Sound absorption performance of underwater anechoic coating in plane wave normal incidence condition

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Sound absorption performance of underwater anechoic coating in plane
wave normal incidence condition
To cite this article: Z Zhang et al 2019 IOP Conf. Ser.: Mater. Sci. Eng. 522 012001
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2018 4th International Conference on Smart Material Research
IOP Conf. Series: Materials Science and Engineering 522 (2019) 012001
IOP Publishing
doi:10.1088/1757-899X/522/1/012001
1
Sound absorption performance of underwater anechoic
coating in plane wave normal incidence condition
Z Zhang, L Li, Y Huang and Q Huang*
State Key Laboratory of Digital Manufacturing Equipment and Technology, School of
Mechanical Science and Engineering, Huazhong University of Science and
Technology, Wuhan 430074, PR China
*Corresponding author - qbhuang@hust.edu.cn
Abstract. On account of the theory of non-uniform waveguide and the wave propagation
theory in layered media, a complete theoretical model is established for predicting the sound
absorption performance of multi-layer underwater composite anechoic coating. Furthermore,
the correctness of the theoretical calculation is verified by the finite-element simulation results.
Then, the velocity nephograms corresponding the three absorption peak frequencies of the
overburden are analyzed to reveal the sound-absorbing mechanism.
1. Introduction
With the increasing demand for acoustic performance of modern underwater equipment, some
measures related to vibration damping and noise reduction continue to emerge. Among them, the
installation of the underwater anechoic coating on the outer surface of the equipment is an important
move, which has the effects of reducing the acoustic target strength of structure itself, suppressing the
structural sound radiation, and improving the sound insulation of internal noise. Therefore, it is
necessary to study the sound absorption properties of the underwater anechoic overburden.
The research methods for the acoustic properties of sound-absorbing coatings vary depending on
their structural forms. For example, Stepanishen and Strozeski [1] analyzed the reflection-transmission
performance of a layered viscoelastic lining by transfer matrix method (TMM). Gao et al. [2] proposed
a Generalized Multiscale Finite-Element Method (GMsFEM) to study the wave propagation
characteristics in non-homogeneous coatings. Haberman et al. [3] and Bonfoh et al. [4] studied the
material and acoustic properties of periodically scattering composite sound-absorbing structures
containing spherical inclusions and ellipsoidal coated particles, respectively, through self-consistent
theory and generalized self-consistent theory. Tao [5] established a complete two-dimensional
theoretical model for predicting the acoustic performances of an overburden embracing periodic
cylindrical or conical cavities. In addition, for anechoic coatings with complex cavity structures such
as spherical, ellipsoidal, super-ellipsoid and non-spherical cavums, Ivansson [6] investigated the
sound-deadening effect by finite element method (FEM). Furthermore, Zhong et al. [7] also adopted
the FEM to explore the low-frequency sound-absorbing property of a viscoelastic overburden
embodying periodic horizontal cavities with infinitely long eccentric cylinders.
Therefore, this paper proposes a multi-layer underwater composite anechoic coating, which is
formed by embedding two heterogeneous linings containing periodic cavity structures in two
homogeneous plates. Meanwhile, we introduce the theory of non-uniform waveguide and combine the
wave propagation theory in layered media to investigate the sound absorption performance of the
present coating.

2018 4th International Conference on Smart Material Research
IOP Conf. Series: Materials Science and Engineering 522 (2019) 012001
IOP Publishing
doi:10.1088/1757-899X/522/1/012001
2
2. Theoretical modeling
2.1. Structural elaboration
The underwater anechoic coating shown in Fig. 1 consists of four rubber layers, labeled from top to
bottom as layers m with thicknesses hm (m = 1,2,3,4), respectively, wherein layers 1 and 4 are
homogenous linings, and layers 2 and 3 are heterogeneous overburdens containing periodic cavity
structures. The coating divides the service water environment into two upper and lower semi-infinite
spaces, namely the water and backing conditions in Fig. 1(a), marked as layers 0 and 5, respectively.
Moreover, the two diameters, small upper diameter, and large lower diameter, of the truncated cone
cavity in layer 2 are respectively r1 and r2, while the diameter of the cylindrical cavity in layer 3 is r2,
as illustrated in Fig. 1(b). In addition, the basic structure of the blue dotted square in Fig. 1(b) is a
periodic unit having the length-width dimension of lb. Furthermore, a global Cartesian coordinate
system is set up on the upper surface of the coating, where the x- and y-axes are along the array
directions of periodic cavities, and the z-axis is perpendicular to the xy-plane downward. When a plane
wave having a sound pressure amplitude of pi impinges on the upper surface of the overburden
normally, the reflected and transmitted waves with amplitudes pr and pt are derived respectively.
x
z
ywater
backing
pipr
pt
AA
(a)
r1
r2
A-A
(b)
x
z
y
lb
lb
unit cell
Figure 1. Schematic diagram of the present coating: (a) assembly drawing and (b) sectional view.
2.2. Matrix synthesis and absorption calculation
For a single homogeneous lining similar to layer 1 or 4, the relationship between sound pressure and
particle velocity on the front and rear interfaces is [8]:
   
