A09:
PHYSICAL ACOUSTICS

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0183

Acoustic Metalens For Seismic Wave Control
Sang-Hoon KIM1; Dae-Jin JUNG2; Ik-Chang CHOI3
1

Division of Marine Engineering, Mokpo National Maritime University, 58628, R. O. Korea

2

Department of Ocean Civil Engineering, Mokpo National Maritime University, 58628, R. O. Korea

3

Department of Ocean Civil Engineering, Mokpo National Maritime University, 58628, R. O. Korea

ABSTRACT
A seismic wave is a kind of acoustic wave and then it behaves reflection, refraction, and transmission. Seismic
acoustic metalens which change the direction of a seismic surface wave and reduces the amplitude and groundacceleration was suggested. It is based on a huge two-dimensional acoustic Eaton lens that has a diameter of
100 m, that is, the order of the wavelength of the seismic surface wave. A 1/100 reduced scale modelexperiment was carried out in the ground. The seismic lens group which was composed of 4 acoustic Eaton
lenses was buried in the ground in a form of 2x2 square format. A seismic Rayleigh wave was generated bya vibrating footing machine for 1~3 minutes, and the acceleration before and after passing the lens group was
measured in the three perpendicular directions. The effectiveness by the comparison of the acceleration
before and after passing the lens group was discussed.
Keywords: Seismology, Metalens, earthquake acceleration

1. INTRODUCTION
Seismic waves are a kind of inhomogeneous acoustic wave with various wavelengths. There are
two types of seismic waves: P (Primary) and S (Secondary) waves are body waves, and R (Rayleigh)and L (Love) waves are surface waves (1,2). Surface waves are generated when body waves arrive at
the surface of the earth and the epicenter is the main point of the generation. Surface waves travelmuch slower than body waves about 1∼3 km/s and have a longer wavelength of the order of 100 m
and the amplitudes decrease exponentially with the depth.
R wave can exist only in a homogeneous medium with a boundary and has transverse motion. On
the other hand, L waves are polarized shear waves guided by an elastic layer which causes horizontal
shifting of the Earth during earthquakes. The amplitude of the surface waves decays slower than thatof body waves and then causes the most destruction of earthquakes. Earthquake engineering is related
to attenuate surface waves
The seismic surface wave is a kind of an elastic wave, and then it should behave reflection,refraction and transmission as other acoustic waves. Our motivation to reduce the amplitude of theseismic wave is to develop an acoustic lens to control the seismic waves. Recently a meta-lens grouphas reported numerical simulation of a seismic lens to control the incoming seismic surface wave (3).
Their choice was an acoustic Luneburg lens which focuses the incoming wave to the edge on the otherside of the lens, but the effect was very limited and has some problems to apply to a real system. Our
choice of the seismic lens is a group of acoustic Eaton lens (4,5). It is an attenuator which creates anartificial seismic shadow zone that reduces the amplitude of the wave exponentially.
We tested the attenuating effect of surface wave by a group of acoustic Eaton lens es. Four lensesof 1/100 reduced scale model was buried in the ground and R wave was created by a vibrating machinefooting as a seismic surface wave source. Then we measured the acceleration of the surface wave by
earthquake accelerometers before and after passing the seismic lens group.
shkim@mmu.ac.kr
jdj2689@naver.com
3
ichoi@mmu.ac.kr
12

2. THEORY OF THE SEISMIC LENS
Eaton lens is a gradient index lens in which the refractive index varies from one to infinity. It wasdeveloped by Eaton in 1952 for a 180-degree deflecting angle. Later, it was extended to arbitrary
deflecting angles which can change the wave trajectories in any direction.
The refractive index of the typical 180-degree Eaton lens is (4,5)(1)
2n( r ) =−1 .

r

The r is the normalized radius from the center of the lens, that is, the range of the r is 0 < r < 1,
and n = 1 for r ≥ 1. It has a singularity in that the RI goes to infinity at the center of the lens. And thenthe lens creates a stop-band of waves at specific frequency rages depending on its geometry.
A numerical simulation of the 2x2 two-dimensional (2D) acoustic Eaton lens format is shown in
Figure 1. The stop-band is an artificial seismic shadow zone (ASSZ). To create ASSZ in the seismic
field, the diameter of one lens must larger than the wavelength of the incoming wave as (4,5)(2)
D.
However, this relation gives a serious difficulty in model experiment because the wavelength ofseismic surface wave is order of 100m and too large to test in a field lab.

Figure 1 – The seismic shadow zone produced by the 2D four seismic lenses

Passing one two-dimensional (2D) seismic lens, the amplitude of the outgoing wave is reduced toabout 70% of the incoming wave, and it corresponds to -0.15 of the Richter scale. To reduce theRichter scale 1, we need 7 seismic lenses of the diameter of 100 m in series. It means that the widthof the seismic barrier composed of seismic lenses is larger than 700 m at least. Instead of the amplitudedifference, we measure the acceleration difference in the front (before passing) and back (after
passing) side of the lens group by earthquake accelerometers.

3. EXPERIMENT OF THE SEISMIC LENS
We tested the Figure 1 in the field with 1/100 small scale. Four 2D acoustic Eaton lenses made ofcement concreate were manufactured. The reduced scale lens has the diameter of 100 cm and the3
thickness of 20 cm. The density of the soft soil in the ground is about 1.8 g / cm , and that of the lensis heavier than the background soil about 50%. The four lens made a seismic lens group as in Figure
1.

The lens group was buried in the ground of soft soil and R wave was created by a vibrating machinefooting 2m in front of the lens group surface for 1~3 minutes. Among the two surface waves we chose
R wave and do not consider L waves here. The acceleration by the vibration was measured in thethree-direction; longitudinal (x-), transverse (y-), and perpendicular (z-) direction just 1m distance inthe front (before passing) surface and back (after passing) surface of the lens group. The frequency of
the R wave is about 25~60 Hz and the propagating surface wave velocity is about 100~300 m / s .
Then, the wavelength of the R wave is 2~12 m and much larger than the diameter of the lens.

acceration(m/s^3)y-7.00E-02
0.00E+00-7.50E-02

1.00E+01

2.00E+01

3.00E+01

4.00E+01

5.00E+01

6.00E+01

7.00E+01-8.00E-02y-8.50E-02-9.00E-02

time(s)acceration(m/s^3)y4.00E-02
3.00E-02
2.00E-02
1.00E-02
0.00E+00
0.00E+00y
1.00E+01

2.00E+01

3.00E+01

4.00E+01

5.00E+01

6.00E+01

7.00E+01

time(s)Figure 2 – The transverse acceleration in the front (up) and back (down) of the lens group

The upper graph is the acceleration before passing the lens group and the down one is after passingthat. The difference of the acceleration between front and back of the lens group was a little. One
possible cause is that the wavelength of the surface didn’t satisfy the ASSZ condition in Eq. (2).

4. SUMMARY
A seismic lens group was made by four 2D acoustic Eaton lenses with 1/100 reduced scale andburied in the ground. R wave was created near the lens group and the acceleration before and after
passing the seismic lens group in the ground was measured by earthquake accelerometers in threedirections; longitudinal, transverse and perpendicular. The difference of the acceleration between thefront and back sides of the group was very a little and less than our expectation. We have a plan torepeat the experiment with a larger seismic lens to satisfy the Eq. (2). The consecutive modelexperiment of the small scale will be reported soon.

ACKNOWLEDGEMENTS
This research was supported by "Regional Innovation Strategy(RIS)" through the National
Research Foundation of Korea(NRF) funded by the Ministry of Education(MOE)(2021RIS -002).

REFERENCES
1. AKI K, Richards PG. Quantitative Seismology. 2nd ed. New York, USA: University Science Books, 2002.
p.
2. Udías A. Principles of Seismology. 2nd ed. Cambridge, UK: Cambridge, 2010. p. 183-204.
3. Colombi A, Guenneau S, Roux P, Craster RV. Transformation seismology: composite soil lenses forsteering surface elastic Rayleigh waves. Sci Rep. 2016;6:25320(1-9).
4. Kim S-H. Retroreflector Approximation of a Generalized Eaton lens. J Modern Optics. 2012;59(9):839842.
5. Kim S-H. Analytic solution of the generalized Eaton lens. J Modern Optics. 2021;68(3):143-145.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0289

Defect-based acoustic-like spectrometer for sensing applications inbiomedicine and materials science using phononic crystals
Delfino REYES1,2,*; Hyeonu HEO 2,3; Arup NEOGI 3,4,*
1

Faculty of Science, Autonomous University of the State of Mexico, Toluca, Mexico
2
3
4

University of North Texas, Texas, USA

Penn State University, Pennsylvania, USA.

University of Electronic Science and Technology of China, Chengdu, China

ABSTRACT
Here, defects induced in a perfect phononic crystal composed of steel cylinders (0.8mm radii) squarely
disposed of in water (2mm lattice-parameter) were spatially disposed to build an acoustic-like spectrometerthat has an input port-defect, which demultiplex the incoming-wave at a second defect in two waves. The
demultiplexed waves follow two similar paths being mixed at a certain defect-point, leaving the phononic
structure through a defect output. The designed acoustic-like spectrometer allows a new transmission bandin the gap interval, in which, a strong absorption peak is induced when breaking the symmetry by placing ahollowed glass tube in one of the two arms. The absorption peak frequency (≈436 kHz) is shifted to different
frequencies when changing the material inside the tube. The shift is sensitive to the type of material andconcentration. Both, experimental and theoretical results are discussed to propose the defect-based structure
for biomedical and material sensing applications, using sugar (glucose), and copper nanofluids as target
analytes.
Keywords: Phononic crystal, acoustic spectrometer, defect modes, sensing

1. INTRODUCTION
Since their discovery, Phononic crystals (PnCs) have been proposed as acoustic demultiplexers (1),lenses (2), sensors (3), and others. Phononic structures, neither 1D nor 2D designs, for sensing
applications are a growing topic, based on resonant effects caused by breaking the symmetry in the
periodic devices by introducing defects due to an analyte holder ( 4). On the one hand, for 1D sensors
are composed of two periodically sandwiched materials in which one of the materials serves as ananalyte (5), modified using only the central part as an analyte holder (6). On the other hand, thesensing applications of 2D PnCs were explored as hollowed metal plates, filling the hollows with theliquid analyte of interest (7), followed by similar perforated plates with a central cavity hole breaking
the symmetry, inducing a resonance peak whose frequency depends on the analyte features (8, 9). The
sensing applications of PnCs have included the design of Mech -Zehnder interferometer-types within
the perfect PnCs, using one of the arms to place the analyte holder (10,11).
The use of PnCs as sensors is mainly based on the fact that the holder containing a reference
induces a resonant mode at a specific frequency. The presence of the analyte induces a shifting,widening, or thinning of the resonant peak and can be correlated with their material properties. The
sensing probes include different aqueous mixtures samples, organic solvents, types of gasoline,
peroxide, and among others, either using an ID crystal or a 2D one. Most of the reported PnCs-based
sensors are hollowed solid plates (7-11). In this paper, we are reporting a defect-based acoustic
spectrometer for sensing nanofluids; defects in a perfect PnC were induced to build a Mach -Zehnderinterferometer-type arrange. In one of the arms, a glass tube was stacked, working as the holder for
dreyes.fc@gmail.com.mx, * Corresponding author
heo.hyeonu@psu.edu
3arupn@yahoo.com, * Corresponding author
12the nanofluid analyte. A deep resonant absorption peak was experimentally recorded at 436.70 kHz,
induced by the presence of the holder when filled with water, which is sh ifted when laser-ablated Cubased nanofluids are placed in, showing frequency-shifting depending on the ablation time. It wasalso found an aqueous magnetic nanofluid also induces frequency-shifting when compared with purewater.

2. BRIEF RESULTS
The perfect PnC has been already experimentally and theoretically reported.
It supports an acoustic bandgap from 380–480 kHz. Figs. 1-a) and b) containthe simulated and actual acoustic spectrometer (AS); the red arrow indicates the
position where the analytes were placed. Figs. 2-a) are the experimental and -b)
theoretical transmission spectrums, respectively, for the acoustic spectrometer(green line), which clearly shows the induced transmission band inside the band
structure due to the presence of the defects array. The induced transmission wasrecorded between 408 kHz and 460 kHz approximately, which is the result of
the collective coupling between induced resonant modes due to the presence ofthe defects array.
In Figs. 2-a) and b), the blue lines correspond to the transmission spectrumsof the acoustic spectrometer with the steel cylinder kept fixed in the sixth
position, as shown in Fig. 1. It can be seen it is composed of at least four deep
peaks, referred to as resonant absorption peaks. The presence of the not removed Fig. 1-a) Simulated andsteel cylinder induces the demultiplexed waves to travel both across nonsymmetric paths, inducing a dephasing where they are mixed. This dephasing -b) actual AS.
yields to a partial destructive
interference. The resonant peak
was also kept when the nonremoved cylinder was replaced
by a hollow glass tube, a Pasteur
pipette, which was implemented
as analyte container. It was
experimentally observed that theresonant frequency is shifted
when the analyte in the container
was changed (sugar solutions or
different nanofluids), compared Fig. 2-a) experimental and -b) theoretical transmission spectra for the ASwith that when it is filled withincluding the intentionally not removed steel cylinder
pure water.
In this extended abstract, only partial results when the pipette
was filled with a commercial magnetic nanofluid. Fig. 3 contains
the transmission spectrum for the acoustic spectrometer without
any analyte as a green line and the corresponding when the pipette
was filled with pure water (black line). It can be seen that thepresence of the pipette holder induces the already referredabsorption peak at 436.80 kHz. However, when the sample in theholder was the magnetic nanofluid, the absorption peak was
modified and shifted to shorter wavelengths, centered at 435.32
kHz. The blue-shifting induced by the nanofluid is due to its
different density compared with water, which have different soundvelocities by itself. This brief result is strictly indicating the acoustic
Fig. 3 Magnetic nanofluid sensing.
spectrometer is sensible to the change in the material inside thepipette container, giving the reason to propose the designed acoustic structure as possible sensor using
the observed resonant absorption peak.

CONCLUSIONSIn conclusion, the design of a defect-based acoustic spectrometer performed in a phononic crystal
is reported, in which a resonant absorption peak was induced due to the presence of a defect point, ananalyte holder, is proposed for sensing applications. It was found the absorption peak is modified inshape and blue-shifted when the water filling the holder was replaced by the magnetic nanofluid,which indicates the AS is sensible to the analyte placed the holder tube.

ACKNOWLEDGEMENTS
H.H. and A.N. thank the support from NSF through EFRI-1741677. D.R Thanks to NSF and UAEMex
for the financial support to carry out the stay at UNT to perform the experimental work.

REFERENCES
1. Motaei F, Bahrami A. Eight-channel acoustic demultiplexer based on solid-fluid phononic crystals withhollow cylinders. Photonics and Nanostructures - Fundamentals and Applications. 1996; 39:100765.
2. Walker EL, Jin Y, Reyes D, Neogi A. Sub-wavelength lateral detection of tissue approximating masses
using an ultrasonic metamaterial lens. Nature Communications. 2022; 11:5967.
3. Villa-Arango S, Torres R, Kyriacou PA, Lucklum R. Fully-disposable multilayered phononic crystal liquid
sensor with symmetry reduction and a resonant cavity. Measurement. 2017; 102:20-25.
4. Lucklum R, Li J. Phononic crystals for liquid sensor applications. Meas. Sci. Technol. 2009; 20:124014.
5. Khateib F, Mehaney A, Aly AH. Glycine sensor based on 1D defective phononic crystal structure. Opticaland Quantum Electronics. 2020; 52:489.
6. Mehaney A, Nagaty A, Aly AH. Fuel Phononic Crystal Sensor for the Determination and Discriminationof Gasoline Components. Plasmonics. 2021; 16:1755.
7. Mehaney A, Ahmed II. Acetone sensor based 1D defective phononic crystal as a highly sensitive biosensor
application. Optical and Quantum Electronics. 2021; 53:97.
8. Zaki1 SE, Mehaney A, Hassanein HM, Aly AH. Fano resonance based defected 1D phononic crystal forhighlySci. Rep. 202; 10:17979.
9. Imanian H, Noori M, Abbasiyan A. Highly efficient gas sensor based on quasi-periodic phononic crystals.
Sens. Actuator: B. Chemical. 2021; 345:130418.
10. Ke M, Zubtsov M, Lucklum R. Sub-wavelength phononic crystal liquid sensor. J. of App. Phys. 2011;
110:026101.
11. Gharibi H, Khaligh A, Bahrami A, Badri H, A very high sensitive interferometric phononic crystal liquid
sensor, J. Mol. Liq. 2019; 296:1118.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0290

Local defects-engineering in solid/liquid phononic crystals fordemultiplexing and high Q-cavity applications
Delfino REYES1; David MARTINEZ2; Miguel MAYORGA1, Arup NEOGI3
1
2

Faculty of Science, Autonomous University of the State of Mexico, Toluca, Mexico

Faculty of Science, PhD in Science Program, Autonomous University of the State of Mexico, Toluca, Mexico
3

