The use of a serial system of cylindrical and concentric Helmholtz resonators
for Low Frequency Sound Absorption
Alexander DELL(1) , Anton KRYNKIN(2)(1) University of Sheffield, United Kingdom, AJDell1@Sheffield.ac.uk(2) University of Sheffield, United Kingdom, a.krynkin@sheffield.ac.uk

01/06/2018
Abstract
A theoretical and numerical study of sound propagation in a serial system of cylindrical and concentric Helmholtz
resonators is conducted. By altering the geometry of each subsequent Helmholtz resonator it is possible to generate very low frequency resonances. Using the transfer matrix method an analytical formulation for the acoustic
absorptive properties of the this serial configuration is proposed. An optimization method is developed to tune
the system for the absorption of sound waves at specific frequency ranges. Through this study it is shown thatusing an optimized configuration of Helmholtz resonators in the low frequency range, sound absorption canbe achieved at sub-wavelength sample thicknesses. The results of the analytical study are validated throughnumerical means.
Keywords: Helmholtz resonator, Absorption, Optimisation

1INTRODUCTION

Typically used sound absorbers consist of bulky porous/ fibrous materials which is often impractical for low
frequency sound absorption due to the mass-density law [2]. To overcome this, acoustic metamaterials consisting of Helmholtz resonators are have been utilised to achieve broadband sound absorption at sub-wavelength
thicknesses [3]. Of particular relevance to this paper is the extensive research that has been undertaken on cylindrical and concentric Helmholtz resonators using the Transfer Matrix Method (TMM) [5, 9, 7, 6], highlighting
how this relatively simple analytical method can be a useful tool for the analysis and optimisation of various
configurations of Helmholtz resonators.
This report sets out a theoretical model to analyse the absorption characteristics for a serial system of N cylindrical, concentric Helmholtz resonators, extending the work conducted on duel resonators in [10]. This also
provides a simple and computationally inexpensive method in which optimisation can then be implemented tomaximise the absorption at desired frequencies. A comparison is then made between the results obtained analytically and those produced by numerical analysis for validation. This is done for the case of three serial
Helmholtz resonators.

2GEOMETRY

An axisymmetric view of the studied geometry can be seen in figure 1. Here it is evidenced that the geometry
consists of a waveguide in which the end is loaded with a serial system of Helmholtz resonators. These areplaced end-to-end and the Nth resonator cavity results in velocity termination. The plane wave propagates fromleft to right. It is worth noting that Rt denotes the radius of the waveguide, Rni denotes the neck radius for theith HR and Rci denotes the cavity radius for the ith HR. Lni and Lci denote the length of the neck and cavity
for the ith HR, respectively.

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Figure 1. Geometry for seriel system of N Helmholtz resonators

3THEORETICAL MODEL

3.1 Transfer Matrix Method
The theoretical modelling is performed by using the transfer matrix method, which relates the initial soundpressure, p, and normal acoustic particle velocity, v, at the start (x=0) and end (x=L) of the system of Helmholtz
resonators. The transfer matrix, T, is derived under the assumption that only plane waves propagate in thesystem. From this the reflection and transmission coefficients can be calculated. The system is written ash ih ihpp
T=T= 11v x=ov x=LT21

T12T22ih ipv x=L(1)Where T is the product of the transfer matrices for each section of each HR within the system,Ni
T = ∏ Mni M∆lMci

(2)

i=1

The transmission matrix for the ith HR neck and cavity take the following forms, respectively.
"Mni =#cos(kni Lni ) iZni sin(kni Lni )isin(kni Lni )cos(kni Lni )Zi(3)#cos(kci Lci ) iZci sin(kci Lci )isin(kci Lci )cos(kci Lci )
Zi

(4)

n"
Mci =

ci =ω
Where kn,cri
ρn,ci
κn,ci = 1and Zn,cSin,cqi ρ i are the effective wavenumber and characteristic impedance of theκn,cn,ciii
ith HR neck or cavity, κn,cand ρn,c
are the effective bulk modulus and density, respectively, and Sn,cis thecross sectional area.
The transmission matrix that accounts for the end corrections of the neck of the ith HR is written as4481

iM∆l="10iZni kni ∆l
1#(5)Where ∆l is arrived at from the addition of two correction lengths ∆l = ∆l1 + ∆l2 written as!3 #rni
rni∆l1 = 0.82 1 − 1.35 i + 0.31 i
rni
rcrc

(6)

∆l2 = 0.6rni(7)

The first length correction, ∆l1 , is due to pressure radiation at the discontinuity from the neck to the cavityof the HR [4], while the second length correction, ∆l2 , comes from the radiation at the discontinuity from theneck to the surrounding medium [1].
To determine the entrance impedance for a system of resonators you multiply the final T matrix by [1, 0]T which
accounts for the velocity termination at the end of the final resonator. From this the impedance can simply befound as follows
Z=T11
Px=0
=
vx=oT21(8)Similarly, the reflection coefficient is also determined by the final Transmission matrix and can be found using
the following expression
R=T11 − T21 Z0
T11 + T21 Z0(9)from which the absorption coefficient can be computed by
α = 1 − |R|2(10)3.2 Visco-thermal Losses
The propagation of acoustic plane waves in a circular cross section of radius r, with viscothermal losses, isgiven by the following complex frequency dependant density and bulk modulus, as described in [8].
ρ = ρ0κ = K02J1 (rGr )1−
rGr J0 (rGr )!