 
1, 2, 2,
pre
1, 2, 2,
cos i sin
isin cos( )
m m m m m m
m m m
m
mm
m m m
mm
mm
k h c k h
p p p
A
kh
v v v
kh
c



     


     

     


, (1)
where subscript m represents the m-th layer in the entire overburden, while subscript 1 and 2 indicate
the front and rear ends of the coating; superscript pre denotes the pressure state; ρ, c, h, k, A are
respectively the density, complex sound velocity, thickness, complex wavenumber, and transfer matrix
of the homogeneous layer; i (i2 = -1) is the imaginary unit.
Thus, the transfer matrix of a homogeneous layer in the force state is

2018 4th International Conference on Smart Material Research
IOP Conf. Series: Materials Science and Engineering 522 (2019) 012001
IOP Publishing
doi:10.1088/1757-899X/522/1/012001
3
pre
m e m
A S A
, (2)
with
1
11
b
eb
s
Ss



, (3)
where symbol ⊗ indicates the Kronecker product between matrices,
2
bb
sl
is the sectional area of
one periodic element.
In layer 2 with periodic truncated cone cavities, the propagation of sound waves in the coating is
similar to that in a highly viscous liquid with variable cross-section waveguides, so the wave equation
is [9]
22
2
10
sk
s z z
z


  
  


, (4)
where ξ and s are the particle displacement and cross-sectional area of the coating, respectively, and
both are all function of z. Therefore, the expression is a nonlinear equation, which can be transformed
into a linear equation to obtain the analytic solution only when the medium cross-sectional area s(z)
satisfies the following formula:
 
2
ss

 
(μ = constant). (5)
Since the total pressure and particle velocity on the front and rear interfaces in the heterogeneous
layer 2 are continuous, the cone-shaped waveguide
 
s z az b
is used to approximate the
truncated cone cavity, and the transfer relations of F1, F2, v1, and v2 can be expressed as
2,11 2,12
12
2
2,21 2,22
12
bb
FF
B
bb
vv

   


   
   

, (6)
with
   
   
 
 
12
2,11 2 2 2 2
2 2 2 1
2
2 2 1 2 2 2 1 2 1
2,12 2 2 2
2 2 2 1 2 2 2
2
2,21 2 2
2 2 2 1 2
2
2,22 2 2
12
1
cos 1 sin
i cos 1
2 sin 1 1
isin
1
cos
ss
b K h K h
s K h s
c s s K h s s s
b K K h
k h s s K h s
k
b K h
K c s s
s
b K h
sK




  








   



     
   


   



   




 
122
22
1 sin
sKh
hs









, (7)
where
22
22
Kk


;
22
11b
s l r


and
22
22b
s l r


are the upper and lower interface
cross-sectional areas of layer, respectively.
However, for the inhomogeneous layer 3 containing periodical cavities, which is a special case of
layer 2, with
22
1 2 2b
s s l r

  
and K3 = k3, so the transfer matrix of sound pressure and particle
velocity on both end faces of layer 3 is
   
   
3 3 3 3 2 3 3
333 33
3 3 2
cos i sin
isin cos
k h c s k h
Bkh kh
cs







, (8)
Moreover, the global transfer matrix of the present coating in the force state can be assembled as
follows:

2018 4th International Conference on Smart Material Research
IOP Conf. Series: Materials Science and Engineering 522 (2019) 012001
IOP Publishing
doi:10.1088/1757-899X/522/1/012001
4
   
 
 
1 1 2 2 3 3 4 4
T A h B h B h A h   
. (9)
Then, the reflection and absorption coefficients at the incident interface of the coating are:
in w
in w
ZZ
rZZ


, (10)
1rr


  
, (11)
with
 
11 5 12 5
21 5 22 5
in b
T F T v
Zs T F T v


, (12)
where superscript * denotes the conjugate of the reflection coefficient, Zw = ρwcw is the characteristic
impedance of the aqueous medium in the 0-th layer.
3. Model validations and discussions
3.1. Simulation verification
The value of the present theoretical model is determined by both the calculation accuracy and
computation rate, in which the model validity is an indispensable basis. For the underwater anechoic
coating as shown in Fig. 1, the materials of layers 1 and 4 are selected as high-elastic rubber (HER),
with densities ρ1 = ρ4 = 1300 kg/m3, Young’s moduli E1 = E4 = 4.96×107(1+i) Pa, Poisson ratios υ1 = υ2
= 0.47, and thicknesses h1 = 0.5h4 = 15 mm. Moreover, the material properties of ordinary
conventional rubber (OCR) in layers 2 and 3 are ρ2 = ρ3 = 1600 kg/m3, E2 = E3 = 2.98×108(1+0.4i) Pa,
υ1 = υ2 = 0.37, h2 = 3h3 = 30 mm, and r1 = 0.5r2 = 6 mm. The density and sound velocity of water in
the incident medium are ρw = 1000 kg/m3 and cw = 1500 m/s, respectively, while the backing condition
is rigid.
Then, a fully coupled finite-element model of the present anechoic coating shown in Fig. 2 is
established by COMSOL software in accordance with the following four steps: first, using the
perfectly matched layer (PML) to simulate the semi-infinite water space. Second, employing the
Floquet periodicity boundaries on the x- and y-direction planes in one cell to elaborate an infinite
lining. Again, approximating the rigid backing by the fixed constraint boundary with vibration velocity
of zero. Finally, treating the fluid-solid interfaces between fluids such as water or air and rubbers by
the acoustic-structure boundaries.
(b)
z
xy
Fixed
constraint
Floquet
periodicity
PML
S
Water
HER
Air cavity
OCR
OCR
Air cavity
HER
(a)
Figure 2. Acoustic-structure fully coupled finite-element model of the present coating: (a)
multiphysics geometry model and (b) mesh model.