University of Electronic Science and Technology of China, Chengdu, China

ABSTRACTDefects-induced in perfect phononic crystals (PnCs) allows propagating defect-modes, forbidden in theperfect structure, whose frequencies are within the phononic bandgap interval. These new allowed modes
travel following the performed defect paths to release the acoustic energy at specific spatial points. Defects
induced in a PnC, composed of steel cylinders (0.8mm radii), squarely disposed of in a water matrix (2mmlattice parameter) allowed to engineer defect-based acoustic demultiplexers and optimized high-Q cavities
for ultrasonic waves control. On the one hand, a demultiplexer containing one defect-input port and two
output arms is able to allow the efficient transmission of 398 kHz by one output port, inhibiting it throughthe second one, while 450 kHz, experiments the opposite behavior. The efficient propagation is temperaturetunable when incorporating a PNIPAM thermosensitive hydrogel at each defect. On the second hand, thespatially optimized defects-position as horizontal and diagonal ones along with the propagation axis, increase
the quality factor (Q) by over 16 times for the diagonal with respect to the horizontal defect-disposition.
Weak-coupling modes for the diagonal configurations explain the observed Q factor increase. Both, theory
and experiment results were validated. Efficiently defects-dispositions allow interesting defect-based
phenomena which can be useful in PnCs-based ultrasonic technologies.
Keywords: Phononic crystal, defects, Q-factor

1. INTRODUCTION
Phononic crystals (PnCs) are artificial periodic structures composed of sound or mechanicalscatters periodically disposed of in a matrix; the main feature of these structures is that b othcomponents, scatters and matrices have different elastic properties. The difference in material
properties gives place to the formation of phononic bandgaps, which are frequency intervals notallowed to travel across the crystal (1). The acoustic bandgap can be controlled by the spatial
disposition of the components of the PnCs, external fields which change the material properties of
one of both components, or due to the presence of defects in the phononic structure (2, 3). Defects in
PnCs can be reached by changing the material properties of their components or position in specificpoints, or removing scatters. The presence of defects, as impurities in semiconductors, allows defects
or impurities modes whose frequencies are inside the bandgap range. Def ects in phononic structures
have been performed as local or arrays to reach defect-based acoustic waveguides or demultiplexers,
high Q-factor, or harvester energy structures (4-6).
A defect alone, reached by removing a scatter, induce a resonant mode, wh ile arrays of defects
yield to collective modes which can open the bandgap, inducing new transmission bands, whose wideand position inside the gap depends on the defect shape, density, or material. Here, the optimized
disposition of defects in a phononic crystal is shown as useful for the high Q-factor structures or
tunable acoustic waveguides whose guided frequencies depend on the temperature due to the inclusionof a thermosensitive PNIPAM hydrogel. Theoretical and experimental results are included .
dreyesc@uaemex.mx, migmayorga@gmail.com
devonia777@gmail.com
3
arupn@yahoo.com
12

2. BRIEF RESULTS
A solid/fluid phononic crystal performed withstainless steel cylinders of 1.6 mm diametersquarely disposed of (19×19) in water with a lattice
parameter of 2 mm was used to design an optimized
defects-based high Q-cavity structure. Defects
were induced by removing cylinders neither along
the propagation waves axis (horizontal) nor
vertical or diagonal to this. Experimental results(5), confirmed with theoretical ones, show that anarray of five defects in whatever direction allows
the propagation of different modes. Fig. 1 contains
the experimental transmission spectrums of thethree defects combination, horizontal, vertical, anddiagonal. It can be seen for the horizontal
configuration (blue line), five resonant peaks can Fig. 1 Effect of the defect’s configuration on the allowed
be seen, while for the diagonal array, a narrow resonant modes (5).
transmission peak was recorded; this behavior was
also theoretically obtained (inset). The different transmission, widths of the new transmission peaks,
allow for calculating the Q factor, founding this is over fifteen times larger for the diagonal disposition
compared with the horizontal ones (5).
Fig. 2 contains the transmission spectrum
when the defects were removed along the
propagation axis for five defects when these arecomposed by 2×2 defects arrays filled with
PNIPAM hydrogel. These arrays shown for five
defects a two well defined minibands are induced(4), which is the result of the different allowed
modes coupling. From Fig. 2, it can be seen when
the defects are composed by PNIPAM hydrogel,extra modes between 460-480 kHz were induced,
depending on the number of defects; this behavior
indicates the presence of the hydrogel allows theFig. 2 Effect of the defect’s number (2×2) on the allowed propagation of new modes, which are in fact
sensible to the temperature, due to theirresonant modes when PNIPAM is included in the defect.
thermosensitive nature (not shown here). In a
demultiplexer composed two arms, one of they containing the hydrogel allows the tuning of
demultiplexed waves with the temperature.

3. CONCLUSIONSWe show that induced defects inside a perfect phononic crystal induce the propagation of different
resonant modes, which depends on their spatial disposition. A diagonal disposition allows to increase
the Q factor, while the inclusion of PNIPAM hydrogel induce new modes which are in facttemperature-dependence due to their intrinsically thermosensible behavior, which could be useful tobuild tunable demultiplexers.

ACKNOWLEDGEMENTS
A.N. thank the support from NSF through EFRI-1741677. D.R Thanks to NSF and UAEMex for thefinancial support to carry out the stay at UNT to perform the experimental work. D.M. thanks to
CONACYT for the financial grant.

REFERENCES
1. Vasileiadis T, Varghese J, Babacic V, Gomis-Bresco J, Urrios D, Graczykowski B. Progress andperspectives on phononic crystals. Journal of Applied Physics. 2021; 129:160901.
2. Liu J, Guo H, Wang T. A Review of Acoustic Metamaterials and Phononic Crystals. Crystals. 2022; 10(4):305.
3. Chen ZG, Wu Y. Tunable Topological Phononic Crystals. Phys. Rev. Applied. 2016; 5:054021.
4. Reyes D, Walker E, Zubov Y, Heo H, Krokhin A, Neogi A. All-acoustic signal modulation and logic
operation via defect induced cavity effects in phononic crystal coupled-resonator acoustic waveguides.
New. J. of Phys. 2019; 21: 113012.
5. Reyes D, Martínez D, Mayorga M, Heo H, Walker E, Neogi A. Optimization of the spatial configurationof local defects in phononic crystals for high Q cavity. Front. Mech. Eng. 2020; 6:592787.
6. Lee G, Lee D, Park J, Yang Y, Kim M, Rho J. Piezoelectric energy harvesting using mechanicalmetamaterials and phononic crystals. Commun. Phys. 2022; 5:94.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0989

Enhancement of Acousto-Optic Interaction
by using Phononic and Photonic Crystals
Junyong An and Wonju Jeon1
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Korea

ABSTRACTIn this study, we propose a metamaterial composed of phononic and photonic crystals to enhance acoustooptic interaction. The metamaterial confines acoustic and optical waves with different wavelength scales ina cavity where the two waves strongly interacted with each other. To enhance the interaction further, weinvestigate the effective material properties of photonic crystals with the aid of quasi-static homogenization
theory and modifiy the geometrical constitution of the photonic crystals.
Keywords: Phononic crystal, Photonic crystal, Acousto-optic interaction

1. INTRODUCTION
The acousto-optic (AO) interaction where acoustic wave affects the behavior of optical wave hasa relatively short history compared to optics and acoustics, but the interaction has received
attention due to various applications (1-2). Several researchers have investigated the enhancementof AO interaction to improve performance of AO devices, and recently, the enhancement using aphoxonic crystal with characteristics of phononic and photonic crystals has been studied (3-5).
The acoustic and optical waves with similar wavelengths were confined in a cavity between the
mirror-symmetric phoxonic crystals and strongly interacted with each other in the cavity. For
example, 1550 nm light that was commonly used in telecommunications interacted with acoustic
wave at GHz frequencies. However, this high-frequency acoustic wave was attenuated in thephoxonic crystals and cavity. Papanikolaou et al. (5) investigated the effect of acoustic attenuationon the confinement of acoustic waves and the interaction between sound at 10.86 GHz and1555.18 nm light. Media loss reduced the amplitude of the sound confined in the cavity,
weakening the AO interaction.
Acoustic attenuation due to media loss can be negligible when using sound with frequencies
in the hundreds of MHz, but a conventional phoxonic cavity cannot confine both 1550 nm lightand sound with a much longer wavelength than the light. Therefore, we intend to overcome thelimitation of the conventional phoxonic cavity by introducing structural hierarchy, which recently
has been used to control waves (6-7), into the phoxonic cavity. This is composed of phoxonic
crystals on the scale of the wavelength of sound, photonic crystals on the scale of the wavelengthof light, and a cavity. The two waves with different wavelength scales can be confined in thecavity, resulting in enhancement of the AO interaction in the phoxonic cavity with structural
hierarchy.

2. RESULTS AND DISCUSSIONS
The acoustic and optical transmittance spectra of single- and two-scale phoxonic cavities areevaluated using the transfer matrix method (TMM) and are shown in Figs. 1(a) and 1(b). The solid
blue and brown lines indicate the spectra of the single- and two-scale phoxonic cavities,
1

Email address: wonju.jeon@kaist.ac.kr

respectively. Due to the Fabry-Perot resonance peaks exist at 0.59 GHz and 193.5 THz within the
acoustic and optical bandgaps. The blue arrows indicate the acoustic and optical cavity modes.
The acoustic and optical confinements at cavity mode frequencies are examined by evaluating
their displacement and electric fields. In Fig 1(c), the displacement fields of the single- (solid
blue line) and two-scale (solid orange line) are shown. The amplitude of the acoustic wave ismaximized in the cavities of both phoxonic cavities and the maximum amplitude in two-scalecavity is slightly smaller than the single-scale one. Figure 1(d) shows the electric fields at the
optical cavity mode frequency for the single-scale (solid blue line) and two-scale (solid orangeline) cavities. The maximum amplitude of electric field in the cavity is 20 times larger than thatof the single-scale phoxonic cavity.

Figure 1 – (a) Acoustic and (b) optical transmittance spectra of the phoxonic cavities. (c) Displacement
fields in the phoxonic cavities at the acoustic cavity mode frequency. (d) Electric fields in the phoxonic
cavities at the optical cavity mode frequency.

REFERENCES
1. D. A. Smith and J. J. Johnson, Sidelobe suppression in an acousto-optic filter with a raised-cosineinteraction strength, Appl. Phys. Lett. 61 (1992), 1025.
2. J. A. Clark, Whole field measurements of acousto-optic refraction, J. Sound Vib. 46 (1976), 306–310.
3. I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, Enhanced
acousto-optic interactions in a one-dimensional phoxonic cavity, Phys. Rev. B 82 (2010), 174303.
4. M. Oudich, S. El-Jallal, Y. Pennec, B. Djafari-Rouhani, J. Gomis-Bresco, D. Navarro-Urrios, C. M. S.
Torres, A. Martínez, and A. Makhoute, Optomechanic interaction in a corrugated phoxonic nanobeam
cavity, Phys. Rev. B 89 (2014), 245122.
5. N. Papanikolaou, I. E. Psarobas, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, Light
modulation in phoxonic nanocavities, Microelectron. Eng. 90 (2012), 155–158.
6. K. Y. Lee and W. Jeon, Hierarchically structured metamaterials with simultaneously negative massdensity and Young’s modulus by using dynamic homogenization, J. Appl. Phys. 124 (2018), 175103.
7. K. Y. Lee and W. Jeon, Hierarchical phononic crystals for filtering multiple target frequencies ofultrasound, Sci. Rep. 10 (2020), 8070.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0228

Acoustic Meta-lenses for Bioimaging
Yuqi Jin1, Teng Yang1, Delfino Reyes1, Ezekiel Walker2 and Arup NEOGI3
1

University of North Texas, USA
2

3

Echnovus Inc., USA

University of Electronic Science and Technology of China, China

ABSTRACTThe metamaterial properties of periodic phononic structures can be used to design and realize ultrasoniclenses that can be used for manipulating the propagation of acoustic waves. The negative index or hyperbolic
properties of these novel meta-lenses can be used to collimate, focus and filter acoustic waves for practicalapplications. We have used two-dimensional periodic structures with different surface shapes or structures ofthe lens for tailoring the wavefront of the incident and transmitted waves through the periodic structures.
These lenses have been used for deep-tissue imaging with subwavelength resolution. The dispersion of the
waves can be also manipulated by using combinatory lenses for biomedical imaging. We demonstrate theuses of these novel lenses for imaging soft and hard tissues including fatty tissues, bones etc.
Keywords: Acoustic metalens, Ultraonic detection, Deep tissue imaging, super-resolution imaging

1. INTRODUCTION
The dispersion and propagation of the acoustic waves through a periodic phononic crystal can be
tailored by controlling the periodicity, and surface features which modifies the equifrequency contourof the phononic structures. Lenses can be designed with propagation characteristics that are
significantly different than the conventional laws of geometric wave propagation in a medium.
Ultrasound serves as a popular imaging and therapeutic modality due to its relative non -toxicity,
practical mobility, and low cost. In both non-destructive and medical applications, the ability tocontrol the spatial sound pressure field in a material is critical for evaluation. These evaluations areused for diagnostics, with diagnostic accuracy from an image directly dependent on an accuratecharacterization of the size, shape, and contrast of materials comprising a composite1. Images thatenjoy high fidelity of the distribution of materials with varying contrast are generally considered ofhigher quality. Ultrasonic imaging quality is both dependent on the minimum sizes detectable by aparticular system, fundamentally a function of wavelength, and the min imum contrast detectable.
Contrast and spatial reconstruction of a signal to produce an image is dependent on achieving highsignal-to-noise ratios by controlling focusing and reducing multiple scattering effects.
Due to the longer wavelength of sound waves compared to electromagnetic waves such as X-Raysor electron waves, the ultrasonic imaging technology suffers from a lack of resolution. These can be
circumvented using phased-array approaches that have shown to approach higher resolution
depending exclusively on the electronic signal processing. More recently phononic metamaterials
have led to the super-resolution spatial dimensions that use acoustic wave engineering without any
electronic or digital image processing techniques. Holey structures, coupled bound modes, andnegative index flat lenses have led to super-resolution as high as λ|50. The group velocity dispersion
functions can be tailored using metamaterials to temporally compress phonons for super-resolution
imaging. However, super-resolving power is generally restricted to the evanescent near-field of thedevice or within a few wavelengths of the surface, making the structures impractical for many
applications.
We present the use of geometric wave mechanics to design and realize phononic lenses using theproperties of the metamaterial for long range collimation or sub-wavelength resolving power of theselenses for the control of low-frequency ultrasound that can be used for biomedical applications. We
have designed both singlet and doublet meta-material lenses that demonstrate both a far-field focal

point and ultra-long collimation characteristics past the Fresnel zone. These lenses can be tuned bothmechanically or infilled with phase-changeable polymer material. The doublet lens demonstrates botha far-field focal point and ultra-long collimation characteristics past the Fresnel zone. The focus tothe collimation behavior of the lens can be adjusted by a simple linear mechanical translation of oneof the lens units. The doublet lens can focus the sound wave beyond 38λ away from the object, which
is farther than any existing ultrasonic transducer or metalenses lens. It can also collimate a long andnarrow collimating ultrasonic beam over 70λ.
The focal point of phononic crystal meta-lens can also be tuned with a polyvinyl alcohol-based
poly n-isopropyl acrylamide (PVA-PNIPAm) bulk hydrogel polymer. A volumetric phase transition of
the hydrogel can lead to a temperature-dependent sound velocity and density within the phononic lensmedium. The temperature variation from 20 °C to 39 °C changes the focal length of the tunabl e solidstate lens by 1 cm in the axial direction. This thermo-reversible tunable focal length lens was used ina monostatic setup for one- and two-dimensional mapping scans in both frequency domain echointensity and temporal domain time-of-flight modes.

2. NEAR-FIELD & SUPER-RESOLUTION IMAGING USING METALENS
2.1 Detection of objects within phantom tissues using a singlet acoustic meta-lens:
The singlet metal lens considered was a 2D square lattice phononic crystal of stainless-steel rods
in water ambient (Fig. 1). The periodicity and the lattice parameters were considered such that thecollimation of the ultrasonic waves only occurs for narrow, limited bands of frequency in the second
band of the crystal, far above the homogenization limit. In additi on, the lens does not function as
described in the axis parallel to that of the cylindrical scatterers. Outside these bands of frequencies,
the ML causes the transient beam to diverge. For ultrasound, the length of the Fresnel zone is also
termed the near-ﬁeld. This lens shown in Figure 1 was used to detect objects embedded in phantom
tissues. A single element transducer at 576 kHz was used for the experiment. The collimation due tothe metalens results in a divergence that is 8 times less than the emitting transducer. This
collimation of the ultrasonic wave results in a 3.6x greater lateral resolution beyond the Fresnelzone. Lateral dimension and resolving power were resolved down to 0.50λ for hard objects, with asoft tissue object to a slightly higher value of 0.73λ. [1]Figure 1 Phononic crystal based lens used for near-field imaging

2.2. Thermally tunable metalensThis phononic crystal lens in Figure 1 can be in-filled with a polymer and externally actuated fortuning the focal length of the lens without any electromechanical component. [2] Poly-vinyl alcohol-based poly n-isopropyl acrylamide (PVA-PNIPAm) hydrogel is a polymer that undergoes avolumetric phase transition at ~32–33 °C. The elasticity of the polymer changes due to the phasetransition and can result in over a 10% change in the velocity of sound in the PVA -PNIPAm. Thisvariance in sound velocity can be used to thermally tune the effective mechanical properties of thephononic lens interstitially filled with PVA-PNIPAm, and consequently achieve a tunable focal
length lens. We have performed both frequency domain echo-intensity and temporal domain time-offlight based one- or two-dimensional scans in a monostatic setup to study the temperature dependent
tunability of the lens. [3]Figure 2 shows the change in the focal length of the lens due to the change in temperature from 20 C
to 39 C. The increase in the temperature from room temperature (20 °C), changes the focal length ofthe lens from 10.29λ to 8.82λ from the transducer surface (Figure 2). This all-acoustic technique isutilized without any advanced signal processing technique and can laterally resolve 1.03 ± 0.15 λ. inone/two-dimensional echo-intensity mode and 2.35 ± 0.28λ in the axial direction using time-offlight mode.

