An analytical model for broadband sound transmission loss of a Finite Single
Leaf Wall using a two degree of freedom resonant Metamaterial
Javier Hernan VAZQUEZ TORRE(1) , Jonas BRUNSKOG(2) , Vicente CUTANDA HENRIQUEZ(3) ,...
(1) DTU, Denmark, jhevaz@elektro.dtu.dk(2) DTU, Denmark, jbr@elektro.dtu.dk(3) DTU, Denmark, vcuhe@elektro.dtu.dk

Abstract
Acoustic metamaterials (AM) have emerged as an academic discipline within the last decade. Metamaterials can
exhibit high transmission loss at low frequencies despite having low mass per unit area. This paper investigates
the possibility of using AMs for increasing the sound insulation of finite single leaf walls (SLW), focusing
on the coincidence effect problem. Formulas are derived using a variational technique for the forced soundtransmission of finite SLW with a coupled array of two degree of freedom resonators. An analytical model ispresented and the effects of the band gap in sound transmission and radiation are analyzed and compared tothe single degree of freedom case. Moreover, numerical simulations verify the two degree of freedom model.
Finally, some conclusions are drawn regarding the effectiveness of the proposed model, possible applications,and future work.
Keywords: Sound Insulation, Transmission, Metamaterial

1

INTRODUCTION

The importance of sound insulation has increased in cities with the ever-growing population. Buildings are incloser proximity to each other. Also growing number of vehicles has given rise to noise pollution in cities,
which in-turn has necessitated sound insulation in buildings. In offices, it is essential to keep the noise levelat the minimum to enhance employee efficiency [1, 2, 3]. In schools, audio comfort is one of the primaryconditions necessary for an effective learning environment.[4].
Acoustic metamaterials have emerged as an academic discipline within the last decade. The definition may
be broadly interpreted as systems or materials that display (as a whole) extraordinary properties not found innatural materials with respect to sound and vibration characteristics, such as negative apparent mass and/or bulk
modulus. Metamaterials can show high transmission loss (TL) at low frequencies despite having low mass
per unit area [8]-[11]. They owe this behavior to internal subwavelength periodic structures. One of the mostimportant characteristics of the AM is the so-called band gaps (BG), a frequency region where wave propagation
is not possible. This property shows great promise to be a good tool to be used in sound insulation, absorption,and even radiation. Sound insulation of walls in buildings or vehicles is a broadband problem, and for a single
homogeneous structure, the sound insulation is mainly given by the mass per unit area of the wall, which leaves
not much room for improvement [6, 7].
This paper investigates the possibility of using AMs for increasing the sound insulation of single leaf walls,
focusing on the coincidence effect problem. The approach utilized in this paper is the same as Brunskog’s
when investigating the forced sound transmission of single leaf walls using a variational technique [14] anda continuation of the authors work in [21]. In the present research formulas are derived for the forced soundtransmission of a finite single leaf wall with a coupled array of two degree of freedom resonators. An analyticalmodel is presented for this case, and the effects of the band gap in sound transmission and radiation are
analyzed. The developed model is restricted to the low frequency range where the wavelengths of the wall ismuch longer than the periodic distance of the resonators. Numerical simulations are carried out to check the
analytical model results and also parameter optimization is done. Moreover, a comparison between the SDOF

2091

Figure 1. (a) A finite wall of dimensions a × b coupled with a series of mass-spring resonators located inside arigid baffle in the x − y plane, at z = 0. (b) Simplified diagram of a small section
and the 2DOF is presented and conclusions drawn.

2

THEORYThis section will present the theory utilized in this paper. The variational formulation of the problem usedthroughout this study is based on Brunskog’s work [14] and is extended for the case described in the following
section.
2.1 Problem description
Consider a finite thin plate with mass per unit area m00p lying in the x − y plane coupled with periodicallyattached resonators as seen in figure 1a. The plate is located inside a rigid baffle at z = 0. For z < 0 theacoustic field consists of an incident plane wave, a reflected plane wave, and one scattered field due to the
motion of the finite wall. On the positive side of z only the transmitted wave is present (pt ). The resonators
have mass m00r , stiffness s00 , and mass moment of inertia jr00 , all per unit area. Structural damping of the spring is
considered by assigning the inherent losses to the spring element. For harmonic motion this can be represented
by a complex stiffness s00 = s00 (1 + iηs ) where ηs is the damping loss factor and s00 is the real part of thecomplex spring constant. The transverse displacement of the resonators is wr and the plate w p . The distance toa reference point is l p . It is of interest to analyze the transmission through this structure and the influence ofthe resonators.
2.2 Model Development
The following expression can be developed to describe the model4∇ wp − wp

kb4 +l 2p Jr00 ω 2 s00 (1 + iηs )m00r ω 2 s00 (1 + iηs )
2
−
k
x
2
B0 [m00r (ω0 − ω 2 ) + iηs s00 ]B0 [Jr00 (ωθ2 − ω 2 ) + l 2p iηs s00 ]

!
= 0.