2J1 (rGk )1 + (γ − 1)
rGk J0 (rGk )

(11)
!
(12)

qq−iωρ0 Pr0Where Gr = −iωρandG=, in which ρ0 is the density, K0 = γP0 is the bulk modulus, γ is thekηηratio of specific heats, P0 is the atmospheric pressure, Pr is the Prandtl number and η is the dynamic viscosity.

4Optimisation

In order to achieve maximum absorption for specified frequencies, whilst adhering to geometrical constraints, amethod of optimisation was used to output the optimal dimensions required. This was implemented by first creating a square wave with a value of 1 around the target frequencies. Then by limiting the absorption coefficient4482

function to the frequencies in which the square wave is 1, the optimal dimensions are outputted by minimising
the following objective function.
Ob jective f unction = area(square wave) − area(α)(13)

Using this method for target frequencies of 100Hz, 300Hz and 500Hz with a maximum radius of 50mm for thewaveguide and cavities (noting that the radius of the necks is always less) the following optimised absorption
coefficient for three nested Helmholtz resonators is produced. The total length of this system is 7.05cm which
equates to a sample thickness ratio of λ /49 at 100Hz, where an absorption of 0.77 is achieved.

Figure 2. Optimised absorption coefficient for three Helmholtz resonators in series
It is worth noting that each subsequent resonator in the series is resonating at a lower frequency than the last.
This is achieved, in part, by the reduction in neck radius of each subsequent resonator. This is evidenced intable 1 which contains the optimised geometry for this specific case.
Table 1. Optimised geometry dimensions for serial array of three Helmholtz resonators
NRn (mm)
Rc (mm)Ln (mm)Lc (mm)

516.40
50.0
5.40
22.0

2
2.90
45.6
4.50
15.6

3
1.20
49.6
2.2
20.8

Numerical Validation

In order to provide numerical validation for the analytical method a model was produced in COMSOL multiphysics 5.0 using the dimensions obtained in the optimisation process. This utilises the finite element method(FEM) to calculate for the complex pressure within a specified geometry. The modelling of viscothermal losses
was achieved by using the ’circular duct’ model within the ’narrow regions’ feature of the acoustic module4483

within COMSOL. This was applied specifically to the neck regions of each HR. The absorption coefficient wasascertained using the two microphone method. The absorption coefficient for both the numerical and analyticalmethods can be seen in figure 3.

Figure 3. Numerical validation for the optimised absorption coefficient for three Helmholtz resonators in series
From figure 3 it is evident that there is excellent agreement in regards to the resonant frequency of eachHelmholtz resonator. However, there is evidently a discrepancy in regards to the amplitude of the absorption
peaks. By comparing the numerical results to the analytical it can be seen that the low frequency resonatoris over-performing, achieving perfect absorption, with the latter resonators underperforming. Whilst this is apositive results in regards to low frequency absorption it also highlights that the modelling of visco-thermal
losses is not completely consistent between the two methods. Regardless of this fact, the analytical model isstill validated but further experimental work would be required to fully determine how the interplay of losses
affects the absorption peaks.

6Conclusions and Future WorkIn this paper, a simple and effective analytical method of modelling a system of N cylindrical, concentric nested
Helmholtz resonators is outlined. Upon implementation into an optimization algorithm to maximise the absorption at specific frequencies given geometrical constraints, an analytical absorption coefficient of approximately
0.77 at 100Hz is obtained. With the overall thickness of the system being 7.05cm the 100Hz absorption occurs
at a sample thickness ratio of λ /49. Further large absorption peaks are obtained at the desired frequencies of300Hz and 500Hz. This shows it is possible to achieved low frequency sound absorption at sub-wavelength
sample thicknesses using this configuration.
Numerical validation highlights excellent agreement on the resonant frequencies of each resonator, proving theTMM matrix accounts for the coupling that occurs between resonators. However, there is some disagreementin the amplitude of the absorption peaks which points to minor inconsistencies in the modelling of the viscothermal losses. In the numerical model perfect absorption is obtained at 100Hz (an over-performance) and thetwo higher frequency resonators under-perform in regards to absorption. Overall the model is still valid but
further understanding of the losses is still required.
Further work needs to be conducted in providing experimental validation to the results attained here to solid-4484

ify the theory as correct. Additionally, more work can be done on understanding the mechanism behind thecoupling of Helmholtz resonators in this configuration, which could prove useful for low frequency sound absorption. Also, a larger system of resonators (3+) can be analysed and porous media introduced which could
lead to more broadband low frequency absorption instead of the individual resonator peaks seen here.

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