Figure 2 Polymer filled temperature tunable solid-state- metamaterial lens

2.3. Mechanically Tunable lens
The next class of the lens we have designed is a class of doublet lens system shown in Figure 3. It
is composed of two 2D phononic crystal metamaterial lenses. It is similar to a telescopic or
microscopic lens system with the eye-piece and the objective. However, the meta-material properties
make it more complex. The first element, which is a plano-convex lens, was used to diverge theincident wave. The second element separated by a distance d is– a flat meta-lens that is used to focus
the divergent beam as a conventional metamaterial lens. (Figure 3) The advantage of focusing thedivergent source beam from the first lens is that it enables a long distant focal point. As in aconventional optical lens, the second element's focusing point in the lens is analogous to the f -numberof the optical system. The ratio of the focal length to the diameter of the entrance pupil width (or theclear aperture) can be controlled in this configuration by modifying the distance between the two lenselements. The width of the source beam is limited by the aperture of the piezo electric transducer
source. The frequency of operation is shown by the red-dashed line in Figure 3b for the optimized
eigenfrequency of the propagating waves.

Figure 3.

A mechanically tunable doublet lens (A) with its corresponding dispersion of the phononic structure (B) and thecollimating or focusing features (C), (D) and (E)Figure 3 (C)-(E), shows the change in the characteristics of the doublet system as the distance
between the two elements changes. One can modify the doublet’s behavior from the focusing to thecollimation operation as the distance between the two elements is reduced. Using this lens
subwavelength spatial resolution has been achieved. The detection limit was 0.26 λ in the monostatic
setup and 0.62 λ in a bistatic experimental setup. This lens demonstrates super-resolution detection
capabilities at distances of 42 λ and can enable ultrasonic diagnostics deep within a material or abiological tissue. [4]3. CONCLUSIONS
The work represents the application of a metamaterial for spatial characterization andsubwavelength resolution in a biosystem beyond the Fresnel zone limit. A thermally tunablephononic crystal lens has been achieved using phase-changeable polymers with a sensitive
volumetric phase transition temperature. We have also demonstrated a phononic crystal-based
doublet lens with mechanically interchangeable focusing to a collimating capability that can be used
for bioimaging.

ACKNOWLEDGEMENTS
This work was supported by the National Science Foundation under EFRI grant no. 1741677.

REFERENCES
1. EL Walker, Y Jin, D Reyes, A Neogi, Sub-wavelength lateral detection of tissueapproximating masses using an ultrasonic metamaterial lens, Nature Communications 11(1), 1-13 (2020)
2. E. L. Walker, D. Reyes-Contreras, Y. Jin, and A. Neogi, Tunable Hybrid Phononic Crystal
Lens Using Thermo-Acoustic Polymers, ACS Omega, 4, 15, 16585–16590 (2019)
3. Y. Jin and A. Neogi, Ultrasound Imaging by Thermally Tunable Phononic Crystal Lens,
Int. J. Mol. Sci., 22(15), 7966; https://doi.org/10.3390/ijms22157966. 2021
4. T Yang, Y Jin, TY Choi, N Dahotre, A Neogi, Mechanically tunable ultrasonic metamaterial
lens with a subwavelength resolution at long working distances for bioimaging, Smart
Materials and Structures 30 (1), 015022 (2021)

ABS-0022

Interface Characterization by Ultrasonic Spectroscopy - Linear andNonlinear Methods
Laszlo Adler
Taine McDougal Professor Emeritus
Ohio State University, USA

ABSTRACTTo characterize complex materials from ultrasonic measurements the frequency dependence of the wave
interaction with interfaces is analyzed. These ultrasonic spectroscopic studies included delamination in solids,
liquid-solid and solid-solid interfaces. Both metallic and non- metallic materials are investigated with solidstate as well as with adhesive bonds. The interfaces maybe random or periodically rough. The experimental
system utilized broad band transducers in the frequency range from 1 to 20 Mhz. For adhesive joint
measurements a nonlinear method was also applied with the addition of a high power transducer. Using
imperfect boundary conditions, a theory was developed which allows to compute both interfacial and bulk
properties of layers and interfaces from the received spectrum. Theoretical predictions agree well withexperimental results.
Keywords: Interface, Bonds, Spectroscopy

1.

INTRODUCTION

One of the most challenging problems in Ultrasonic Nondestructive Evaluation is to characterize aninterface between two materials. The interface may a flaw or delamination in the material, a bonded structure
like solid-state or adhesive bond just to mention a few. Since the frequency dependence of ultrasonic waves
interaction with interfaces Ultrasonic Spectroscopy1 is useful tool to characterize interface studies. UltrasonicSpectroscopy is a study of the ultrasonic waves resolved into their Fourier frequency components. In this paper
we will discuss three examples of the frequency spectra of ultrasonic wave interaction with interfaces.
a.

Interface characterization between liquids and solids with periodically rough interfaces.

b. Characterization of Solid-State bond of layered material.
c.

Evaluation of Adhesive bonds between two solids. Linear and Nonlinear angle beam ultrasonicspectroscopy.

2. INTERFACE EVALUATION AT LIQUID AND PERIODICALLY ROUGH SOLID
INTERFACE
An ultrasonic wave propagates in liquid toward a solid with periodic surface as shown schematicallyon figure 1. In the liquid in addition to the specular reflection a number of diffraction orders are generated dueto the periodic surface. In the solid both longitudinal and transverse waves are produced.

Figure 1 - Coordinate system used to describe interaction of ultrasonicwaves with periodic liquid-solid interface.

The ultrasonic spectroscopy system used in this study is shown on figure 2. A wide-band ceramictransducer (bandwidth up to 14 MHz) emits a short RF pulse. The scattered signal is received either by thesame pulse-echo transducer or by another wide-band transducer (pitch-catch). The received signal is amplified
and gated out in order to select the portion of the signal to be processed. The time domain signal can then be
fed to an analog spectrum analyzer to obtain the amplitude spectrum or be sampled and converted to digital
information and processed by calculation to obtain the spectrum via FFT (Fast Fourier Transform). A typical
spectrum of a normally reflected signal from a perfectly smooth brass surface is shown in figure 3.

Figure 2 - Experimental system for ultrasonic spectroscopy

Figure 3 - Power spectrum of the transducer output

The spectrum shown in figure 3 is used as a reference to that of the scattered spectrum from theperiodically rough surface. In figure 4 the spectrum of a normally incident reflected signal is shown from a
grating of 250 µ periodicity on brass. The most significant features of this spectrum are the two distinct
minima occurring at 6 and at 8 Mhz. The energy which is not reflected back at these frequencies to thereceiver is diffracted along the interface This phenomenon is similar to the so-called Woods anomalies. Wood(2) observed missing components from the reflection spectrum (colors) of light from a diffraction grating. Thisanomaly was treated by Lord Rayleigh (3) where he developed a dynamic theory of grating and foundconditions that the missing frequency corresponds to a diffraction order directed along the grating (90degrees). Theoretical work (4)was developed using a numerical method to obtain the amplitudes of thediffracted waves. Theoretical prediction agrees well with experimental data to predict the reflected spectrum
from a water-brass corrugated surface. The minima at 6 and 8Mhz correspond to the Scholte waves (velocity
1500 m/sec) and to the Rayleigh waves (velocity 2160 m/sec) respectively.

Figure 4 - Comparison of theory and experiment for reflected the ultrasonicspectrum from a water - brass interface

3.
3.1

CHARACTERIZATION OF SOLID-STATE BOND AT LAYERED SUBSTRATE
Generalized Lamb Waves

Most previous work to evaluate bond quality between two dissimilar materials used normal incidence
longitudinal waves with very limited success. In this work we will describe the use of generalized Lamb waves(5) to evaluate bond quality of steel layer bonded to an aluminum substrate. In figure 5 dispersion curve of thevarious modes plotted for a steel layer on an aluminum substrate immersed in water using rigid boundary
condition. Such condition is valid for a good bonding.

Figure 5 - Theoretical dispersion curves for various guided modes. Steellayer on aluminum substrate

The behavior of the lowest mode is very interesting. The phase velocity first decreases with frequency
then riches a minimum value at .5 Mhz and then increases. Changing the boundary condition between the steel
layer and the aluminum substrate the lowest mode is calculated for various values of bonding as shown on
figure 6. For “good” bond the theoretical prediction agrees very well with the experimental result obtained asshown ob figure 8. The agreement is especially good for the lowest mode.

Figure 6 - Theoretical dispersion curve of the lowest velocity- mode forvarious bond qualities

3.2

Experiment
The experimental set up is shown on figure 7. A broadband transducer (15 MHz bandwidth) emitting apulse. A signal is received by the same transducer if the frequency and incident angle is such that a leaky wave
is generated in the sample. The wave is reflected at the right hand corner. The received signal is gated andspectrum analyzed. See figure 1 for details. The velocity of a given mode is calculated using Snell’s law.

Figure 7 - Experimental technique

The experiment was repeated for various bond conditions of the lowest velocity mode and comparedto theory as shown on Figure 9. The resulting curves are shown for the lowest mode where the experimental
results agree well with theoretical prediction for the two extreme cases, for “good” bond and for “weak” bond.

Figure 8 - Theoretical/experimental comparison for dispersion curves for good bond
between a steel layer and an aluminum substrateFigure 9 - Experimental dispersion data for the lowest velocity mode

4.
4.1

EVALUATION OF ADHESIVE BONDS BETWEEN TWO SOLIDS
Angle Beam Ultrasonic Spectroscopy

In the Angle Beam Ultrasonic Spectroscopy (6) approach obliquely and normally incident
ultrasonic beams are combined. The two angle measurements allow decoupling the effects of the bondline thickness. The schematics of the method is shown on figure 10. The head accommodates transducers
for the normal and oblique measurements. The ultrasonic wave excited by the transducer is reflected fromthe interface toward a reflector and returns back to the transducer. The procedure used to scan the qualityof the thin adhesive layer embedded between two composite plates is twofold. At each point of the scan,
the normal and angle beam time domain signals reflected from the layer are recorded. Then they are
analyzed in the Fourier domain and processed using our algorithms to obtain the quality of the bond line.

y-axis motor
x-axis motor

Scanning Head

Scan Area

Adhesive Joint

Figure 10 - Transducer head

4.2

Nonlinear/Parametric Angle Beam Ultrasonic Spectroscopy

To enhance interfacial property measurement and to relate the measurement to local bond integrity
degradation we use stress modulation at the interface with the low frequency (7) transducer. We combine low
frequency vibrations of the bonded structure with overlapped linear pulse echo dual beam ultrasonic reflection
measurements. For a good bond the interfacial stiffness is independent of low frequency stress under ourexperimental conditions. In the case of a poor bond, the interfacial stiffness decreases under the vibration
tension. The mechanisms of this decrease may be different. As a result, the interfacial stiffness changes during
compression and tension periods of the low frequency cycles leading to a low frequency modulation of thespectral minima of reflected ultrasonic signals. Figure 11 illustrates our approach schematically.

Figure 11 - Angle beam ultrasonic spectroscopy of adhesive joints under thickness
resonance parametric excitation

A tension/compression cycle is applied to the bond. The low frequency stress applied on the bond line isalternately in compression or tension. As illustrated in Figure 11 (on the right), for an imperfect bond thespectral minimum of the ultrasonic signal reflected from the adhesive layer shifts under the loading. The bondlayer properties are evaluated during cycling using angle beam ultrasonic spectroscopy for different states ofthe stress of the bond line. This is achieved by synchronizing the low frequency continuous excitation of thestructure with the high frequency ultrasonic pulse triggering used for the spectroscopic measurement. The
schematic of the experimental setup is shown in Figure 12. The bonded sample is excited by the shaker andthe vibration amplitude on the sample is measured by an accelerometer. Electrodynamic transducers covering
the frequency range 50 Hz-30 kHz is used for shaker. For perfect bonds no mixing of vibrations with pulsed
ultrasonic signals is expected; for weak interfaces the boundary springs change under load leading to
amplitude and frequency modulation of ultrasonic signals by low frequency vibrations. The angle beam
ultrasonic head includes three 10 MHz wide band transducers. The recorded high frequency signals weredigitized with a 12 bit 125 MHz digitizer followed by signal gating and FFT.

Figure 12 - Schematic of the experimental setup for stress modulated ultrasonicspectroscopy

To illustrate the relation between the frequency modulation amplitude and adhesive bond degradation, themodulation resonance spectra for three different bond conditions are shown in Figure 13. As one can see thefrequency modulation amplitude increases as bond degradation increases. For degraded bonds, the resonancemodulation amplitude is significant and the signal-to-noise ratio is larger than 33 dB. For bonds with intermediatebond strength, the resonance modulation amplitude is reduced. For “good” bonds, the modulation amplitude is
reduced to near the noise level.

Figure 13 - Experimental resonance modulation spectra for diﬀerent bond properties

5.

CONCLUSIONS

It was demonstrated that the frequency dependence of ultrasonic waves interaction with interfaces
between two materials, provides important characteristics for the interface. At a water-periodic interface bothRayleigh and Scholte surface waves velocities can be obtained from the reflected spectra. The frequency
dependence of the lowest generalized Lamb modes generated on a solid layer bonded to a solid substrate canbe used to evaluate the quality of the sold-state bond. A nonlinear/parametric angle beam ultrasonicspectroscopy method can be used to differentiate from poor to good adhesively bonded solids.

REFERENCES
1. Fitting, D. E. and Adler L.: Ultrasonic Spectral Analysis for Nondestructive Evaluation, 1981.p. 1-6
Plenum Press, New York,
2. Wood, R.W. On a remarkable case of uneven distribution of light in diffraction grating spectrum Philos.
Mag. 1902. H 24, p.396
3. Lord Rayleigh: The Theory of Sound (revised) Dover Publication (1945)
4. Clays, I.M., Leroy, O. Jungman, A., Adler, L. Diffraction of ultrasonic waves from periodically-roughliquid solid surfaces. J. Appl. Phys. 1983 54 10 p. 5657-62.
5. Adler, L, de Billy, M., Quentin, G., Talmant, M., Nagy, P. A Study of Generalized Lamb Waves in SolidState Bonded Aluminum-Steel Samples. J. Acoust. Soc. 1988 Supplement 1 S84
6. Adler, L., Rokhlin, S. I., Mattei, C., Blaho, G., Xie, Q., Angle beam ultrasonic spectroscopy system forquantitative inspection of adhesive bonds, D. O. Thompson and D. E. Chimenti (Eds.), Rev. Prog. QNDE
1999 18B: 1553-1559
7. Rokhlin. S., Wang. I. Xie. B. Yakovlev. V. and Adler. L. Modulated Angle Beam Ultrasonic Spectroscopy
for Evaluation of Imperfect Interfaces and adhesive Bonds Ultrasonics 2004 42 p. 1037-47

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0377

Airborne Ultrasonic Phased Arrayusing Custom-made High-intensity Transducers
Takayuki HOSHI; Yoshiki O-OKA
Pixie Dust Technologies, Inc., Japan

ABSTRACTWe developed an ultrasonic phased array that can drive 271 custom-made transducers individually andgenerate more powerful ultrasound at a focal point than a phased array consisting of commercially-available
transducers. The ultrasonic sound pressure at a focal length of 100 mm was measured, and it was confirmedthat the sound pressure level of 40 kHz component exceeded 170 dB. This is about 3 dB larger than thephased array with the same number of commercially-available transducers. We also observed the generationof nonlinear acoustic phenomena, such as non-contact tactile sensation due to acoustic radiation pressure andacoustic streaming, generated by this high-intensity phased array.
Keywords: Phased array, Focused ultrasound, High-intensity ultrasound

1. INTRODUCTION
When forming a focal point using ultrasonic phased array (1), the ultrasonic sound pressure at thecenter of the focal point depends on the number of transducers used and the intensity of the ultrasonicwaves emitted by each transducer (assuming a fixed transducer size). In other words, to achieve higher
sound pressure, it is necessary to either increase the number of transducers or increase their intensity.
Increasing the number of transducers has already been attempted. For example, Mizutani et al. (2)reported a phased array with 996 transducers that had a maximum sound pressure level of 171 dB atthe center of the focal point with a focal length of 250 mm. Hoshi (1) reported a similar phased arraywith 285 transducers that had a maximum sound pressure level of 162 dB at the center of the focal
point with a focal length of 200 mm. Although a direct comparison cannot be made due to the different
array sizes and focal lengths, it has been demonstrated that higher sound pressure levels can beachieved by increasing the number of transducers.
On the other hand, the intensity of ultrasonic waves emitted by each transducer has been treated asa fixed value because of the limited variety of commercially-available products (Nippon Ceramic
T4010A1/T4010B4 or Murata Manufacturing MA40S4S, with a diameter of 10 mm) and their similar
specifications. If more powerful ultrasonic transducers were available, higher sound pressure could
be achieved without changing the size of the phased array.
In order to improve the output power by the latter method, we fabricated a prototype phased arrayusing custom-made high-intensity transducers (TAM2101 manufactured by Tamura Corporation). We
also prepared a phased array of the same type using T4010B4 for comparison. This paper reports theresults of these experiments conducted to compare these two prototype devices. This paper is partiallybased on the proceeding of Japanese-domestic meeting (3) and contains additional results.

