Sound Insulation in a Hollow Pipe with Subwavelength Thickness
Sound Insulation in a Hollow Pipe with Subwavelength Thickness
OPEN Suppression of the transmission of undesired sound in ducts is a fundamental issue with wide applications in a great variety of scenarios.Yet the conventional ways of duct noise control have to rely on mismatched impedance or viscous dissipation, leading the ducts to have ventilation capability weakened by inserted absorbers or a thick shell to accommodate bulky resonators. Here we present a mechanism for insulating sound transmission in a hollow pipe with subwavelength thickness, by directly reversing its propagating direction via anomalous reflection at the flat inner boundary with well-designed phase profile. A metamaterial-based implementation is demonstrated both in simulation and in experiment, verifying the theoretical prediction on high-efficient sound insulation at the desired frequencies by the resulting device, which has a shell as thin as 1/8 wavelength and an entirely open passage that maintains the continuity of the background medium. We have also investigated the potential of our scheme to work in broadband by simply cascading different metamaterial unit cells. Without the defects of blocked path and bulky size of existing sound insulators, we envision our design will open new route to sound insulation in ducts and have deep implication in practical applications such as designs of ventilation fans and vehicle silencers.
Elimination of the transmission of undesired noise is of fundamental significance and potential application in the
field of acoustics1?3. Among the numerous applications of sound insulation, duct noise control4?6 is of particular
interests from both physics and engineering points of view. The past several decades have witnessed
considerable efforts dedicated to the development of both active and passive methods for noise control. The methods of
active noise control7 need to introduce another sound source with antiphase, while passive methods1?6 provide a
simple and low cost solution requiring no power supply. Conventionally, the sound insulation in ducts has to rely
on reflection effect by mismatched impedance such as with the expansion-chamber-type muffler or by viscous
dissipation such as with fibrous duct lining in air distribution systems5. However, the insertion of porous media
in the sound path obviously impair the crucial ventilation function of the ducts and, on the other hand, the usage
of Helmholtz resonators8?11 or membranes4,6 may make them impractical for many important applications, esp.,
sound insulation in hollow and thin-shelled pipes with no space for decorating resonators with bulky size under
its inner surfaces. It is therefore of both physical and practical significance to explore new mechanisms producing
sound insulation to break through these limitations in conventional methods.
In this article, we propose to insulate acoustic transmission in hollow pipes by directly manipulating the
propagation direction with specially-engineered phase profile at the inner boundary, fundamentally different
from the traditional methods that depend on either impedance mismatch or viscous effect. This should be a
universal mechanism that applies to ducts with different geometries, as long as the profile of phase delay on the inner
boundary can be controlled as desired. The desired phase profile is produced by employing the recently-emerged
metasurfaces12?16 which have potentials to achieve novel phenomena such as extraordinary
refraction/reflection17?19, beam forming20?23, asymmetric transmission24,25, and vortices generation26. An acoustic metasurface
(AM) based implementation of our proposed scheme is demonstrated both numerically and experimentally.
Good agreement is observed between the numerical results and the experimental data, with both showing that
the designed insulator effectively blocks the transmission of incident wave within the prescribed frequency range
despite its flat shape and open configuration. We anticipate our finding to constitute a significant step beyond
the existing duct noise control methods that generally suffers from blocked sound path and bulky size, therefore
bringing promising applications in a great variety of practical scenarios ranging from ventilation fans to vehicle
silencers with need for open passage.
The basic idea of our proposed scheme is schematically shown in Fig.?1, which aims to bounce back the incident
wave in an intuitive manner. For simplicity without losing generality, we consider a two dimensional (2D) case.