(1)

This modified Helmholtz equation is the same as the one developed for the SDOF resonators case [21] with an
extra term now including the rotation of the mass of the resonators. For simplicity this approximate
expressionp
was developed considering q
rotation only around the x axis. The natural resonance is ω0 = s00 /m00 and theangular resonance is ωθ =

2092

00 00
l 002
p s / jr . From Eq 1, it can be seen that the modified wavenumber isskmod ≈ 4 kb4 +l 2p Jr00 ω 2 s00 (1 + iηs )m00r ω 2 s00 (1 + iηs )2−k,x 0 00B0 [m00r (ω02 − ω 2 ) + iηs s00 ]B [Jr (ωθ2 − ω 2 ) + l 2p iηs s00 ](2)

2.3 The wall impedanceRe-writing and developing Eq. 1 it is possible to get an expression for the wall impedance. It can be interpreted
as the impedance operator of the plate plus a term controlled by the resonators. Assuming that the travelingwave is in the form of e−i(kx X+kyY ) , kx and ky being wavenumbers, the wall impedance is reduced to
2
B0 2
z≈
kx + ky2 + iωm00p −iω

l 2p Jr00 ωs00 (1 + iηs )m00r ωs00 (1 + iηs )
2
−
k
x
2
m00r i(ω0 − ω 2 ) − ηs s00
Jr00 i(ωθ2 − ω 2 ) − l 2p ηs s00

!
,

(3)

In order to include the losses in the plate, the loss factor η p is included as an imaginary part of the Young’s
modulus E −→ E(1 + iη p ) and by that also to the bending stiffness B0 −→ B0 (1 + iη p ).
2.4 Radiation impedance
The formula of the radiation impedance of a finite plate utilized in this paper was derived and explained in aprevious study [14], so here it is only presented for clarity
ik
zf =
2πS

√
2
2
e−ik κ +%4 cos(kµx κ) cos(kµy %) p
× (a − κ)(b − %)dκd%0
κ 2 + %2

Z aZ b0(4)where a and b are the dimensions of the plate in the x and y direction, S is the area of the plate (S = ab), k
is the wavenumber on air, µx = sin(θ ) cos(ϕ), and µy = sin(θ ) sin(ϕ), θ is the evaluation angle and ϕ is theazimuth angle.
2.5 Effective mass
It is of interest to develop an expression of the effective mass (also referred to as apparent mass) of the proposedmodel. It is straightforward to do so from the wall impedance
m00e f f ≈l 2p s00 (1 + iηs )iJr00
im00 s00 (1 + iηs )2m00p + 00 r2−kx
imr (ω0 − ω 2 ) − ηs s00
Jr00 i(ωθ2 − ω 2 ) − l 2p sηs!
(5)

An approximate expression can be found if losses in the spring are neglected
m00e f f ≈m00p +

l 2p s00
s00
2
−
k
x
ω02 − ω 2
ωθ2 − ω 2!
(6)

In this form it is much easier to understand the behavior of the effective mass. For low frequencies the mass ofthe plate and resonators are added. This means that, in this frequency region, the proposed model is effectively
working as a wall with mass equal to the sum of the plate and resonators. If this frequency region is also
below the angular resonance, the mass moment of inertia is subtracted making the effective mass smaller. This
expression show a complex interaction between the resonances, even in this simplified model.
2.6 Diffuse field transmission
The same approach utilized in Brunskog’s paper [14] will be used in this study. For the sake of simplicity, thederivation of the equations will not be repeated here. The diffuse field transmission using Paris formula reads

2093

τ=

1
π

Z 2π Z π/2004ρ 2 c2 ℜ z f }sin θ dθ dϕ|z + 2ρcz f |2(7)

where ρ is the density of air, c is the speed of sound in air, θ is the incidence angle, ϕ is the azimuth angle,z f is the radiation impedance and z is the wall impedance, equation 4 and 3 respectively in this paper. It is
assumed that the resonators do not contribute to the sound radiation. In the following sections transmission loss
(TL) will be used to analyze the proposed model. It is defined as
R = 10 log

3

1
τ

(8)

RESULTS AND ANALYSIS

A test case will be presented in this section to analyze the analytical model and the sound insulation behaviorof the proposed structure. A brick wall with dimension 2 × 3 × 0.05m is used. Material properties are shown in
Table 1. The stiffness of the springs are selected specifically to tune the resonant frequency of the resonators
above the coincidence frequency of the plate. As known from previous research [21], this is the optimized
design in terms of sound insulation. The mass ratio M is one. The mass moment of inertia and the distance tothe reference point are selected to tune the angular resonance below the coincidence frequency.
Table 1. Material properties
Property
Young’s modulus
Density
Poisson’s ratio

Value
17 × 109 [Pa]2000[Kg./m3 ]0.2This case is considering losses in the springs and the plate. Losses are η = 0.03 for both. Results are presented
in Figure 2.
The effect of the angular resonance is barely visible as it has been damped. As expected the general behavior of the structure remains the same as the SDOF model. For frequencies above the band gap the TL ofthe metamaterial is lower than the one of just the plate. So there seems to be a trade-off between the two
approaches.