2. PROTOTYPES OF ULTRASONIC PHASED ARRAY
The prototype ultrasonic phased array consists of 271 transducers with a diameter of 10 mm
densely arranged in a hexagonal area (Figure 1 (a)). These transducers can be driven by independent
signals and make a focal point by controlling their phase differences. The transducers are driven by asquare wave of from 10 to 30 V p-p by changing the power supply voltage. Additionally, the prototype
has a function to control the amplitude of the output ultrasonic wave in 256 steps by PWM control ofthis square wave.

(a)

(b)

Figure 1 – (a) Prototype ultrasonic phased array and (b) Measurement setup using a fiber-optic microphone
The frequency characteristics of the custom-made high-intensity transducer used in this study(Tamura Corporation TAM2101) and a commercially-available transducers (Nippon Ceramic
T4010B4) were measured using TSP signals with increasing frequency and an amplitude of 30 V p-p.
The results are shown in Figure 2, which shows that the custom-made transducer emits 1.63 times (4.2dB) larger ultrasonic waves than the commercial transducer at 40 kHz.

Figure 2 – Frequency characteristics of the custom-made and commercially-available transducers

3. COMPARATIVE EXPERIMENTS
Experiments were conducted to compare the prototype phased arrays equipped with the custommade and commercially-available transducers (3). First, the ultrasonic sound pressure at the center ofthe focal point was measured. In addition, the acoustic radiation pressure and acoustic streaming wereobserved. The results are reported below.
3.1 Ultrasonic Sound Pressure
The phased array with a focal length of 100 mm and a microphone for measurement were fixedwith a jig in an anechoic chamber, and the ultrasonic sound pressure at the center of the focal point
was measured (Figure 1 (b)). Preliminary measurements with the microphone moved in the depth
direction confirmed that the distribution of ultrasonic sound pressure reached its maximum value at
95 mm from the phased array, not 100 mm, so the microphone was fixed at 95 mm in this measurement.
The sound pressure level at the center of the focal point for this measurement was expected to beover 170 dB. Since this exceeds the maximum rating of a typical measurement microphone, the opticalfiber microphone (4) was employed as a microphone capable of measuring high frequencies and highsound pressure. This microphone detects sound waves as a change in the refractive index of air bymeasuring the light reflected at the tip of the optical fiber. Since the increase or decrease of thereflected light due to the refractive index change is minute, the amplitud e of a specific frequency
component (40 kHz in this case) is obtained by FFT. Since the aperture of the optical fiber is as small

as 1 mm, the sound field of 40-kHz ultrasonic waves with a wavelength of 8.5 mm can be measured
almost undisturbed.
The amplitude of the 40-kHz rectangular driving signal was changed by supply voltage (between
10 and 30 V p-p ) or PWM control (under 10 V p-p ), and the sound pressure of the fundamental component(40 kHz) was measured using the optical fiber microphone. The measurement results are shown in
Figure 3. The sound pressure level at the center of the focal point differed by about 3 dB between thecustom-made and commercially-available transducers, and it was confirmed that the phased arrayusing the custom-made transducers achieved the maximum sound pressure level of 172 dB whendriven by a 30-Vp-p rectangular driving signal.

Figure 3 – Focal point sound pressure characteristics against driving voltages (equivalent values for PWM)
It is known that the sound pressure level at an ultrasonic focal point increases as the intensity of
the ultrasonic waves emitted by the ultrasonic device increases, but gradually deviates from a linear
relationship. This is due to the distortion of the waveform of the high-intensity ultrasonic waves (5),which creates harmonic components and weakens the fundamental components (40 kHz in this case).
It is also possible that the nonlinearity in the output characteristics of the transducer is also a factor.
Here we check whether the measurement results in Figure 3 maintain linearity. For that purpose,the horizontal axis (driving voltage) is shown in logarithmic form in Figure 4. If the linearity is
maintained, the graph should look like the dotted lines, but the actual measurement results deviate
from the dotted lines when the sound pressure level exceeds about 160 dB. Although a deviation fromthe linear line was originally visible in the prototype equipped with commercial ly-available
transducers, the one equipped with custom-made transducers confirmed that this trend continued up
to a higher sound pressure level (at least over 170 dB).

Figure 4 – Focal point sound pressure characteristics against driving voltages (equivalent values for PWM)
in a log scale. The dotted lines are linear extrapolation

The distortion of the waveform of ultrasonic waves at the focal point was observed by a 1/4-inch
microphone (6) that can measure high-frequency and high-intensity sound waves, calibrated with ACO
4158 N without protection grid. Differently from the optical fiber microphone used above, thismicrophone can capture a broad frequency range and give us a sound pressure waveform that is close
to the original one.
The sound pressure waveforms at the focal point generated by the phased array consisting of thecustom-made transducers were recorded. Figure 5 shows the recorded waveforms for the driving
voltages of 6, 18, and 30 V p-p . It is observed that the waveform distortion progresses and differs fromthe original 40-kHz sinusoidal wave as the drive voltage is increased.

(a)
(b)
(c)
Figure 5 – Recorded waveforms at the focal point for driving voltages of (a) 6, (b) 18, and (c) 30 Vp-p
3.2 Acoustic Radiation Pressure
Nonlinear acoustic phenomena are also prominent at the ultrasonic focal point. As an example ofsuch nonlinear effects, the force generated by acoustic radiation pressure was measured using anelectronic scale (A&D EJ123B-JA).
The phased array with a focal length of 100 mm and the maximum power was placed above theelectronic scale. The distance from the ultrasonic emitting surface to the electronic scale was fixed at
95 mm. The reading of the electronic scale was observed to fluctuate slowly over several seconds, so
the reading was recorded 10 times at 1-second intervals and the average value was obtained. As aresult, a value of 3.5 gf was obtained for the custom-made transducers and 1.4 gf for the commerciallyavailable transducers.
3.3 Acoustic Streaming
We attempted to observe the acoustic streaming near the focal point. A hot-wire anemometer(Ckeyin GM8903) and a vane anemometer (OTraki RZ818) were used as simple observation methods.
The focal length and output were set to 100 mm and maximum, respectively. The anemometer washeld at the focal point. Because the readings fluctuated with time, the anemometer was recorded 20
times every 2 seconds and the average value was obtained. As a result, values of 8.3 m/s for thecustom-made transducers and 0.8 m/s for the commercially-available transducers were obtained forthe hot-wire method. For the vane method, the values were 3.9 m/s for the custom-made transducers
and 2.3 m/s for the commercially-available transducers. It is possible that the heat emitted by thedevice in the hot-wire method and the radiation pressure in the vane method may have influenced theresults. To eliminate these effects, other measurement methods (e.g ., PIV) should be employed.

4. CONCLUSIONS
A prototype phased array was fabricated using custom-made high-intensity ultrasonic transducers
that emits 4.2 dB larger ultrasonic waves than the case of commercially-available transducers. The
sound pressure level of the fundamental component (40 kHz) at the focal point was measured when
the focal length was set to 100 mm, and 172 dB was achieved at the maximum, which is about 3 dB
higher than that of the case of commercially-available transducers. We noticed that the sound pressurelevel at the focal point deviated from linear, which might be caused by waveform distortion and thenonlinearity of the output characteristics of the transducer.
We also attempted to observe nonlinear phenomena caused by high-intensity ultrasonic waves.
First, the force generated by the acoustic radiation pressure was measured using an electronic scale.

The force generated by the acoustic radiation pressure was measured using an electronic scale. Forcevalues of 3.5 gf and 1.4 gf were obtained for the custom-made and commercially-available transducers,
respectively. Next, anemometers were used to observe the acoustic streaming. Different values wereobtained with a hot-wire anemometer (8.3 m/s and 0.8 m/s, respectively) and a vane anemometer (3.9m/s and 2.3 m/s, respectively) for the custom-made and commercially-available transducers. In bothcases, the values for the custom-made transducers were larger. However, due to the large time
variability and poor reproducibility, these results were not reliable and should be considered as
reference. For accurate measurement in further investigation, a measurement method is required thatcan distinguish between acoustic radiation pressure and acoustic streaming, and ideally eliminate theinfluence of disturbance factors.

REFERENCES
1. Hoshi T. Noncontact tactile display using airborne ultrasound. Proc IDW; 3-5 Dec 2014; Niigata, Japan. p.
1529-33.
2. Mizutani S, Fujiwara M, Makino Y, Shinoda H. Thresholds of haptic and auditory perception in midair
facial stimulation. Proc IEEE HAVE; 3-4 Oct 2019; Bandar Sunway, Malaysia. p. 1-6.
3. Hoshi T, O-oka Y. Development of ultrasonic phased array using high-intensity transducers. IEICE Tech
Report. 2022; 122(36):6-7. (in Japanese)
4. Hoshi T, O-oka Y. Linearity and directivity of optical fiber microphone for 40-kHz airborne ultrasound.
Proc ICA; 24-28 Oct 2022; Gyeongiu, Korea. to be published.
5. Duck F A. Acoustic saturation and output regulation. Ultrasound Med Biol. 1999;25(6):1009-18.
6. Toda Y, Kamakura T, Adachi H, Nomura H, Hayashi H, Yasuno Y. Prototype and its characteristics ofcellular polypropylene microphone for ultrasound waves. IEICE Tech Report. 2015;115(385):37-40. (inJapanese)

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0494

A review of jet crackle
Kent L. GEE1
1Brigham Young University, Provo, Utah, USA

ABSTRACT
Jet crackle is an unusual sound. Embedded in the broadband noise from high-speed jets are irregular popping
events that give a distinct impulsive quality to the acoustic radiation. Crackle has been associated with noisefrom laboratory-scale jets, military jet aircraft, rockets, and even explosive volcanic eruptions. This paper
reviews historical and recent research into a phenomenon that fundamentally combines nonlinear acoustics
and psychoacoustics through the formation, propagation, and reception of shocks. Source and propagationrelated effects are described using various examples from the literature.
Keywords: jet noise, shock, rocket noise, nonlinear propagation, crackle

1. INTRODUCTION
Crackle is a perceptual phenomenon associated with supersonic jet noise; it has been labeled as anannoying and dominant characteristic of the total noise.1,2 First discussed extensively in the context
of the Concorde Olympus 593 engines in 1975, Ffowcs Williams et al.1 described crackle as “sudden
spasmodic bursts of a rasping fricative sound . . . It is a startling staccato of cracks and bangs and its
onomatope, ‘crackle,’ conveys a subjectively accurate impression.”
Although some of its conclusions have been refuted, the seminal work of Ffowcs Williams et al.
has significantly guided subsequent supersonic jet noise investigations. This paper briefly reviews
historical and recent research into crackle as a phenomenon that exists at the intersection of supersonicjet noise radiation, nonlinear acoustic propagation, and psychoacoustics. Examples from different jet
aeroacoustic sources are discussed.

2. WHAT IS CRACKLE?
It is important to recognize that crackle is a perceptual phenomenon, although researchers havesought physical features of the pressure waveform that result in its perception. Ffowcs Williams etal.1 identified waveform shock-like features as the cause of the crackle percept, but unfortunately
attempted to link it to the pressure skewness. While pressure skewness is a physical feature in thenoise from supersonic jets, Papamoschou and Debiasi3 questioned this as a suitable measure of crackle.
Gee et al.4 demonstrated that pressure skewness is not a sufficient condition for crackle by developing
waveforms that had the same spectrum, pressure skewness, and kurtosis as a measured F -18 waveform
at afterburner, but did not crackle. Their conclusion was that, if crackle was the perception of shocks
embedded in the noise, the statistics of the time derivative should be used as its quantifier. Swift etal.5 followed the investigation by Gee et al. with a signal processing study that proved that pressure
skewness was not a necessary nor sufficient condition for crackle to be present – a non-skewed
waveform with shocks crackled but one with the same spectrum but with randomized phase did not.
Signal processing investigations aside, it was only recently that the first subjective test s were 6,7 carriedout. Listeners evaluated F-35 noise waveforms as belonging to different perceptual categories ranging
from “smooth noise” to “intense crackle.” The investigation conclusively showed that while pressure
skewness was not correlated with jet noise crackle perception, the skewness of the pressure waveform
time derivative correlated strongly with crackle perception. To summarize, crackle is resulting soundquality of shocks embedded in a jet noise waveform. To date, it is best quantified by the skewness ofthe pressure waveform derivative.
1

kentgee@byu.edu

3. SOURCES OF CRACKLE
As mentioned, the original subject of crackle was the Olympus 593 Concorde engines. Crackle has
been investigated in numerous military aircraft noise studies, from the F -184,8 to the F-229 to the F3510,9,11,12 to the T-7A13. In addition to static and flyover measurements of crackle-related shocks,
uninstalled engines have also been investigated by Schlinker et al.14 and Greska and Krothapalli.15
Martens et al.16 investigated crackle (using pressure skewness as a metric) in the context of lowbypass jet engine noise control.
Although crackle is not perceived the same way in supersonic laboratory-scale jets because therelatively small nozzles push the high-frequency spectrum into the ultrasonic range, the shocks thatare responsible for the crackle percept have been studied. Examples include Schlinker,17 Krothapalli
et al.2 , Petitjean et al.,18 and Mora et al.19 More recently, notable crackle-related studies include thoseby Baars and Tinney,20 Fievet et al.,21 and Murray and Lyons. 22 The shocks leading to a perception of
crackle have been observed in both heated 19 and unheated 20,23 laboratory jets. Computational studies
that examine shocks in the heated supersonic jet flow 24, in a shear layer near field 25,26 and in an
unheated Mach 3 jet near-field 27 also been performed.
Crackle is not limited to low-bypass jet engines. It also occurs in static rocket firings and in spacevehicle launches. Several online discussions reference the “popping” noise rockets make during liftoff.
Studies of shocks in large rockets include Refs. 28-31. In rocket launches I have observed, distinct
crackle gives way to a low-frequency thumping that persists for several minutes. Additionally, lowfrequency crackle-like phenomena have been observed in at least two explosive volcanic eruptions.
In addition to a comparative analysis of Nabro Volcano with a military jet and a rocket by Fee et al.32 ,Matoza et al.33 describes the cause of audible “booming” in Alaska from the 2022 Hunga Volcano
eruption – at a distance of ~10000 km – as being shocks undergoing old-age nonlinear decay. Whilemost of the eruption energy is in the infrasound regime, the booming that occurred over a period of
about 30 min appears to be a form of crackle with a peak frequency so low that individual events areeasily distinguishable.

4. PHYSICAL CAUSES OF CRACKLE
What are the origin(s) of the shocks that cause crackle? Although this topic is not entirely resolved,
there are different explanations that merit ongoing investigation. Ffowcs Williams et al. describedcrackle as effectively a source phenomenon because they supposed nonlinear propagation to be to oweak, but this was contradicted by concurrent work by Blackstock. 34 Later works show clearly theformation and strengthening of shocks through nonlinear propagation into the far field .10,30,9,35 The
weakening/thickening of these shocks eventually may lead to a transition from a crackling sound to
more of a popping or booming sound.
This is not to say that source or near-field phenomena are unimportant. In a numerical study,
Nichols et al.24 described the existence of large pressure gradients existing insi de and outside theplume. Anderson and Freund 25, Buchta et al.26 , and Pineau and Bogey 27 describe coalescence of nearshear layer shocks. Experimentally, near-field jet noise coalescence has been studied by Fievet et al.21
and Willis et al.36 Murray and Lyons22 investigated the propagation direction and convective Mach
number associated with shocks radiated near the nozzle. Reichman et al.35 have described small-scale
shocks existing near the F-35 while larger time-scale shocks evolved from nonlinear propagation. It
is unknown if the small-scale near-field shocks observed by Reichman et al. are the same near-nozzle
shocks described as crackle-related in laboratory-scale jet studies, but it might help to resolve priorconfusion in describing crackle as a source versus a propagation phenomenon.
Recent studies by Vaughn et al. on F-35 near-field data also help elucidate physical causes. First,
an event-based beamforming method 11 was used to examine the apparent origin of the events. Both
Mach wave radiation and large-scale noise radiation were found to contribute to crackle-causing
shocks. Additionally, nonlinear reflection phenomena 37 strengthened shocks near the ground. Finally,
there is relatively high correlation 12 between the large-scale near-field events and the shocks
responsible for the far-field crackle percept.

5. FUTURE RESEARCH
Crackle is a fascinating phenomenon that has interested me since I was a doctoral student at Pe nn
State, studying nonlinear propagation of jet noise. Although we conclusively understand that it is theshocks that create the crackle percept and that the waveform derivative statistics (specifically thederivative skewness 37) form a convenient metric for quantifying it, there is so much more tounderstand – from the aeroacoustics to the psychoacoustics. How is crackle impacted by Mach number
and temperature ratio? What of nonideal vs ideal expansion? What of the crackle perception of rocket
noise at great distances? What is the effect of hearing protection on crackle? Are there other metrics 38that could describe other perceptual aspects of crackle? Hopefully this brief review is of service tothe research community, but I invite readers to reach out if there are questions I can answer or if a
discussion would be helpful.