Given that the proposed structure needs to keep open and contains no layered materials in the propagating path,
one has to rely on the interaction between the incident wave and the flat inner surfaces of the pipe. In this context,
the most straightforward way to reflect back the incident wave might be bending its direction of propagation
twice to form a U-shape trajectory as shown in Fig.?1(a) and (b) which display the desired trajectory for the wave
incident in the upper part of the pipe, due to the incapability of a flat surfaces to directly block the transmission of
wave travelling across it. However such a design apparently needs anomalous reflection at the inner boundaries
on both sides, which goes beyond the conventional concept of equivalent incident and reflected angles and has to
be governed by the generalized Snell?s law which is deduced by Fermat?s principle27?29. By engineering the
microstructure on flat surface to produce appropriate phase gradient, the direction of reflected wave can be predicted
by sin ?r = sin ?i + (?/2?)(d?/dx), where ?r(?i) is the reflected (incident) angle, ? is the wavelength of sound
wave and d?/dx is the phase gradient. Note that in our design the propagation direction of incident wave needs to
be bent by approximately ?/2 when travelling along the two boundaries. By making the phase gradient term equal
to ?k with k being the wave vector of incident wave, the reflected angle can be controlled to be nearly zero,
leading the reflected wave to impinge on the opposite boundary normally. On the opposite surface with the same
phase profile, the reflected wave would be bent again to propagate along a direction almost reverse to the original
incidence direction due to the symmetry of the structure, preventing the incident acoustic energy from passing
through the system. Such theoretical analysis predicts that by properly modulating the reflection phase profile on
the inner surface of its shell, regardless of the thickness of the shell itself, the proposed structure would have the
potential to yield high-efficient sound insulation. Typical numerical results in Fig.?1(c) and (d) show respectively
the scattered sound pressure fields yielded by the two opposite boundaries with the aforementioned phase
profiles, which are mimicked by two arrays of unit cells characterized by ideal effective acoustical parameters
(marked by black boxes in the figures). Here the phase distribution along the two boundaries is discretized for
facilitating the experimental realization which will be demonstrated later. The results verify the occurrence of the
expected anomalous reflections on the two boundaries that eventually bend the wave vector of the incident wave
approximately to the reverse of the original direction. This suggests the potential of the combination of the two
boundaries to serve as an effective sound insulator for blocking the incident wave, as will be proven both
theoretically and experimentally later.
In what follows we will demonstrate a practical implementation of our scheme by employing the
recently-emerged AMs capable of providing discrete phase with thickness much smaller than the working
wavelength. A basic building block of the designed acoustic metasurface is shown in Fig.?2(a), which is constructed
by four identical thin rigid plates in air (length d1 and width w1), leaving the space with a width of w2 between
the plates as a channel for effectively delaying the propagation of sound inside it30. It should be noted that, since
acoustic waves, as scalar waves, can propagate within the channels freely, the delayed phase of reflected waves on
the metasurface can be retrieved. Dependence of reflected wave phase on the width d1 is plotted in Fig.?2(b) with
red solid line. Despite the subwavelength scale of the overall thickness of such a labyrinthine unit which is chosen
as l = 0.128? in this study, a sufficient phase shift can be achieved by coiling up space. In our design, the structural
parameters are chosen as w1 = 0.1 cm, l = 1 cm and d = 1 cm. Through carefully selecting the values of d1 for the
eight units to cover the 2? span with a discrete phase step of ?/4, as marked by the eight black dots in Fig.?2(b),
the desired phase gradient can be produced on the metasurface. To further verify the ability of the designed AM
to produce discrete phase shifts, the reflected waves by these eight units are shown in Fig.?2(c). The stripes refer to
the pressure filed patterns at the same time instant. Here the x axis represents the reflected phase delayed by the
eight different unit cells which are arranged along the y axis while the z axis is the meaning of pressure amplitude.
The peak of the pressure field can shift up to a wavelength, making it possible to give rise to the desired discrete
phase shifts by the eight units.
Then we numerically calculated the distribution of acoustic pressure of the scattered wave generated by an
AM implemented by the above-mentioned labyrinthine metastructures, whose structural parameters are well
tuned so as to yield specific phase profiles shown in Fig.?1. Simulated results for two cases of grazing incidence
and normal incidence, shown in Fig.?3(a) and (b) respectively, are almost identical with the results obtained for
an ideal model displayed in Fig.?1, showing that the metamaterial-based AMs produce anomalous reflection
exactly along the predicted direction. As a result, by arranging such unit cells periodically in a row to realize the
required phase distribution, we form a hollow pipe capable to reflect back most of the incident acoustic energy as
expected. Figure?3(c) demonstrates the simulated acoustic pressure field in the pipe at the frequency of 4346 Hz.
A comparison between the transmissions as functions of frequency for systems with different periods is shown in
Fig.?3(d), showing the possibility to further improve the performance of the insulator through simply increasing
the period numbers, but at the cost of enlarging the length of the whole structure, as will be demonstrated later.
Experiments are designed and carried out to verify the realization of sound insulation in the proposed hollow
pipe with shells much thinner than the operating wavelength. A sample with four periods of labyrinthine units
on each side is fabricated with acrylonitrile-butadiene-styrene (ABS) plastic via three dimensional (3D)
printing technique (Stratasys Dimension Elite, 0.177 mm in precision) as shown in Fig.?4(a). In both the simulation
and experiment, the transmissions are obtained by integrating the sound power along the cross-section of the
waveguide structure. Figure?4(c) displays the experimental results of transmission coefficients of the proposed
hollow pipe. Numerical simulations are also performed for a quantitative comparison, with corresponding results
shown in Fig.?4(c) as well. Good agreements are observed between the simulated and measured results, with both
demonstrating that the propagation of incident wave in the fabricated sample is virtually blocked with the
maximal reduction in transmission coefficients achieved at the designed frequency 4346Hz. The slight discrepancy
should come from the imperfect sample fabrication and non-zero reflection at the end. As a consequence, the
proposed scheme is verified effective to insulate sound at the desired frequency with sound path totally open to
other entities like flows or lights and with no need for bulky-sized resonators decorated under the inner surfaces.
