overline{r}=0})$
representing the incident velocity, at the origin. In the limit as
$b\rightarrow 0$ we find that

$\displaystyle{\alpha}_{11}$
$\displaystyle\sim\frac{2\mathrm{i}}{(kc_{\mathrm{p}})^{2}h_{\varepsilon}}\left[-1+(k\bar{b})^{2}\left(\frac{1}{2}-\gamma_{\mathrm{e}}+\frac{\mathrm{i}\pi}{2}-\log\left(\frac{k\bar{b}}{2}\right)\right)\right]-\frac{\pi\mathrm{i}(k\bar{b})^{2}}{4},\quad{\alpha}_{12}\sim\frac{4\rho\bar{b}}{c_{\mathrm{p}}kh_{\varepsilon}},$
(35a) $\displaystyle{\alpha}_{21}$
$\displaystyle\sim-\frac{4\mathrm{i}\bar{b}}{c_{\mathrm{p}}^{2}k^{2}h_{\varepsilon}},\quad{\alpha}_{22}\sim-\frac{2\pi\mathrm{i}\rho\bar{b}^{2}}{c_{\mathrm{p}}k}+\frac{8\rho\bar{b}^{2}}{c_{\mathrm{p}}kh_{\varepsilon}},\quad{\alpha}_{33}\sim-\frac{2\pi\mathrm{i}\rho\bar{b}^{2}}{c_{\mathrm{p}}k}.$
(35b)

Thus, if we were to describe our scatterer in terms of an ideal point-source
monopole and dipole placed at the origin, we would find that the presence of
the aperture (i.e., the formation of a resonator) can be understood to cause
these amplitudes to depend upon both the incident pressure and the incident
velocity defined at the origin. Such cross-coupling is not observed in the
analogous expressions for non-resonant cylinders: this is readily seen by
considering the aperture closing limit $\varepsilon\rightarrow 0$ where
$h_{\varepsilon}\rightarrow-\mathrm{i}\infty$ in (34) which then recovers
established results [14, S59,S64]; in fact, the matrix in (33) is strictly
diagonal with ${\alpha}_{12}={\alpha}_{21}=0$ and
${\alpha}_{22}={\alpha}_{33}$ for a closed Neumann cylinder.

(a) Subfigure 6 list of figures text

(b) Subfigure 1 list of figures text

Figure 7: Reference calculations for the configuration considered in Melnikov
et al. [14]: LABEL:sub@fig:figq1 the scattered potential $\phi_{\mathrm{sc}}$
as defined in (5) at $k=k_{\mathrm{H}}=40.3589$ m-1 and LABEL:sub@fig:figq2
the (lossless) scattering efficiency $Q_{\mathrm{sc}}$ (defined from (21)). In
both figures we use $\bar{b}=20$ mm, $2\bar{\ell}=12$ mm,
$h=2\bar{\mathbcal{m}}/2\bar{\ell}=5/6$, and $\theta_{\mathrm{inc}}=\pi$; the
result for a single closed Neumann cylinder of the same radius is superposed
for reference (dashed red). Inset figure: resonator geometry (not to scale).

For the purposes of comparison, we present the scattered field profile
$\phi_{\mathrm{sc}}$ and scattering efficiency $Q_{\mathrm{sc}}$ for the
resonator geometry considered in Melnikov et al. [14] in Figure 7. Here, we
observe relatively strong enhancements in both the scattered field and
scattering efficiency at the Helmholtz resonance $k_{\mathrm{H}}$, as
expected, and only comment that there could be room for obtaining a stronger
response through an optimisation search of the type considered in Figure 3d.

Despite the emergence of off-diagonal terms in the polarisability tensor,
earlier work by the authors and others [13, 28], has found that the dispersion
equation for a two-dimensional square array of cylindrical Helmholtz
resonators possesses the usual structure for linear media,
$\bar{k}_{\mathrm{B}}^{2}\rho_{\mathrm{eff}}^{-1}-\omega^{2}B_{\mathrm{eff}}^{-1}=0$
at low frequencies, where the subscript eff denotes effective quantities and
$\bar{\mathbf{k}}_{\mathrm{B}}$ denotes the dimensional Bloch vector. An
example of a Willis coupling form for an infinitely extending medium is given
in [26, Eq. (2.68)], which along an arbitrarily chosen high symmetry direction
may take the form

$\bar{k}_{\mathrm{B}}^{2}-\omega(\zeta_{\mathrm{eff}}+\xi_{\mathrm{eff}})\bar{k}_{\mathrm{B}}-\omega^{2}(\rho_{\mathrm{eff}}B_{\mathrm{eff}}^{-1}-\xi_{\mathrm{eff}}\zeta_{\mathrm{eff}})=0,$
(36)

where $\zeta_{\mathrm{eff}}$ and $\xi_{\mathrm{eff}}$ are Willis coupling
tensor elements. Upon introducing the conventional decomposition
$\zeta_{\mathrm{eff}}=\kappa_{\mathrm{eff}}+\mathrm{i}\chi_{\mathrm{eff}}$ and
$\xi_{\mathrm{eff}}=\kappa_{\mathrm{eff}}-\mathrm{i}\chi_{\mathrm{eff}}$,
where $\kappa_{\mathrm{eff}}$ and $\chi_{\mathrm{eff}}$ are the reciprocity
and chirality parameters, respectively [29], the fact that the resonator
geometry itself has at least one mirror plane of symmetry (i.e., is nonchiral)
means that $\chi_{\mathrm{eff}}\equiv 0$. Similarly, if reciprocity were not
preserved $\kappa_{\mathrm{eff}}\neq 0$, then a linear term in
$\bar{k}_{\mathrm{B}}$ would be present in (36). Since the effective
dispersion equation corresponding to a two-dimensional array of Helmholtz
resonators does not possess linear $\bar{k}_{\mathrm{B}}$ terms at low
frequencies [13], and the resonator is nonchiral, we deduce that Willis
coupling is therefore absent in bulk. Hence, it is unclear how the Willis-like
coupling observed in the polarisability tensor (33) for a single resonator
implies the presence of Willis coupling effects in bulk, as is implied
elsewhere, and would therefore encourage the use of alternative nomenclature
(i.e., Willis-like depolarisability) for this behaviour. For reference, should
$\kappa_{\mathrm{eff}}=0$ then (36) takes the form

$\bar{k}_{\mathrm{B}}^{2}-\omega^{2}(\rho_{\mathrm{eff}}B_{\mathrm{eff}}^{-1}-\chi_{\mathrm{eff}}^{2})=0,$
(37)

from which it is clear how Willis coupling coefficients could be misattributed
to the effective phase velocity; since our resonator geometry is nonchiral
however we discount that possibility here.

## Acknowledgements

I.D.A. gratefully acknowledges support from the Royal Society for an Industry
Fellowship with Thales UK. This work was also supported by EPSRC grant no
EP/R014604/1 whilst I.D.A. held the position of Director of the Isaac Newton
Institute Cambridge. The authors acknowledge, with thanks, the invaluable
assistance of R. Assier of the University of Manchester in the development of
the matched asymptotic expansion methodology for problems of this class.

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