REFERENCES
1. J. E. Ffowcs Williams, J. Simson, and V. J. Virchis,“‘Crackle’: An annoying component of jet noise,” J.
Fluid Mechanics, 71, 251-271, (1975).
2. A. Krothapalli, L. Venkatakrishnan, and L. Lourenco, “Crackle: A dominant component of supersonic jet
mixing noise,” AIAA paper No. 2000-2024.
3. D. Papamoschou and M. Debiasi, “Directional suppression of noise from a high-speed jet,” AIAA J., 39,
380-387, (2001).
4. K. L. Gee, V. W. Sparrow, A. A. Atchley, and T. B. Gabrielson, “On the perception of crackle in highamplitude jet noise,” AIAA J., 45, 593-598 (2007).
5. S. H. Swift, K. L. Gee, and T. B. Neilsen, “Testing two crackle criteria using modified jet noise
waveforms,” J. Acoust. Soc. Am. 141, EL549-EL554 (2017).
6. K. L. Gee, P. B. Russavage, T. B. Neilsen, S. H. Swift, and A. B. Vaughn, “Subjective rating of the jet
noise crackle percept,” J. Acoust. Soc. Am. 144, EL40-EL45 (2018).
7. P. B. Russavage, T. B. Neilsen, K. L. Gee, and S. H. Swift, “Rating the perception of jet noise crackle,”
Proc. Mtgs. Acoust. 33, 040001 (2018).
8. Tam, C. K. W., Spyropoulos, J. T., Aubert, A., and Powers, R., “Crackle in the noise of high-performance
aircraft,” AIAA paper 2018-3306.
9. K. L. Gee, T. B. Neilsen, A. T. Wall, J. M. Downing, M. M. James, and R. L. McKinley, “Propagation of
crackle-containing noise from military jet aircraft,” Noise Control Eng. J. 64, 1-12 (2016).
10. K. L. Gee, T. B. Neilsen, J. M. Downing, M. M. James, R. L. McKinley, R. C. McKinley, and A. T. Wall,“Near-field shock formation in noise propagation from a high-power jet aircraft,” J. Acoust. Soc. Am.
133, EL88-EL93 (2013).
11. A. B. Vaughn, K. L. Gee, S. H. Swift, K. M. Leete, A. T. Wall, J. M. Downing, and M. M. James, “Source
localization of crackle-related events in military aircraft jet noise,” AIAA J. 59, 2251-2261 (2021).
12. A. B. Vaughn, K. L. Gee, K. M. Leete, and J. M. Downing, “Near to far field correlation of crackle-related
events in military aircraft jet noise,” Proc. Mtgs. Acoust. 42, 045008 (2020).
13. K .M. Leete, A. B. Vaughn, M. S. Bassett, R. D. Rasband, D. J. Novakovich, K. L. Gee, S. C. Campbell,
F. S. Mobley and A. T. Wall, “Jet Noise Measurements of an Installed GE F404 Engine,” AIAA 20211638.
14. R. H. Schlinker, S. A. Liljenberg, D. R. Polak, K. A. Post, C. T. Chipman, and A. M. Stern, “Supersonicjet noise source characteristics & propagation: Engine and model scale,” AIAA paper No. 2007-3623.
15. B. Greska and A. Krothapalli, “On the far-field propagation of high-speed jet noise,” Proceedings of
NCAD2008, paper NCAD2008-73071 (2008).
16. S. Martens, J. T. Spyropoulos, and Z. Nagel, “The effect of chevrons on crackle – Engine and scale model
results,” Proc. ASME Turbo Expo 2011, Paper GT2011-46417 (2011).
17. R. H. Schlinker, “Supersonic jet noise experiments,” Ph.D. Dissertation, Dept. of Aerospace Engineering,
Univ. Southern California, Los Angeles, CA, August 1975.
18. B. P. Petitjean, K. Viswanathan, and D. K. McLaughlin, “Acoustic pressure waveforms measured in highspeed jet noise experiencing nonlinear propagation.” Int. J. Aeroacoustics, 5, 193-215 (2006).
19. P. Mora, N. Heeb, J. Kastner, E. J. Gutmark, and K. Kailasanath, “Effect of heat on the pressure skewness
and kurtosis in supersonic jets,” AIAA J., 52, 777-787 (2014).
20. W. J. Baars and C. E. Tinney, “Shock structures in the acoustic field of a Mach 3 jet with crackle,” J.
Sound Vib., 333, 2539-2553 (2014).
21. Fiévet, R., Tinney, C. E., Baars, W. J., and Hamilton, M. F., “Coalescence in the Sound Field of aLaboratory-Scale Supersonic Jet,” AIAA Journal, Vol. 54, No. 1, 2015, pp. 1–12.
22. Murray and Lyons
23. K. L. Gee, T. B. Neilsen, and A. A. Atchley, “Skewness and shock formation in laboratory-scale
supersonic jet data,” J. Acoust. Soc. Am., 133, EL491-EL497 (2013).

24. J. W. Nichols, S. K. Lele, F. E. Ham, S. Martens, and J. T. Spyropoulos, “Crackle noise in heatedsupersonic jets,” J. Eng. Gas Turbines and Power, 135 (5), 051202-051202-7 (2013).
25. A. T. Anderson and J. B. Freund, “Source mechanisms of jet crackle,” AIAA paper No. 2012-2251.
26.D. Buchta and J. Freund, “The near-field pressure radiated by planar high-speed free-shear-flow
turbulence,” J. Fluid Mech., 832, 383-408 (2017).
27. P. Pineau and C. Bogey, “Numerical investigation of wave steepening and shock coalescence near a cold
Mach 3 jet,” J. Acoust. Soc. Am.149, 357-370 (2021).
28. S. A. McInerny, “Launch vehicle acoustics part 2: Statistics of the time domain data,” J. Aircraft, 33, 518523 (1996).
29. McInerny, S. A., and Ölçmen, S. M., “High Intensity Rocket Noise: Nonlinear Propagation, AtmosphericAbsorption, and Characterization,” Journal of the Acoustical Society of America, Vol. 117, No. 2, 2005,
578–591.
30. K. L. Gee, R. J. Kenny, T. B. Neilsen, T. W. Jerome, C. M Hobbs, and M. M. James, “Spectral andstatistical analysis of noise from resusable solid rocket motors,” Proc. Mtgs. Acoust., 18, 040002 (2013).
31. M. C. Anderson, K. L. Gee, C. F. Cunningham, and G. W. Hart, “Analysis of Nonlinear Acoustic
Propagation from an Atlas V Launch Vehicle,” Proc. Int. Congress. Acoust. 2022, Oct. 2022.
32. D. Fee, R. S. Matoza, K. L. Gee, T. B. Neilsen, and D. E. Ogden, “Infrasonic crackle and supersonic jet
noise from the eruption of Nabro Volcano,” Geophys.Res. Letters, 40, 2780-2785 (2013).
33. R. S. Matoza, D. Fee, J. D. Assink, A. M. Iezzi, D. N. Green, K. Kim, L. Toney, T. Lecocq, S.
Krishnamoorthy, J.-M. Lalande, K. Nishida, K. L. Gee, M. M. Haney, H. D. Ortiz, Q. Brissaud, L. Martire,
L. Rolland, P. Vergados, A. Nippress, J. Park, S. Shani-Kadmiel, A. Witsil, S. Arrowsmith, C. Caudron,
S. Watada, A. B. Perttu, B. Taisne, P. Mialle, A. Le Pichon, J. Vergoz, P. Hupe, P. S. Blom, R. Waxler, S.
De Angelis, J. B. Snively, A. T. Ringler, R. E. Anthony, A. D. Jolly, G. Kilgour, G. Averbuch, M. Ripepe,
M. Ichihara, A. Arciniega-Ceballos, E. Astafyeva, L. Ceranna, S. Cevuard, I.-Y. Che, R. De Negri, C. W.
Ebeling, L. G. Evers, L. E. Franco-Marin, T. B. Gabrielson, K. Hafner, R. G. Harrison, A. Komathy, G.
Lacanna, J. Lyons, K. A. Macpherson, E. Marchetti, K. F. McKee, R. J. Mellors, G. Mendo-Pérez, T. D.
Mikesell, E. Munaibari, M. Oyola-Merced, I. Park, C. Pilger, C. Ramos, M. C. Ruiz, R. Sabatini, H. F.
Schwaiger, D. Tailpied, C. Talmadge, J. Vidot, J. Webster, D. C. Wilson, “Atmospheric waves and global
seismoacoustic observations of the January 2022 Hunga eruption, Tonga,” Science, 377, 95-100 (2022).
34. Blackstock, D. T., “Nonlinear Propagation of Jet Noise,” Third Interagency Symposium on University
Research in Transportation Noise, U.S. Dept. of Transportation, Univ. of Utah, Salt Lake City, UT, 1975,
pp. 389–397.
35. B. O. Reichman, K. L. Gee, T. B. Neilsen, S. H. Swift, A. T. Wall, J. M. Downing and M. M. James,“Acoustic Shock Formation in Noise Propagation During Military Aircraft Ground Run-Up Operations,”
AIAA Journal, Published Online:25 May 2022 https://doi.org/10.2514/1.J060307
36. Willis, W. A., Cormack, J. M., Tinney, C. E., & Hamilton, M. F. (2022). Reduced-order models of
coalescing Mach waves, AIAA paper 2022-1792.
37. B. O. Reichman, M. B. Muhlestein, K. L. Gee, T. B. Neilsen, and D. C. Thomas, “Evolution of the timederivative skewness of nonlinearly propagating waves,” J. Acoust. Soc. Am. 139, 1390-1403 (2016).
38. S. H. Swift, K. L. Gee, T. B. Neilsen, J. M. Downing, and M. M. James, “Exploring the use of timesensitive sound quality metrics and related quantities for detecting crackle,” Proc. Mtgs. Acoust. 30,
040003 (2017).

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0515

Investigation of the thermal-acoustoelastic effect on 6061-T6
aluminum alloy using stepwise coda wave interferometry
Xu ZHANG1; Lei CHEN1; Gang LIU1; Fumin QIN1
1

China University of Petroleum (East China), China

ABSTRACTIn recent years, nonlinear ultrasonic thermal modulation has become a potential nonlinear ultrasonicnondestructive testing approach due to its high sensitivity for early performance degradation of materials.
The objective of this research is to investigate the nonlinear acoustic characteristics of materials driven by
the thermal effect. This paper conducts thermal cycling experiments on 6061-T6 aluminum alloy in various
temperature ranges. The homogeneous thermal strain of the material was ensured by adjusting the heating(cooling) rate. The stepwise coda wave interferometry was used to analyze the variation law of the relativechange in ultrasonic wave velocity with temperature. Finally, the thermal-acoustoelastic coefficient isdeveloped, which is related to third-order elastic constants of materials and is projected to become an essential
evaluation index for early material performance degradation. The results reveal that thermal strain in 6061T6 aluminum alloy causes a substantially bigger change in ultrasonic velocity than in a conventional uniaxial
loading experiment, and that shear wave velocity is more susceptible to temperature change than longitudinal
wave velocity. There is indeed a hysteresis effect between the relative change of ultrasonic velocity andthermal strain in a thermal cycle, which is connected to the temperature range.
Keywords: Thermal-acoustoelastic effect, Coda wave interferometry, Thermal modulation

1. INTRODUCTION
In recent decades, 6061-T6 aluminum alloy has been widely used in mechanical (manufacturing ofstructural components in aircraft, automobile and nuclear industries) and civil engineering due to its
excellent mechanical properties, corrosion resistance and weldability. Nevertheless, due to the
manufacturing process and operational environment, damage to the material is inevitable and will
eventually leads to structural failure.
Research has shown that the changes in material properties (microcracks, performance degradation,etc.) are closely related to the nonlinear mechanical response of the material (1-3). Therefore, theevolution of material nonlinearity is of great interest for nondestructive evaluation and structural
health monitoring. Nonlinear ultrasonic techniques have attracted considerable attention due to theirhigh sensitivity to nonlinear properties of material.
The nonlinearity of materials is divided into classical nonlinearity and non-classical nonlinearity.
The classical nonlinearity is generally related to lattice defects, dislocations, and non-harmonicity (4),while the non-classical nonlinearity is mainly related to damage such as microcracks in materials.
Based on the principle of nonlinear effects, nonlinear ultrasonic techniques can be classified into
nonlinear resonance acoustic spectroscopy (NRAS) (5, 6), second harmonic generation (4, 7), andnonlinear wave modulation spectroscopy (NWMS) (8). the NRAS (9) method measures the resonancefrequency shift when a sample is excited at different strain levels. The frequency shift is caused bymaterial softening at high strain levels. This method is usually only applied to small samples. Higher
harmonic generation (10) is a phenomenon wherein the ultrasonic waveform is distorted by thenonlinear response of the material, and then higher harmonic waves are generated. However, thismethod makes it difficult to exclude the influence of the nonlinear sources in the system during thetest process, and cannot accurately locate the position of structural damage. The NWMS method inputs
two elastic waves with frequencies f 1 and f 2 into the material, which produces sidebands (f 1 ± f 2) when
two signals pass through a damaged region.
1b20060015@s.upc.edu.cn

In summary, although these nonlinear ultrasonic techniques have achieved varying degrees ofsuccess in characterizing material nonlinearity, there are still some limitations that hinder theapplication of these methods in practical engineering.
Temperature is often an environmental variable that needs to be excluded from the experimentproducer, and even control experiments with reference samples are arranged to compensate for thetemperature effect. However, several recent literature reports evidenced that, the thermal strain
generated by temperature can be used as the driving force to excite nonlinear behaviors of material
and modulate ultrasonic waves propagating in the material. Sun et al. investigated the variation ofultrasonic wave velocity in microcracking damaged concrete samples induced by ambient temperatureand defined a thermal modulation coefficient to represent the sensitivity of wave velocity totemperature effects. It was found that the thermal modulation coefficient increased with the degree ofdamage to the concrete samples. The results of the study demonstrate the potential of the thermo acoustoelastic effect in the non-destructive evaluation of materials.
In this paper, we propose using thermal cycling experiments in different temperature ranges toinvestigate the thermo-acoustoelastic effect of 6061-T6 aluminum alloy by analyzing the relativechange in ultrasonic velocity with temperature. Based on the theory of thermo-acoustoelasticity andthe results of thermal modulation experiments, we develop a thermoacoustic elasticity coefficient
related to the third-order elastic constants to characterize the nonlinearity properties of the material.
This paper is organized as follows: In Section 2, the theory of thermo-acoustoelasticity is provided.
Section 3 describes the experimental materials, setup, and measurements, respectively. Section 4
presents the experimental results. Finally, the conclusions and discussions are drawn in Section 5.

2. THEORY OF THERMO-ACOUSTOELASTICITY
Thermo-acoustoelasticity is a nonlinear phenomenon of interest in ultrasonics because it explainschanges in wave speed of bulk waves as a function of applied thermal str ain. This section introduces
and reviews the theory of thermo-acoustoelasticity (11, 12), where the thermoelastic equations of
motion in a prestressed thermoelastic medium are derived.
2.1 Thermo-acoustoelastic constitutive equations
In thermo-acoustoelasticity theory, the medium is regarded as a hyperelastic body, i.e. all work
done on the body by external forces is stored and can be recovered when the forces are removed. The
linear elastic theory is no longer applicable for hyperelastic materials, and the third-order elastic
tensor of materials is needed to characterize the nonlinear constitutive relation. The acoustoelasticstrain energy function 𝑊 can be approximately written as
1
1(1)
𝑊 = 𝐶𝛼𝛽𝛾𝛿 𝐸𝛼𝛽 𝐸𝛾𝛿 + 𝐶𝛼𝛽𝛾𝛿𝜖𝜂 𝐸𝛼𝛽 𝐸𝛾𝛿 𝐸𝜖𝜂 − 𝛽𝛼𝛽 𝐸𝛼𝛽 𝛥𝑇
23!
Where 𝐶𝛼𝛽𝛾𝛿 , 𝐶𝛼𝛽𝛾𝛿𝜖𝜂 , 𝛽𝛼𝛽 , and 𝛥𝑇 𝐸𝛼𝛽 are the second-order elastic tensor, third-order elastic
tensor, thermoelastic tensor, and temperature change, respectively.
For a hyperelastic medium without internal constraints, the constitutive relationship can beobtained with
𝜕𝑊
1
𝑆𝛼𝛽 =
= 𝐶𝛼𝛽𝛾𝛿 𝐸𝛾𝛿 + 𝐶𝛼𝛽𝛾𝛿𝜖𝜂 𝐸𝛾𝛿 𝐸𝜖𝜂 − 𝛽𝛼𝛽 𝛥𝑇
(2)
𝜕𝐸𝛼𝛽
2
Where 𝑆𝛼𝛽 is the second Piola-Kirchoff stress tensor.
For isotropic medium, the second-order elastic tensor 𝐶𝛼𝛽𝛾𝛿 , the third-order elastic tensor(11)𝐶𝛼𝛽𝛾𝛿𝜖𝜂 , and the thermoelastic tensor 𝛽𝛼𝛽 can be written as, respectively𝐶𝛼𝛽𝛾𝛿 = 𝜆𝛿𝛼𝛽 𝛿𝛾𝛿 + 𝜇(𝛿𝛼𝛾 𝛿𝛽𝛿 + 𝛿𝛼𝛿 𝛿𝛽𝛾 )(3)𝐶𝛼𝛽𝛾𝛿𝜖𝜂= (2𝑙 − 2𝑚 + 𝑛)𝛿𝛼𝛽 𝛿𝛾𝛿 𝛿𝜖𝜂 + (2𝑚 − 𝑛)(𝛿𝛼𝛽 𝐼𝛾𝛿𝜖𝜂 + 𝛿𝛾𝛿 𝐼𝜖𝜂𝛼𝛽 + 𝛿𝜖𝜂 𝐼𝛼𝛽𝛾𝛿 )(4)𝑛+ (𝛿𝛼𝛾 𝐼𝛽𝛿𝜖𝜂 + 𝛿𝛼𝛿 𝐼𝛽𝛿𝜖𝜂 + 𝛿𝛽𝛾 𝐼𝛼𝛿𝜖𝜂 + 𝛿𝛽𝛿 𝐼𝛼𝛾𝜖𝜂 )2(𝛿𝛼𝛾 𝛿𝛽𝛿 + 𝛿𝛼𝛿 𝛿𝛽𝛾 )(5)𝐼𝛼𝛽𝛾𝛿 =
2
𝛽𝛼𝛽 = (3𝜆 + 2𝜇)𝛼 𝑇 𝛿𝛼𝛽(6)Where 𝜆 and 𝜇 are Lamé constants, 𝑙, 𝑚, and 𝑛 are Murnaghan constants, 𝛼 𝑇 is the isotropic
coefficient of thermal expansion, and 𝛿𝛼𝛽 is the Kronecker delta.