Next we will discuss the extendibility of the working bandwidth of our designed sound insulator with thin
shell and open configuration. The proposed structure has been proven both theoretically and experimentally
effective to work at predesigned frequency, blocking the transmission of incident wave without sacrificing the
continuity of background medium. This suggests the possibility of producing sound insulation in a more wide
frequency range by cascading several metasurface unit cells with different eigen-frequencies. When more unit
cells are added to extend the working bandwidth of the designed insulator, the above theoretical analysis requires
that the phase gradient on each AM must follow the rule of d?/dx = ?k where the value of k depends on the
corresponding operating frequency. For simplifying the design and fabrication of the device, here we choose to
fix the phase difference between adjacent units as a constant of ??/4 and simply change the spatial interval to
adapt to the different working frequencies. We verify such a possibility via numerical simulation on the frequency
dependence of transmission efficiency of hybrid structures composed of different numbers of components with
eigen-frequencies chosen from 4312, 4346, 4370, 4400 and 4446 Hz respectively to cover the target frequency
range with the fewest types of units. For a comparison, three particular cases are selected: f1 = 4346 Hz, f2 = 4346,
4370, 4400 Hz and f3 = 4312, 4346, 4370, 4400 and 4446 Hz. The corresponding numerical results are plotted in
Fig.?5. It can be seen that the series connection of different parts of pipes, engineered to give rise to desired phase
modulation around their respective eigen-frequencies, helps to effectively broaden the working bandwidth of the
resulting device. This would be significant for various practical applications of sound insulators that may need to
control broadband noises.
In conclusion, we have presented a mechanism for insulating sound transmission in a hollow pipe with
subwavelength thickness. Through carefully designing the structure parameters, we can block propagating wave in the
waveguide while leaving an open channel allowing other substances to pass. Instead of relying on the mismatched
impedance or viscous dissipation used in the traditional designs of sound insulators with blocked path and bulky
size, here we use the metasurface that has vastly decreased weight and thickness but is capable to provide phase
gradient needed for manipulating the direction of incident wave. Furthermore, the dimension of the device in
terms of the wavelength can still be further reduced by increasing the coiling ratio of the labyrinthine
metastructure. Theoretical analysis, numerical simulations and experimental results have shown the effectiveness of our
proposal. With the unique advantages of thin shell, light weight and easy fabrication, our designs should open
new route to design of sound insulation with great application potentials in a variety of practical scenarios such
as duct noise control in architectural fields and mechanical fields.
Numerical simulations. Throughout the paper, the numerical simulations are conducted by the finite
element method based on commercial software COMSOL MultiphysicsTM 5.1. The background medium is air
whose mass density and sound speed is ?0 = 1.21 kg/m3 and c0 = 343 m/s, respectively. Mechanical parameters
of ABS plastic are mass density ?A = 1180 kg/m3 and sound speed cA = 2700 m/s, which are the parameters of the
3D-printed materials in the experiments. The viscous effect has been ignored in simulations, corresponding to
the experimental situation, where the thickness of viscous boundary layer, approximating to 0.024 mm for lower
frequency limitation, is about 62.5 times smaller than the space of w2 between the plates.
Acoustic measurements. The measurement is performed in the anechoic chamber in order to eliminate
the undesired reflected waves. A 1/4-inch microphone (Br?el&Kjaer type-4961) is used for measuring the sound
field in the scanned region. A loud speaker is placed 2 m away from the pipe to obtain a plane wave incidence,
emitting sound wave of frequency from 4200 Hz to 4500 Hz with a step of 10 Hz. Sound absorbing foams are also
set at the exit of experimental insulator.
This work was supported by the National Natural Science Foundation of China (Grant Nos 11634006, 11374157
and 81127901) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher
H.L.Z., B.L., Y.F.Z. and Jing Y. performed the analytical and numerical simulations. H.L.Z. and Y.F.Z. conducted
the simulator design and the experiments. H.L.Z. and B.L. conceived the idea and wrote the manuscript. B.L.
and J.C.C. conceived and supervised the manuscript. Jing Y. and Jun Y. contributed in the analysis. All authors
contributed to the discussions.
Competing Interests: The authors declare no competing financial interests.
How to cite this article: Zhang, H.-L. et al. Sound Insulation in a Hollow Pipe with Subwavelength Thickness.
Sci. Rep. 7, 44106; doi: 10.1038/srep44106 (2017).
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