2018 4th International Conference on Smart Material Research
IOP Conf. Series: Materials Science and Engineering 522 (2019) 012001
IOP Publishing
doi:10.1088/1757-899X/522/1/012001
5
Therefore, based on the two methods of theoretical calculation and simulation analysis, the sound
absorption coefficients of the underwater anechoic overburden are compared as illustrated in Fig. 3.
Obviously, the trends of the two curves are consistent, and they all fluctuate with increasing frequency
and eventually become stable. In addition, the two highly coincident curves rise rapidly to 0.6, which
just shows that the present coating’s design is reasonable and the developed theoretical model has
higher accuracy.
Figure 3. Comparison between the theoretical analysis and simulation result of the present coating.
3.2. Sound-absorbing mechanism exploration
It can be seen from Fig. 3 that the two curves of theoretical calculation and simulation result have
three peak frequencies in the research frequency range of 10Hz ~ 10 kHz. In order to explore the
energy dissipation mechanisms for the absorbing peaks in the sound absorption curve of the present
overburden, the vibration velocity of the coating in the whole research frequency range is simulated
and analyzed. The velocity nephograms corresponding to the three peak frequencies are displayed in
Fig. 4.
Figure 4. Velocity nephograms of the present anechoic coating at three peak frequencies: (a) 464.16
Hz, (b) 3274.55 Hz and (c) 7564.63 Hz.
From this diagram, we can know that the mechanisms for generating the respective sound
absorption peaks are different. For the first peak frequency in the low-frequency band, the
sound-absorbing peak at 464.16 Hz is mainly attributable to the axial piston-like resonance energy
dissipation generated by the layer 1 coupling the truncated cone cavities in layer 2, as shown in Fig.
4(a), the velocity magnitudes of layers 1 and 2 are dominant. As can be seen from Fig. 4(b), the
vibration velocity magnitudes of the heterogeneous layers 2 and 3 are significantly concentrated and
superior to other layers. Therefore, the two sound-absorbing modes, the radial drum-like resonance
energy consumption of the periodic cavity structures in layers 2 and 3, and the waveform conversion
in layer 2, synergistically cause the second sound absorption crests at the frequency 3274.55 Hz in the

2018 4th International Conference on Smart Material Research
IOP Conf. Series: Materials Science and Engineering 522 (2019) 012001
IOP Publishing
doi:10.1088/1757-899X/522/1/012001
6
mid-frequency band. However, for the velocity contour at 7564.63 Hz in the high-frequency range
depicted in Fig. 4(c), a number of distinct velocity stratifications are formed within the anechoic
coating, while the interface between layer 1 and the aqueous medium is the most intense. This is
because the viscous energy dissipation in the anechoic coating plays a major role, so that the sound
absorption coefficient of the entire structure is stable in the high-frequency region.
4. Conclusions
This paper concerns with the sound absorption performance of an underwater composite anechoic
coating. It is composed of four stratified rubber structures, in which the upper and lower layers are
homogeneous linings, while the center two layers are heterogeneous overlays, each containing
periodically truncated cone cavums and cylindrical cavities. On account of the theory of non-uniform
waveguide and the wave propagation theory in layered media, a complete theoretical model is
established for analyzing the acoustic properties of the present coating.
Furthermore, the homogeneous layers 1 and 4 are selected to be high-elastic rubber, and the
ordinary conventional rubber is used in place of the non-homogeneous layers 2 and 3, thereby
completing the following two aspects of research work. One is to verify the correctness of the
developed theoretical model by finite element simulation method. The other is to study and analyze
the velocity nephograms corresponding to the three peak frequencies to explore the absorbing
mechanisms of the sound absorption crests. Thus, the following two main conclusions are drawn: first,
the present theoretical model has higher accuracy. Second, the sound-absorbing mechanisms for the
three absorption peaks of the underwater anechoic coating are different, respectively, by the axial
piston-like resonance energy dissipation, radial drum-like resonance energy consumption coupled
waveform conversion, viscous energy dissipation.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 51575201)
and the Fundamental Research Funds for the Central Universities, HUST, China (Grant no.
2018JYCXJJ039).
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