2.2 Thermoelastic equations of motion
To build the thermoelastic equations of motion in a prestressed thermoelastic medium, we classify
the wave motion of a medium under the influence of temperature into three states (natural, initial, andfinal state). As shown in Fig. 1, coordinates of a material point in the natural, initial and final states
are represented by the position vectors 𝝃, 𝑿, and 𝒙, respectively. A medium in a state free of stresswith constant temperature 𝑇 𝑁 is called the natural state. When the medium undergoes a finite static
thermal deformation with temperature 𝑇 𝐼 , it is then in the initial state. The final state occurs when an
infinitesimal deformation is superimposed on the initial position and the medium has s temperatureof 𝑇 𝐹 . Components of quantities referring to the natural state are given by Greek subscripts,
uppercase Roman subscripts give those referring to the initial state, and those refer ring to the ﬁnal
state by lowercase Roman subscripts. Thus 𝜉𝛼 , 𝑋𝐽 and 𝑥𝑗 are components of the position vectors in
the natural, initial and ﬁnal systems, respectively.

Figure 1 – A medium deformed statically from the natural state (𝝃) to the initial state (𝑿), followed by wave
motion leading to the final state (𝒙)The static deformation of a medium from the natural state to the initial state as denoted by 𝒖𝑰 =𝑿 − 𝝃, the static deformation from the natural state to the initial state is represented by 𝒖𝑭 = 𝒙 − 𝝃.
The difference between these two vectors is the dynamic displacement from the initial to the final
state (13),𝒖 = 𝒙 − 𝑿 = 𝒖𝑭 − 𝒖𝑰
(7)
Based on Pao et al (14), the Green-Lagrange strains 𝐸𝛼𝛽 at initial and final states in the natu
ral coordinate system under the influence of temperature are given as respectively
𝐼,𝐹
1 𝜕𝑢𝛼𝐼,𝐹 𝜕𝑢𝛽
𝜕𝑢𝐼,𝐹 𝜕𝑢𝜆𝐼,𝐹
𝐼,𝐹
(8)
𝐸𝛼𝛽
= (
+
+ 𝜆
)
2 𝜕𝜉𝛽
𝜕𝜉𝛼
𝜕𝜉𝛼 𝜕𝜉𝛽Assuming the superimposed wave motion is infinitesimally small, i.e.,𝐹𝐼𝐼‖𝑢𝛼 ‖ ≪ ‖𝑢𝛼𝐼 ‖, ‖𝐸𝛼𝛽
− 𝐸𝛼𝛽
‖ ≪ ‖𝐸𝛼𝛽‖(9)Then the strain tensor in the wave motion state is given approximately by
1 𝜕𝑢𝛼 𝜕𝑢𝛽 𝜕𝑢𝜆𝐼 𝜕𝑢𝜆 𝜕𝑢𝜆𝐼 𝜕𝑢𝜆𝐹𝐼𝐸𝛼𝛽 = 𝐸𝛼𝛽
− 𝐸𝛼𝛽
= (
+
+
+
)
(10)
2 𝜕𝜉𝛽 𝜕𝜉𝛼 𝜕𝜉𝛼 𝜕𝜉𝛽 𝜕𝜉𝛽 𝜕𝜉𝛼
Combining Eq. (1) and (2), The second Piola-Kirchoff stress at the initial and final states in thenatural coordinate system are given as
1
𝐼,𝐹
𝐼,𝐹
𝐼,𝐹 𝐼,𝐹(11)
𝑆𝛼𝛽= 𝐶𝛼𝛽𝛾𝛿 𝐸𝛾𝛿
+ 𝐶𝛼𝛽𝛾𝛿𝜀𝜂 𝐸𝛾𝛿
𝐸𝜀𝜂 − 𝛽𝑖𝑗 𝛥𝑇𝐼,𝐹
2
The natural state is assumed to have the temperature 𝑇 𝑁 = 25℃. If initial and final states are at
the same temperature ( Δ𝑇 = 𝑇 𝐼 − 𝑇 𝑁 = Δ𝑇 𝐼 = 𝑇 𝐹 − 𝑇 𝑁 = Δ𝑇 𝐹 ), and disregarding the higher orderterms due to the infinitesimal wave motion, then the second Piola-Kirchoff stress tensor in the naturalstate and the incremental second Piola-Kirchoff stress tensor (15) between initial and final states canbe approximated as, respectively
1
𝐹
𝐼
𝐼
𝐼
(12)
𝑆𝛼𝛽 = 𝑆𝛼𝛽
− 𝑆𝛼𝛽
= 𝐶𝛼𝛽𝛾𝛿 𝐸𝛾𝛿 + 𝐶𝛼𝛽𝛾𝛿𝜖𝜂 [𝐸𝛾𝛿𝐸𝜖𝜂 + 𝐸𝛾𝛿 𝐸𝜖𝜂 ] ≈ 𝐶𝛼𝛽𝛾𝛿 𝐸𝛾𝛿 + 𝐶𝛼𝛽𝛾𝛿𝜖𝜂 𝑒𝛾𝛿
𝑒𝜖𝜂
2
𝐼
𝐼
𝑆𝛼𝛽
= 𝐶𝛼𝛽𝛾𝛿 𝑒𝛾𝛿
− 𝛽𝑖𝑗 𝛥𝑇 𝐼
(13)
Where
1 𝜕𝑢𝜖 𝜕𝑢𝜂
1 𝜕𝑢𝛾𝐼 𝜕𝑢𝛿𝐼𝐼𝑒𝜖𝜂 = (+
) , 𝑒𝛾𝛿
= (
+
)
(14)
2 𝜕𝜉𝜂 𝜕𝜉𝜖2 𝜕𝜉𝛿 𝜕𝜉𝛾
The thermoelastic equation of motion in the natural coordinate system is given by
𝜕
𝜕𝑢𝛼
𝜕𝑢𝛼𝐼
𝜕 2 𝑢𝛼𝐼(𝑆𝛼𝛽 + 𝑆𝛽𝛾+ 𝑆𝛽𝛾) = 𝜌𝑁
(15)
𝜕𝜉𝛽
𝜕𝜉𝛾
𝜕𝜉𝛾
𝜕𝑡 2
Where 𝜌𝑁 is the density in the natural state. For homogeneous pre-deformations, the equation of
motion in Eq. (14) simplifies to𝜕 2 𝑢𝛾𝜕 2 𝑢𝛼
𝐴𝛼𝛽𝛾𝛿
= 𝜌𝑁
(16)
𝜕𝜉𝛽 𝜕𝜉𝛿
𝜕𝑡 2
Where
𝜕𝑢𝛾𝐼
𝜕𝑢𝛼𝐼
𝐼
𝐼
𝐴𝛼𝛽𝛾𝛿 = (𝐶𝛽𝛿𝜖𝜂 𝑒𝜖𝜂− 𝛽𝛽𝛿 𝛥𝜃 𝐼 )𝛿𝛼𝛾 + 𝐶𝛼𝛽𝛾𝛿 + 𝐶𝛼𝛽𝜌𝛿
+ 𝐶𝜌𝛽𝛾𝛿+ 𝐶𝛼𝛽𝛾𝛿𝜖𝜂 𝑒𝜖𝜂
(17)
𝜕𝜉𝜌
𝜕𝜉𝜌

3. MATERIALS AND METHODS
3.1 Materials
The experiments were performed with plate samples of two materials: 6061-T6 aluminum alloyand fused quartz. All samples have the same dimensions of size 350 mm×300 mm×4 mm. 6061T6 aluminum alloy sample is used to study the thermal-acoustoelastic effect and the fused quartz
sample is used to calibrate the temperature deviation of the ultrasonic transducer due to its small linear
thermal expansion coefficient. The surfaces in the thickness direction of the samples were polished toensure better coupling with the transducer.
Before the thermal modulation experiments, we measured the Shear (S) and longitudinal (P) wave
velocity of the samples at 25°C using the pulse-echo method. The isotropic thermal expansion
coefficient of specimens in the range of 20 to 100°C was measured using thermal mass analysis (TMA).
The characteristics of the samples are summarized in Table 1.
Table 1 – Measured geometry and physical properties of the samples
Material/Sample

6061-T6
Aluminum alloy

SiO2
Fused quartz

Length×width (mm×mm)

350×300

350×300

Thickness (mm)44Density 𝜌0 kg/m

2691.1

2200

Longitudinal wave velocity (m/s)

6440.15754.2

Shear wave velocity (m/s)

3158.64048.4

Thermal expansion coefficient 𝛼 𝑇 (/℃)

2.1×10-5

5.4×10-7

3

3.2 Experimental Setup
The experimental setup for thermal modulation is shown in Fig. 2, where the sample is placed in
the environmental chamber for ambient temperature control and monitoring. Two ultrasonictransducers were installed at one end of the sample thickness direction (4mm), namely, a 2.25 MHz P
wave transducer (Olympus A106S) polarized in direction 1 and a 2.25 MHz S wave transducer(Olympus V154) polarized in direction 2, which are used to transmit and measure P and S wave. To
improve the transmission efficiency of acoustic energy, B2 glycerin, and SWC-2 shear gel were used
as ultrasound couplants to thinly coat the surface of P wave and S wave transducers respectively.
An Olympus 5072R ultrasonic square pulser/receiver in the pulse-echo mode was used to drive theultrasonic transducer and measure echo waves in direction 1. The pulse repetition frequency was set

as 100Hz. The received echo wave signals were digitized and recorded by a digital oscilloscope(Tektronix DPO 2002B) at a sampling rate of 1 GS/s. Each signal was averaged 512 times to increase
the signal-to-noise ratio (SNR), and the echo wave signals were recorded every 0.2 ℃ during thethermal modulation process. The Open Choice Desktop software can communicate with theoscilloscope and use waveform data capture to transfer the waveform to the computer for subsequent
signal processing.

Figure 2 – Experimental setup used to measure the bulk waveform various temperatures
To quantify the impact of the variability due to the position of the ultrasonic transducers on thethermal modulation test, the thermal modulation test was repeated three times with systematic removal
and repositioning of the two ultrasonic transducers. In each experiment, a sample was heated from
25 °C to the specified temperature (30 °C, 35 °C) and then cooled down to 25 °C at a rate of 0.2 ℃/40 min, a total of two heating and cooling cycles. This rate of temperature change can avoid
temperature gradient and ensure uniform thermal strain in the sample.
3.3

Measurements

3.3.1 Stepwise Coda Wave Interferometry
Coda wave interferometry (CWI) was proposed by Snider et al. (16) as a high-precision method
for evaluating very small wave velocity variations of a medium and is widely used in the field ofgeological engineering. It has also been applied to ultrasonic-based NDT and SHM, to capture subtle
time-lapse variations in the medium due to various impacts of stress (17, 18), humidity (19), andtemperature (16, 20).
Coda waves are formed by multiply scattering in the medium. It is more sensitive to small changes
in the properties of the medium due to its long propagation path compared to direct waves . Fig. 3
shows two P-waveforms recorded for the 6061-T6 aluminum alloy at a temperature of 25 ℃ andheated to 26 ℃, respectively. Note that the time shift in direct arrivals is almost indistinguishable
while the coda waves show an apparent time shift.

Figure 3 – The average full waveforms collected from the 6061-T6 aluminum sample (2 mm) by the P
wave ultrasonic transducer at 25 and 26 ℃
The primary signal processing approach for CWI is to determine the relative velocity change by
the time shift between the coda waves obtained at different times. There are two main methods used
in CWI for estimation of the relative time shift between waveforms, the d oublet method (16) and thestretching method (21). The advantage of the stretching method is that the size of the time window ismuch larger (even the whole signal can be used) than that used in the doublet method, and thecalculation results are more stable, reliable, and precise. Therefore, the stretching method is used to
calculate the relative time shift of code waves in this paper.

(a) Original signals(b) Stretching of the reference signal
Figure 4 – Schematics of signal stretching for coda wave interferometry (CWI)
In the stretching method, a time delay of Scur(t) is modeled by stretching the time axis of Sref(t),schematically illustrated in Fig.4. First, Sref(t) is interpolated at times t(1+ϵ) with various values of ϵfor the signal stretching in the time window of [t1 , t2 ]. Then the stretching factor ( ϵ𝑚𝑎𝑥 ) is obtainedby selecting a ϵ value that maximizes the correlation coefficient ( 𝐶𝐶) between S ref(t(1+ϵ)) and Scur(t)as follows:𝑟𝑒𝑓(18)
𝑆𝜖 (𝑡) = 𝑆 𝑟𝑒𝑓 (𝑡(1 + 𝜖))𝑡𝐶𝐶(𝜖) =
and𝑟𝑒𝑓∫𝑡 2 𝑆𝜖 (𝑡)𝑆 𝑐𝑢𝑟 (𝑡)𝑑𝑡1

1
𝑡2 𝑟𝑒𝑓𝑡2 𝑐𝑢𝑟22(∫𝑡 (𝑆𝜖 (𝑡)) 𝑑𝑡 ∫𝑡 (𝑆 (𝑡)) 𝑑𝑡)2
1
1

(19)

𝛿𝑡
𝛿𝜐
(20)
=−
𝑡
𝜐
𝜖
In our experiments, the correlation between the initial waveform and the subsequent waveform
deteriorates with temperature when the discrepancy from the initial temperature is large. To overcome
𝑖,𝑖−1
this issue, we calculate the stepwise relative time shift 𝜖𝑚𝑎𝑥
between consecutive waveforms 𝑆 𝑖−1
and 𝑆 𝑖 at the adjacent temperature interval (0.2 ℃) and then multiplied to provide a time shift on the
cumulative changes relative to the initial temperature of the experiments. With these modifications,
Stepwise CWI (22) now is expressed as

ϵ𝑚𝑎𝑥 = 𝑎𝑟𝑔𝑚𝑎𝑥(𝐶𝐶(𝜖)) =𝑡𝑖

𝐶𝐶 (𝜖) =and∫𝑡 2 𝑆 𝑖−1 (𝑡(1 + 𝜖))𝑆 𝑖 (𝑡)𝑑𝑡1

1
𝑡𝑡(∫𝑡 2(𝑆 𝑖−1 (𝑡(1 + 𝜖)))2 𝑑𝑡 ∫𝑡 2(𝑆 𝑖 (𝑡))2 𝑑𝑡)2
1
1

(21)

𝑘
𝛿𝑡
𝑖,𝑖−1
(22)
= ∏ (𝜖𝑚𝑎𝑥+ 1) − 1𝑡𝑖=1
3.3.2 Calibration of the ultrasonic transducer
To ensure the accuracy of the experimental results, it is necessary to account for the apparent time
shifts caused by the external deviation of the experimental procedure, including the temperature
response of the ultrasonic transducer. Because of the low thermal expansion coefficient of fused quartz,
the time shift produced by itself should be equal to the same order of magnitude regarding theinﬂuence of thermal expansion. Any deviation from these relative time shifts is expected to originate
from external factors. HereIn this paper, the thermal modulation experiment of fused quartz plates with the same geometricsize is carried out by using the same experimental equipment (23). The heating and cooling cycle in
the temperature range of 25 ~ 40 ℃ is repeated three times. The relative time shift per 0.2 ℃ is
calculated by using Stepwise CWI. The experimental results are shown in Figure 5.

(a) S wave ultrasonic transducer(b) P wave ultrasonic transducer
Figure 5 – The stepwise CWI relative time shift per 0.2℃ with temperature calculated from waveforms
recorded over three cycles of heating and cooling of a fused quartz sample
The data in the figure are the average values of the three repeated experiments. It can be foundfrom Fig. 5 (a) that the positive relative time shift (about 1.16×10 −4) with the increase of temperatureis measured by the S wave transducer, and the negative relative time shift (about 1.27 ×10 −5) with theincrease of temperature is measured by the P wave transducer, which is significantly greater than thethermal expansion coefficient of fused quartz (5.4×10−7). We expect that the observed relative time
shift (Figure 5) may be due to wave scattering/reﬂection within the steel casings of the sensors, andthe sensitivity of the electro-elastic response of the piezo-ceramic transducers to temperature.

4. EXPERIMENTAL RESULTS AND DISCUSSIONSIn this section, we investigate the variation of ultrasonic velocity with tempe rature in thermal
modulation experiments using the stepwise CWI mentioned in Section 3.3.