4

NUMERICAL SIMULATION

In order to validate the results of the analytical model a numerical simulation was carried out using Comsol
software. Solving the diffuse case numerically would take a long time and make the simulation harder to set
up, so a single incidence angle of θ = π/6 was chosen (more incident angles were tested but not shown forthe sake of brevity). The case analyzed in this test is a brick wall as used in section 3 but with dimensions
15 × 3 × 0.04m and losses η = 0.003 in order to be able see the angular resonance. The angular resonanceis tuned below the natural resonant frequency of the system. Results are shown in figure 3. The numericalresult seem to validate the analytical one. The mismatch seen in higher frequencies may be because of meshingissues. This is going to be investigated further in the future.

5

OPTIMIZATIONThe objective of optimization is to find a combination of mass ratio, spring loss factor,spring stiffness, mass
moment of inertia and the distance to the reference point that would provide the best possible sound insulation.

2094

Figure 2. Results for the case considering structural losses of the plate and springs a) wave speed, b) transmission loss, and c) effective mass. Vertical dashed line indicates BG limits of SDOF case and vertical dashed-dot
line the angular resonant frequency

Figure 3. Analytical and Comsol simulation for incidence angle θ = π/62095

Figure 4. Constrained parameter optimization TL results. Vertical dashed line indicates BG limits. Red verticaldashed-dot line indicates angular resonant frequency
However, defining what is better in terms of sound insulation is not a trivial task. ISO standard 717 [19] defines
single-number quantities for airborne sound insulation of building elements such as walls. This provides a solidbase to compare different solutions, at least if considering building structures.
5.1 Cost function
The weighted sound reduction index Rw [19] is the selected value to judge which solution is better. The costfunction used in this paper is

ϕ=

1
Rw

2
(9)

5.2 Constrained Optimization test case
The developed cost function is used to optimize the case presented in section 3. In order to keep the solutions
confined upper and lower bounds were used. The constraints and optimized values are shown in table 2.
Table 2. Constraints used for optimization and optimized values
Property
M
ηs
s00
Jr00
lp

Lower Bound
0.001
0.001
1 × 104
10 × 10−4
10 × 10−3

Upper Bound1
0.5
1 × 1010
1
40 × 10−3

Optimized Value
1
0.42553.477 × 109
0.59760.0254

A genetic algorithm was used for the optimization (Matlab’s “ga” function). In this way most permutations aretested, almost ensuring a good result at the cost of efficiency. At this time, efficiency is not a concern so thisis an acceptable drawback. Figure 4 presents the result of the optimization.

2096

The weighted sound reduction index of the optimized metamaterial solution is Rw = 56dB meanwhile the value
for the wall with equivalent mass is Rw = 46dB. Furthermore, the minimum TL value of the optimized structureis bigger than the minimum value of just the wall. The optimization process seems to have maximized thefrequency range where the transmission loss of the metamaterial is larger than the wall and tuned the BG above
the coincidence frequency as in the SDOF case. The angular resonance was tuned below the band gap. The
coincidence effect and angular resonance are smoothed by the losses in the springs and overall design. This
result is almost identical to the one reached for the SDOF [21]. In other words, no advantage can be seen in
the usage of this model. The feasibility of the optimized values shown in Table 2 are not explored in this study.
On the other hand, for frequencies above the band gap the TL of the wall is higher than the metamaterial.

6

DISCUSSION

The developed analytical model for sound transmission loss of a finite SLW with a coupled array of two degreeof freedom resonators was tested and analyzed. The behavior of the proposed metamaterial structure is on parwith the SDOF model presented in [21]. Around the frequency of the angular resonance there is a fast decrease
in TL followed by an increase, explained by the effective mass. For frequencies above the BG, the mass of theresonators and the plate are moving out of phase, while below they are in phase. If the frequency region below
the BG is also below the angular resonance, the mass moment of inertia is subtracted making the effective mass
smaller. This model show a complex interaction between the resonances, even in this simplified model. The
developed model is restricted to the frequency range where the wavelengths of the vibrations traveling throughthe wall are much longer than the periodic distance between the resonators. This provides a limitation when
designing these types of structures.
The numerical simulation presented in section 5 is an initial step towards the validation of the analytical result.
Further analysis has to be done, including a comparison with experimental data.
Furthermore, it is of interest to discuss the feasibility of the proposed structure as a tool to increase soundinsulation at constant mass per unit area, paying special attention to the coincidence effect problem. The angular resonance does not seem to provide an advantage in terms of sound insulation. The optimization process
finished with an equivalent result to the one obtained for the SDOF case in [21]. It is worth noting that the approximation presented in this study is intended to illustrate the behavior and influence of the angular resonancein sound insulation of this type of structure in an easier manner. The real case would be much more complex
and harder to analyze. In future work it would be interesting to validate the analytical model with experimental
data. In order to create and analyze more complex designs it is pertinent to explore the possibilities brought bynumerical methods techniques.