(a) S coda wave in the time window of[20.3 μs, 21.4 μs](d) P coda wave in the time window of[64.4 μs, 66 μs]

Figure 6 – P- and S- waveforms of 6061-T6 aluminum alloy sample under the thermal modulation heating
experiment showing coda waves shifting right with temperature
Fig. 6 shows the pulse-echo waveforms acquired by the S wave and P wave ultrasonic transducer
during the thermal modulation heating experiment (temperature range is 25 °C to 35 °C) at specifictime windows, where the S-waveforms are selected for [20.3 μs, 21.4 μs] and the P-waveforms are
selected for [64.4 μs, 66 μs]. It can be found from the figure that with the increase in temperature, it
can be clearly observed that there is a positive time shift in the coda wave, indicating that the wave
velocity decreases. In addition, the information on waveform changes with temperature includes not
only time shifts but also amplitude changes, although in this paper only the time shifts of the waveform
are analyzed. Subsequently, the relative changes in S wave and P wave velocity were extracted using
Equations (21) and (22).
Fig. 7 presents the results for relative P wave and S wave velocity changes due to temperaturechange. The data points in the figure are the average values of the three repeated experiments. The
results show that the S wave velocity is more sensitive to changes in temperature than the S wave
velocity, consistent with the findings of Weaver and Lobkis (24). Compared with the acoustoelastic
data of uniaxial experiments by Zhong et al. (25), it can be found that the relative wave velocitychanges due to the thermal effect are greater than that caused by uniaxial loading under the samestrain.

(a) S wave relative velocity change in heating
cooling cycle 1(c) P wave relative velocity change in heating
cooling cycle 1(b) S wave relative velocity change in heating(d) P wave relative velocity change in heating
cooling cycle 2
cooling cycle 2
Figure 7 – Thermal cycle measurements in the 6061-T6 aluminum alloy: relative change of ultrasonic wave
velocity as a function of the temperature. Blue circles indicate decreasing temperature while red circles
indicate increasing temperature
Many previous studies(20, 26) indicate the linear relationship between the relative velocity change
and the temperature change in metals. Figure 7 demonstrates that in thermal cycling experiments on
6061-T6 aluminum alloy, there is a thermal hysteresis between the relative wave velocity change andtemperature, i.e. the slope of the relative change in wave velocity with temperature is not equal
between the heating and the cooling process. A comparison of Fig. (a), (b), (c), and (d) shows that thethermal hysteresis effect is more pronounced for S waves than for P waves and that t he area of thethermal hysteresis region increases as the thermal modulation temperature range increases.
The theoretical derivation in section 2 shows that the relative variation of wave velocity with

thermal strain is related to the third-order elastic constants of materials. In this paper, the averageslope of the relative change in wave velocity with temperature in a thermal cycling experiment isdefined as the thermal-acoustoelastic coefficient (TAEC) and used to characterize the classicalnonlinearity of materials.

5. CONCLUSIONSIn this work, a nonlinear ultrasonic thermal modulation method is developed for the detection ofearly damage in 6061-T6 aluminum alloy. The method uses thermal strain as the driving force forwave velocity variation, designs thermal cycling experiments for different temperature ranges, andanalyzes the acquired pulse-echo signals using stepwise coda wave interferometry to extract therelative change of ultrasonic wave velocity with temperature. The experimental results show that thesensitivity of wave velocity changes due to thermal strain is higher than that of acoustoelastic effects,
and the sensitivity of the S wave to temperature changes is stronger than that of the P wave. It wasalso found that in thermal cycling experiments, both S wave and P wave have thermal hysteresis
effects, and the slope of the relative change in wave velocity with temperature is not equal during theheating and cooling process, where the thermal hysteresis phenomenon is more obvious in transverse
waves, and the larger the temperature range, the larger the area of the thermal hysteresis region
This study shows that the selection of shear wave is beneficial to improve the sensitivity of
nonlinear thermal modulation detection. The TAEC proposed in this paper are related to the thirdorder elastic constants of materials, which can be used to characterize the nonlineari ty of materials
and serve as an important evaluation index for the early performance degradation of materials.

ACKNOWLEDGEMENTS
This study was supported by the Fundamental Research Funds for the Central Universities (No.
20CX02403A), Shandong Provincial Key Laboratory of Oil and Gas Storage and Transportation
Safety Development Fund, the Postgraduate Innovation Project of China University of Petroleum (No.
YCX2021092).

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PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0608

Stepped Plate Parametric Array Loudspeaker: An In-Air Wideband
Ultrasonic Source Transducer
Wonkyu Moon1; Beomseok Oh2;
1

Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang
37673, Republic of Korea

2

Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang
37673, Republic of Korea

ABSTRACTThe stepped-plate parametric array loudspeakers (SPPALs) technology has been developed in VATrans Lab.
of POSTECH since 2008 when we started for the way to evolve a wide-band ultrasonic sound beam radiationtransducer in air from the conventional stepped plate radiator for single-frequency ultrasonic sound beams,
which was first proposed by Gallego et al. In 2010, sound beams at two frequencies are shown to be
successfully radiated from a stepped-plate radiator with modified steps. In 2018, we also succeeded in making
a modified stepped plate radiator that can emit sound beams of wide frequency bandwidth, and then name it
as Stepped Plate Parametric Array Loudspeakers (SPPALs) because it can generate wide-frequency-band
audible sound beams. Until now, the SPPALs technology is still evolving by adopting light and flexible
polymer steps to tolerate the difficulties in the design process for equipped with multi-resonance around theoperating frequency and reduce the power dissipation in piezoelectric stack actuators. In this paper, thedevelopment of SPPALs technology is briefly reviewed and its present status is also presented.
Keywords: Parametric array, Directional loudspeaker, Stepped-plate transducer, Sound beams

1. INTRODUCTION
Many people accustomed to acoustic devices know well about the Parametric Array (PA)loudspeaker that can generate extraordinarily directional audio sounds, but it does not seem to be
widely used in everyday life. One may imagine that the PA loudspeaker (PAL) could be substituted
for the earphones in some circumstances and that it could transfer some pieces of important
information, by use of highly direction sounds, only to a person who stands at a specified spot. Such
imagination cannot be realized because the conventional PALs in the market cannot make it happen.
The characteristics of the current PALs such as its directivity, sound pressure level and reaching rangeof audio sounds, are almost the same. It may be said that the current technology level for the PALs
has not been well developed yet to realize the various wishes using the PALs that can be imagined.
The basic working mechanism of PALs would be well explained by use of the difference frequency
generation phenomenon during the progressive transmission of strong sound beams with more than
two frequencies. The theories can be found in the nonlinear acoustics, which can explain precisely
and predict the PA sounds generated through the interactions between strong sound beams withdifferent frequencies. Once the strong sound beams are identified to generate the highly directional
PA sounds, an appropriate sound beam emitter should be designed and fabricated to realize the PALs.
Therefore, in order to control the acoustic properties of target PA sounds, the Parametric Acoustic
Array phenomenon should be understood in detail and choose some specific conditions for generationof target sounds by considering the required functions and properties. Nowadays the numericalsimulation tools are well developed to predict the PA phenomenon, hence they can be used for concept
design of PALs. However, they are not widely used in the design processes for the PALs probably
because the are not sufficiently user-friendly, hence, the PALs-developing engineers could utilize them
12

wkmoon@postech.ac.kr
obs7869@postech.ac.kr

with ease in the design processes for PALs. The difficulties in learning the nonlinear acoustics, may
provide engineers with another strong reason for rare use of the analytic or numerical tools.
The in-air radiators for strong sound beams must be one component that supplies many difficult
technological hurdles to be cleared. In the conventional PALs, ultrasonic sound beams are usuallyused as the primary waves to indirectly generate the secondary (PA) audible sounds with highdirectivity so that they could be transmitted only to the listeners in a certain direction, since the highly
directional primary ultrasonic sound beams can endow the PA audio sound with similar high directivity.
All PALs in the market use the 40-kHz ultrasonic primary sound beams as the primary ultrasonic beam.
This implies that most of the PA sound generated by the conventional PALs might have almost thesame properties. In addition, the unit drivers for emitting the primary ultrasonic sound beams are
identical in most of the current PALs, and all PALs are formed by use of the massive array of ultrasonicunit drivers. Since Yoneyama first suggested the concept of PALs in 1983, the piezoelectric bimorphbased transducer, used typically for automobile ultrasonic ranging sensors, is normally adopted as theunit radiation driver for the PALs. Since the conventional PALs are fabricated by use of assembling alarge number of unit drivers, the manufacturing cost of conventional PALs should be too high.
Moreover, the special signal processing unit and the power amplifier unit are also required to properly
use the PALs, which makes to PALs are too expensive and have limited applications to become apopular acoustic device in everyday life for its commercialized period of 24 years until today.
In 2018, Hwang et. al in VATrans Lab. of POSTECH, has published a paper that reports theirsuccessful development of a mono-driver PA loudspeaker named as Stepped Plate PA Loudspeaker(SPPALs) by modifying the conventional Stepped Plate Radiator, known as a mono-frequency soundbeam source transducer, suggested and developed by Gallego et. al. The second version of SPPALs
was built in 2019 and reported to the public in 2022. A new fabrication technology has been proposed
and adopted in the study so that the severe difficulties could be released in prediction of the natural
frequencies and their corresponding mode shapes for the design of a required stepped circular plate,which are the inevitable information to determine the locations of the steps on the flat circular plate.
Recently we have designed and fabricated the third version of SPPALs to improve the performance.
In this paper, the history of the SPPALs built in VATrans Lab. is briefly introduced. In thedescription of our development, the design and the fabrication techniques for the SPPALs would betreated in the aspect of how they are interconnected each other, in addition to the background acoustic
theoretical models.

2. CONCEPTUAL DESIGN OF PALS
2.1 Parametric Acoustic Array Phenomenon
The parametric acoustic array, frequently abbreviated as PA, is an acoustic phenomenon thatgenerates extraordinarily directional low-frequency sounds from highly directional sound beams at
higher frequencies radiated from a relatively small aperture [1, 2]. The mechanism of PA is frequently
considered as the difference frequency sound generation through nonlinear interactions between
strong sound beams with two frequencies: A strong sound beam with two frequencies f1 and f2 would
generate secondary sounds with the following other frequencies through nonlinear interactions
between the sounds at the two frequencies during the propagation: f1 − f2 , f1 + f2 , 2 f1 , 2 f2 , 3 f1 ,3 f2 , ! . Figure 1 shows a conceptual illustration of the difference frequency wave generation
phenomenon through the non-linear acoustic interaction. The difference frequency sound would beheard by the human if the primary sounds are ultrasonic. And it would survive after long propagation
since the absorption of ultrasonic sounds would be much higher than that of audible sounds.

Figure 1 – Conceptual diagram of the frequency conversion by non-linear interaction in the medium.

2.2 Theoretical Model for the PA
The Westervelt equation may be the nominal model equation of the acoustic wave behaviors
including the comprehensive nonlinear cumulative effects due to interactions in strong sound beams.
In fact, the Westervelt equation is an approximation of the second-order wave equation by ignoring
the local effects and it includes the nonlinear terms as given in the right hand side of the following
equation in addition to the linear acoustic equation term as given in its left hand side:1 ∂2 p δ ∂ 3 p
β ∂2 p 2,(1)∇2 p − 2 2 + 4 3 = −
c0 ∂t
c0 ∂t
ρ0 c0 4 ∂t 2
where δ is the sound diffusivity and β is the non-linear coefficient [3,4]. Although Equation (1) can
predict the propagation characteristics of waves in all directions, since the attentions are usuallyfocused only on the major part of sound beams in many applications of nonlinear acoustic
phenomenon, the following KZK (Khokhlov-Zabolotskaya-Kuznetsov) equation is preferred byresearchers on the PAL and other applications of nonlinear acoustics:∂ 2 p c0 2
δ ∂3 pβ ∂2 p 2,(2)= ∇⊥ p ++∂z ∂τ 22c0 3 ∂τ 3 2 ρ0 c0 3 ∂ p 2
where ∇ ⊥ 2 = ∂2 ∂r 2 + (1 r ) ⋅ ∂ ∂r . [3]
It is well-known that the analytical or numerical predictions based on Eq. (2) might be valid inside
the circular conical region defined by its aperture less than double of 20 degrees [5].
Several analytic approximate solutions are known for the Westervelt equation as well as the KZK
equation and several numerical simulation packages are also well developed on both. Therefore, thenumerical predictions are possible on the PA sounds for a given sound source condition. The spatial
acoustic pressure distribution of PA sounds generated from a given sound beam is shown in Fig. 2(b),while that of the same frequency sounds as that of the PA is shown in Fig. 2(a) if it is directly radiatedfrom a sound source with an identical aperture size.

Figure 2 – Sound pressure field of (a) conventional source and (b) parametric array source with a same
aperture size and frequency.
However, either the analytic solutions or the numerical simulation tools do not seem to be well
utilized in the design process for PALs. It might be because it usually takes too long time to prepare
oneself for using them. In order to properly prepare oneself an engineer should learn the pretty
complicated nonlinear acoustic theory and how to use the numerical simulation tools on nonlinear
phenomena in acoustics for obtaining the necessary information from simulations for PALs design.
Moreover, the commercial values of PALs are not sufficiently high to get sufficient investment, which
may be the real cause of the too low speed in the PAL technology development.

3. GENERATION OF SOUND BEAMS IN AIR
3.1 Difficulties to overcome for building an in-air sound beam radiator
As mentioned in the above, the PAL requires ultrasonic sound beam radiators in air. The large
mismatch in radiation mechanical impedance in air (415 Rayls) and the internal transducer mechanical
impedance (e.g., 34 MRayls for PZT) [6]. This mismatch in radiation impedances may be overcome

by introducing plate-like structures to magnify the radiation area, but the bending deformation in theplate-shaped structure would hinder the advantage by the introduction, since the frequency offundamental mode of a plate-form structure with a typical size and geometry is less than 20 kHz,
which is the maximum frequency of audible sounds. Specifically speaking, a circular flat plate could
not emit a sound beam at an ultrasonic frequency if it is excited at its center point in the axial direction,
since the the phase differences between the areas separated by the nodal lines would cancel out thesound pressure on the acoustic axis [10]. It is miniaturization that makes it possible to escape fromthe effects by the bending deformation. It is, hence, the reason why in-air ultrasonic radiators would
be composed of small sized unit radiators such as the piezoelectric bimorph type radiator in theultrasonic ranging sensor, which is widely used for automobiles. [6−9]In 1972, Gallego et. al proposed an in-air sound beams source transducer composed of a circularplate with steps at its appropriate locations, a Langevin-type mechanical resonator with several PZT
discs and a horn attached at its one end, which connects the resonator with the circular plate [10]. It
was shown that the transducer, called Stepped Plate Transducer, can emit an ultrasonic sound beameffectively.
3.2 Stepped Plate Transducer: Mono-frequency ultrasonic sound beam radiator
As mentioned above, the bending modes of a circular plate may cause destructive interference ofradiated sound waves due to the inconsistent phases [10]. Gallego et al. proposed a method to
overcome the phase differences due to the vibration modes of a plate structure; appropriate steps are
installed on the circular plate surface with the height of the half-wavelength of the target ultrasonicsound in air so that the out-of-phase motion of one area could radiate sound at the location halfwavelength forward to the propagating direction (see Fig. 3(b)) [10]. After the half period, the soundradiated on the plate surface would meet the sound made from the top of steps, which starts in thehalf-wavelength in front to the propagating direction with the half-period time lag. The sounds fromthe bottom and the top of steps would be in phase, hence, when they meet each other in the position
sufficiently far away from the radiating surfaces. It is the way to overcome the phase differences inthe sounds radiated from the out-of-phase regions in the circular plate due to the mode in the steppedplate radiator.
The concept of the stepped plate transducer has been well realized and used as an effective
ultrasonic sound beam radiator in air more than last 50 years. In 2006, we started investigating thepossibility to adopt the stepped plate transducer as the transducer mechanism for building a new lowcost single-body PA loudspeaker. The first thing to do must be to modify the mono-frequency in-air
sound beam radiating stepped plate transducer so that it could generate multi-frequency sound beams.
It took less than 4 years to attain the first goal in the way to the final remote target.

Figure 3 – Stepped-plate directional ultrasonic radiator: (a) transducer structure and (b) first description of
the phase compensation principle [10].
3.3 Modified Stepped Plate Transducer: A bi-frequency sound beam source
In 2010, Je et al., in VATrans Lab., reported a modified stepped-plate radiator that can radiate
directional ultrasonic sound beams with two frequencies [6]. The basic concept of the proposed
modification is to properly install the steps with two different heights on the appropriate locations on
the circular plate. The locations as well as the heights should be determined by the given twofrequencies for sound beams. The limitation of the approach is that the ratio of the two frequencies
should be a rational number. However, it is not a severe limitation in engineering point of view because
the rational numbers are dense in real numbers. In addition to bi-frequency sound beam generation,
the mono-frequency PA sound can be successfully observed in the experiments reported in the paper[6].
However, since the PAL requires wide-frequency-band sound beam radiation, it seemed to be so
deficient in building a PALs using the stepped plate transducer technology, but obviously it was thethe first step to the final goal, the first SPPALs. It took about 8 years for the first SPPALs to appear
in VATrans Lab.

Figure 4 – Bi-frequency stepped plate radiator: (a) transducer structure, (b) two types of the steps for phasecompensation of two ultrasonic frequencies and (c) radiation patterns of the generated difference frequency
waves.