7

CONCLUSIONS

The analytical model developed in this paper is useful to better understand metamaterials composed of two
degree of freedom resonators and how the different parameters affect their behavior. The proposed structure
is an effective sound insulator but present not perceivable advantage over the SDOF case. The possibility ofusing springs with high losses and tuning the band gap above the coincidence frequency still emerges from theoptimization process as a possible solution to this phenomenon, rendering the effect of the angular resonancealmost null.

REFERENCES[1] Landström, U. Ventilation noise and its effects on annoyance and performance, J.Acoust. Soc. Am., 115(5),2370(A), 2004.
[2] Bowden, E.; Wang, L.M. Relating human productivity and annoyance to indoor noise criteria systems: a2097

low frequency analysis, The 2005 ASHRAE Winter Meeting Transactions, Orlando, 111, pt. 1, 684-692.
2005.
[3] Persson, W.; Rylander, K. Effects on performance and work quality due to low frequency ventilation noise,
J. Sound Vib, 205, 467-474, 1997.
[4] Shield, B. The effects of environmental and classroom noise on the academic attainments of primary school
children, The Journal of the Acoustical Society of America 123, 133, 2008.
[5] Zuccherini, N; et al. Analysis of Direct and Flanking Sound Transmission between Rooms with Curtain
Wall Facades, 6th International Building Physics Conference, IBPC 2015.
[6] Kinsler, L; et al. Fundamentals of acoustics, 4 th ed, New York: John Wiley & Sons, 2000.
[7] Bies, D.; Hansen, C. Engineering Noise Control: Theory and practice, 4 th ed, London and New York,
Spon Press, 2009.
[8] Lu, K.; et al. A lightweight low-frequency sound insulation membrane type acoustic metamaterial, AIP Adv,
6:025116, 2016.
[9] Shengming, L.; et al. Enhanced transmission loss in acoustic materials with micromembranes, App. Acoustics 130, 92-98, 2018.
[10] Jimenez, N.; et al. Ultrathin metamaterial for perfect and quasi-omnidirectional sound absorption, Appl.
Phys. Lett., 109,121902. 2016.
[11] Wang, X.; et al. Acoustic perfect absorption and broadband insulation achieved by double-zero metamaterials, Appl. Phys. Lett, 112, 021901, 2018.
[12] Brunskog. J. A Wave Approach to Structural Transmission Loss in Periodic Structures: Thin Beam Case.
Acta Acustica united with Acustica, Vol 91, 91-102,2005.
[13] Brunskog J. Near-Periodicity in Acoustically Excited Stiffened Plates and its Influence on Vibration, Radiation and sound Insulation. Acta Acustica united with Acustica, Vol 90, 301-312, 2004.
[14] Brunskog J. The forced sound transmission of finite single leaf walls using a variational technique, J.
Acoust. Soc. Am., Vol. 132, No. 3, 2012.
[15] Claeys, C.; et al. On the potential of tuned resonators to obtain low-frequency vibrational stop bands inperiodic panels, Journal of Sound and Vibration 332, 1418-1436, 2013.
[16] Claeys, C.; et al. On the acoustic radiation efficiency of local resonance based stop band materials, Journalof Sound and Vibration 333, 3203-3213, 2014.
[17] Van Belle, L.; et al. Sound transmission loss of a locally resonant metamaterial using the hybrid wave
based – finite element unit cell method, 11th International Congress on Engineered Material Platforms forNovel Wave Phenomena, France, 2017.
[18] Melo Filho, N. et al. Dynamic mass based sound transmission loss prediction of vibrio-acoustic metamaterial double panels applied to the mass-air-mass resonance, Journal of Sound and Vibration 442, 28-44,
2019.
[19] ISO 717-1:2013(E). Acoustics-Rating of sound insulation in buildings and of building elements.
[20] Cremer L.; et al. Structure-Borne Sound, Springer-Verlag, 544-547, Berling, 1988.
[21] Vazquez J.; et al. An Analytical Model for Broadband Sound Transmission Loss of a Finite Single Leaf
Wall using a Metamaterial, Proceedings of internoise 2019, Madrid, 2019.

2098