4. STEPPED PLATE PARAMETRIC ARRAY LOUDSPEAKERS (SPPALs)
4.1 SPPALs: a Wideband Modified Stepped Plate Ultrasonic Emitter
In 2018, Hwang et al. reported their success in constructing a new PAL based on the Stepped Plate
Transducer, which can be seen as an upgraded version of the first in-air PA sound source transducer
developed by Je. The new PAL was named as Stepped Plate Parametric Acoustic Array Loudspeaker(SPPALs) [11]. In addition to the ingenious design methods of the height and locations of the steps
on the circular plate, the multi-resonance structure is also intentionally devised so that the vibration
amplitudes around the operating ultrasonic frequency would be increased by the closely-located multiresonances inside the operating frequency band. The design processes are so complicated and timeconsuming to find the proper configurations of the stepped plate and the Langevin transducer with ahorn since the design parameters are do strongly coupled each other that all design parameters should
be determined simultaneously together. Even after fixing all parameters properly, the temperature
increases in the PZT actuating discs has been found while the SPPALs is operated during the timeshorter than 5 minutes. Despite the several technical issues, it is obviously the first SPPALs everdesigned and built.

Figure 5 – First version of SPPALs. (a) Transducer structure and (b) frequency response in audible range.
The configuration of the first SPPALs is shown in the left side of Fig. 5 and its right side shows
the frequency response function (FRF) of the output PA sounds relative to the input signals. Since thesystem is nonlinear, the shown FRF does not need to be valid for any input signals with arbitrarylevels and frequency contents. However, the acoustic output from it suggests that it could be aloudspeaker for wideband sounds. Since the SPPALs have a simpler structure than the conventional
array-type PALs, it can reduce the manufacturing cost considerably so that it could make the PALs to
become a more popular device widely used in everyday life.
4.2 PC-SPPALs: SPPALs with polymer composite steps
The first version of SPPALs was designed and built based on the design methods proposed by
Hwang. He suggests the method to determine the heights and locations of steps on the circular platesurface such that they could compensate the phase difference due to the vibration modes more
effectively in the wide frequency bands provided that the vibration pattern of plate would not changemuch. Obviously, there is no exact one configuration that satisfies the requirements. It should be amatter of compromises and object-based optimizations. Moreover, it is extremely difficult to predictthe vibration mode shape of a circular plate with arbitrary addition of steps on the surface. Even theresonance frequency would be altered by adding some steps [11]. Hwang suggests a design guide forbuilding a proper stepped plate for a given operating frequency that the stepped plate should be sodesigned that its basic thickness would be sufficiently large; the thickness of plate cannot be smaller
than the maximum step height [11]. This guideline may be adoptable since the wavelength of operating
ultrasonic sound is sufficiently small. And it brings about the low radiation efficiency and the longdesign processes to find out the proper geometry of an appropriate stepped plate for a given operating
conditions. In addition, the entire stepped plate and the driver with the horn should be fabricated as
one piece without any bolt joints. Even the welding is not allowed due to the fact that the structure
should be in highly stressed states during the operation period. In other view point this manufacturing
constraints provide a critical disadvantage in lowering the fabrication cost.
Another problem in the first version of SPPALs was degradation in the performance caused by thetemperature increase in the PZT body. Under the high voltage driving condition, the equivalentmechanical Quality factor of the PZT discs is the lowest among all components of the SPPALs sinceothers are composed of aluminum [12−14]. It leads to a relatively large temperature increase in the
PZT actuating parts to result in the shift in their material constants and in the system resonancefrequency in turn, which eventually causes to reduce the vibration amplitude of the plate and thevolume of sounds.
To overcome the problems described above, we started seeking for the solution in 2017. One
approach to the solution is to fabricate the steps composed of such a light and flexible material, andattach them on the circular flat surface, that the resulting stepped plate would preserve the originalnatural frequencies and the corresponding mode shapes of the flat one. The original purpose of thisapproach was to reduce the design difficulties of the proper stepped plate for a given operating
frequency, which turns out, however, to solve the second problem either. In Figure 6(a), the circularplate is depicted in the photograph with the polymer composite steps and the horn connecting thedriver with the center of plate.

Figure 6 – SPPALs with the composite polymer steps. (a) Transducer structure, (b) mode shapes with andwithout composite polymer steps, and (c) frequency response in audible range, respectively.
The detailed report is published in 2022 on the Polymer-Composite-Stepped Plate PA Loudspeaker(PC-SPPALs) by Kim et. al [15]. The first step in the research work might be to find the required
values for the Young's modulus and mass density so that addition of the steps would not change theplate dynamic property. After finding the values, we need to search for specific real materials
satisfying the requirements. The chosen composite polymer has a density of 1100 kg/m3 and Young’s
modulus of 3.5 GPa [15]. It was shown that the composite polymer steps would make negligible effects
on both the flexural vibration modes and the resonant frequencies. With the help of the chosen material,

Kim et al. successfully minimized the thickness of the plate. Moreover, the relatively higher value of
mechanical damping in the polymer composite material would considerably reduce the energy
dissipation in the PZT actuating parts so that the temperature would not severely change to cause theperformance degradation. In Fig. 6(b), the mode shapes of the circular flat plate (left) and the circularpolymer-composite stepped plate while the FRF is given in Fig. 6(c) of the PC-SPPALs built in the
research work depicted in Reference 15.
4.3 Research on Improvement of performance of SPPALs
Although adoption of the polymer composite steps solves the two serious problems in the firstversion of SPPALs, several shortcomings are still left. One may be the insufficient strength of target
audio sounds. In order to increase the target sound loudness, the vibration amplitude of the steppedplate should be increased. There are various ways to increase the vibration amplitudes of the steppedplate since it is determined by the detailed design of the horn and the Langevin transducer parts. Since
the dimensions of the second version of SPPALs were carefully determined considering themanageability of the design-by-analyse routines in addition to the directivity and audible range of PA
sounds, there is strong needs to maximize the vibration amplitude by optimizing the whole structureof SPPALs including the Langevin transducer, the horn, and the stepped plate, which would respondas a one body after connecting the Langevin transducer with the plate through the horn. The
manufacturing techniques also effectively restricts many options in design parameters.
Other shortcoming is found in the directional pattern of the audible PA sounds. In a certain
frequency band inside the audio region, the PA sounds can be heard outside the target direction suchas 30 degrees off from the acoustic axis of the SPPALs. It could be improved by optimizing the heightprofiles and their locations, but it has not been done yet.
We have focused our attention on increasing the audio sound volume in last 3 years. The first stepto the goal is to develop an appropriate model for concept design of the SPPALs structure, whichshould be sufficiently light for repeated numerical simulations in finding the optimum design
parameters. Oh has developed a pretty simple model called Simplified Vibration Model (SVM) to
predict the dynamic responses of the whole system of PC-SPPALs [16]. In the SVM, the 1-dimensional
longitudinal vibration of a rod is adopted to express the vibration behaviors of the horn part, the head
and tail parts of Langevin transducer, and the piezoelectric actuating part. It is noteworthy that a moreaccurate model for the polymer-composite-stepped plate is developed based on the modified Mindlin
plate theory (MMPT), which is an advanced plate theory to accurately predict the behavior of themoderately thick plate at its higher mode vibrations [17]. A better design of PC-SPPALs can be devisedin the viewpoint of the audio generating performance by use of an objective function defined as
Performance Index (PI), which is related with the ‘target slant frequency response region’ that should
be determined by the closely spaced two resonance frequencies of the circular plate and the driverpart [16]. The fabricated newly-designed PC-SPPALs can achieve 8.2 dB increase in the average soundpressure level (SPL) of audible PA sounds and 10 dB increase of 1 kHz PA sound.
Even though the PA sound volume has been considerably improved compared to the previous
versions of SPPALs, it does not seem to be sufficient. Hence, a critical change in the structure andmechanism of the driver may be required to meet the target PA sound volume. In addition, thedirectional characteristics of SPPALs should be also improved. Therefore, technology development
needs to be continued to improve the performance of SPPALs.

5. CONCLUSIONSIn this paper, the stepped-plate loudspeaker (SPPALs) technology is briefly discussed in its
historical viewpoint and its technological status. The stepped plate transducer, a mono-frequency inair sound beam radiator, has been evolved, through a bi-frequency in-air sound beam emitter and amono-frequency PA sound beam radiator, to the wideband PA sound beam radiator, which implies thatit becomes a Parametric Acoustic Array Loudspeaker. After a first successful prototype of the SPPALs,
two defects were identified. Fortunately, the approach to solve one of the two dissolves both. The
resulting second version of SPPALs is the Polymer-Composite-Stepped Plate Transducer in which thepolymer composite steps would be fabricated on the flat circular aluminum plate using the molding
technique.

Since the SPPALs can be a low-cost PALs, it can make the PALs technology to be more frequently
and more widely used in everyday life. However, the SPPALs technology should be improved morein order to be conveniently used in many potential applications.

ACKNOWLEDGEMENTS
The authors may acknowledge Ingyu Lee for his help in developing the theoretical model. Also,
the authors would like to express their big gratitude to Dr. Junsuk Rho for his financial aids. This
work was partially supported by by the National Research Foundation of Korea (NRF) grant fundedby the Korea government (MSIP) (NO. 2016R1E1A2A02945515, NO. 2021R1F1A1064046).

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Aanonsen SI, Barkve T, Tjo/tta JN, Tjo/tta S. Distortion and harmonic generation in the nearfield of a finite

amplitude sound beam. The Journal of the Acoustical Society of America. 1984;75(3):749-68.
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Averkiou MA, Hamilton MF. Nonlinear distortion of short pulses radiated by plane and focused circular

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Je Y, Lee H, Park J, Moon W. A stepped-plate bi-frequency source for generating a difference frequency

sound with a parametric array. The Journal of the Acoustical Society of America. 2010;127(6):3494-502.
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Yoneyama M, Fujimoto Ji, Kawamo Y, Sasabe S. The audio spotlight: An application of nonlinear

interaction of sound waves to a new type of loudspeaker design. The Journal of the Acoustical Society of America.
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Lee H, Kang D, Moon W. A micro-machined source transducer for a parametric array in air. The Journal of

the Acoustical Society of America. 2009;125(4):1879-93.
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Je Y, Lee H, Been K, Moon W. A micromachined efficient parametric array loudspeaker with a wide

radiation frequency band. The Journal of the Acoustical Society of America. 2015;137(4):1732-43.
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Barone A, Juarez JAG. Flexural Vibrating Free‐Edge Plates with Stepped Thicknesses for GeneratingHigh Directional Ultrasonic Radiation. The Journal of the Acoustical Society of America. 1972;51(3B):953-9.
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Hwang Y, Moon W. A wide-band in-air parametric array source composed of a circular plate with steps anda Langevin transducer with a horn. Applied Acoustics. 2018;141:223-33.
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Hirose S, Aoyagi MAM, Tomikawa YTY. Dielectric loss in a piezoelectric ceramic transducer under high-power operation; increase of dielectric loss and its influence on transducer efficiency. Japanese journal of applied
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Hirose SHS. New method for measuring mechanical vibration loss and dielectric loss of piezoelectrictransducer under high-power excitation. Japanese journal of applied physics. 1994;33(5S):2945.
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Li T, Chen Y, Ma J. Frequency dependence of piezoelectric vibration velocity. Sensors and actuators A:

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Kim C, Hwang Y, Shin K, Moon W. A parametric array loudspeaker featuring a Langevin transducer, acircular aluminum plate, and composite polymer steps. Sensors and Actuators A: Physical. 2022;338:113466.
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Oh B, A design method for improved sound radiation capability of stepped-plate parametric array

loudspeakers, Master's thesis, POSTECH, 2022.
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ABS-0885

Nonlinear damping effects in MEMS-based microspeakers
Anton MELNIKOV(1) , Hermann A.G. SCHENK(1) , Franziska WALL(2) , Bert KAISER(2)(1) Bosch Sensortec GmbH, Germany, anton.melnikov@bosch-sensortec.de(2) Fraunhofer Institute for Photonic Microsystems IPMS, Germany, bert.kaiser@ipms.fraunhofer.de

ABSTRACT
Miniaturized microelectromechanical systems (MEMS) microspeakers are a current development trend for inear audio applications. However, when a transducer shrinks to a size that fits on a microelectronic chip, its
physics begins to differ from the effects governing the macroscopic world. On the one hand, the electrostatic
drive benefits from small electrode gaps, which lead to high driving forces. On the other hand, the fluid dynamics
of the air in the microscopic cavities of a microspeaker chip can cause strong damping forces, including severe
nonlinearities. Adequate damping is considered beneficial for wideband audio applications, however, nonlinear
damping effects introduce unwanted harmonic distortion and should therefore be avoided, or at least taken into
account, throughout the development process. In this presentation we will discuss the source and nature of such
effects and give an outlook on how to reduce them in designs.

Keywords: Sound, Insulation, Transmission

1

INTRODUCTION

Nonlinearities of fuid dynamic damping can be neglected in most applications of mechanical vibrations, since
the displacement amplitudes typically are extremely small, compared to the dimensions of the fluidic domains.
However, considering a MEMS-based transducer, as shown in Fig. 1a, the amplitudes of the air-displacing
structures (e.g. membranes) can be considerable, compared to the microscopic size of the inner cavities of thetransducer. Especially, in the case of lateral transducers [3, 7] (see Fig. 1b) the actuators are densely packed
increasing the usage of the wafer area. In this case, actuator movements can cause nonlinear system responses.
For MEMS transducers, used as microspeakers, and operating at audio frequencies, these nonlinearities need tobe carefully addressed by a suitable design, ensuring a high quality audio reproduction. Such MEMS microspeaker development requires accurate lumped parameter models for the relevant damping mechanisms. Suitablemodels should be compatible with a suite of similar modelling tools, related to other aspects of the device
physics [4, 6, 8] . In this work we investigate a lumped parameter model for the nonlinear fluid dynamic
damping, called squeeze film damping, which is described by the Reynolds equation.

2

SQUEEZE FILM DAMPING AS LUMPED PARAMETER

Squeeze film damping is a viscous effect caused by a fluid, flowing in and out the volume enclosed between
two parallel plates, when the plates move towards or away from each other. This nonlinear effect is welldescribed by the Reynolds equation






∂
∂
∂p
∂H
3 ∂p
3 ∂p
H p
+
H p
= 12µ H
+p
.
(1)
∂x
∂x
∂z
∂z
∂t
∂t
Upon introducing several simplifying assumptions, suitable for ridgid parallel plates, this partial differential
equation can be reduced [2, 10, 9, 1] to∂ 2 ∆p 12µ ∂ H
= 3
,
(2)
∂ z2
H ∂t

a

b

c

Figure 1. a Exemplary MEMS microspeaker based on the nanoscopic electrostatic drive technology. [3] b
Principle of creating sound pressure using the lateral motion of MEMS actuators. [3] c Sketch of the setup as
discussed in this paper, with dimensions.
where ∆p is the change in pressure, H is the gap size, and µ is the dynamic viscosity of the fluid. The resulting
total squeeze film damping force acting on the MEMS actor, is given by the integral of this pressure over the
exposed actuator surface area,Z∆F(t) =∆p(z,t)dA.

(3)

A

In the following we consider the scenario shown in Fig. 1c. A moving plate facing a fixed wall displaces
air in a narrow slit with a single opening over the full length of the slit at z = h/2. This leads to the following
boundary conditions [5]∆p(z = +h/2) = pA ,∂∆p (z = −h/2) = 0,∂z∂∆p (x = +L/2) = 0,∂x∂∆p (x = −L/2) = 0.
∂xHere pA is an acoustic pressure at the slit opening. Using these boundary conditions, the simplified Reynolds
equation can be integrated, leading to the following force model∆F(t) = −4µh3 L dH(t),H(t)3 dt

(4)

where L is the actor length and h is the actor height as shown in Fig. 1c. To calculate the damping force
caused by an externally imposed harmonic movement, we make the approach
H(t) = u0 − û cos(ωt) ,(5)with u0 as the initial gap and û as the amplitude of harmonic oscillation (see Fig. 1c). The change of the slitheight H is negligible for small amplitudes. In this case we obtain a constant damping coefficient
c=

4µLh3
,
u30

(6)

as known from linear oscillator theory. However, in case of MEMS microspeakers with laterally moving actors,
deflections beyond the scope of the linear theory may occur. To obtain the nonlinear damping force relevant tolarger actuator movements, we insert Eq. (5) into Eq. (4) leading to∆F(t) =

4µh3 Lωβf (t) ,
u20

(7)

with
f (t) =sin(ωt)(1 − β cos(ωt))3,(8)

where we have introduced the nondimensional normalized displacement amplitude β = uû0 .

3

HARMONIC DISTORTIONS CAUSED BY NONLINEAR DAMPING

When exposed to the nonlinearity of a squeeze film damping force according to Eq. (7) and Eq. (8) , themovement of a MEMS actuator will be subjected to harmonic distortions. To a certain extend these damping
force distortions will affect the acoustic performance of the microspeaker. The purpose of this section is to
extract the respective harmonic components of the damping force. In a sequel to this study we will analyse
their impact on the beam dynamics and on the acoustic total harmonic distortion.
To analyse the higher harmonics of the damping force, we expand f (t) as a Fourier series∞f (t) = ∑ (an cos(nωt) + bn sin(ωt)).

(9)

n=1

The Fourier coefficients bn are given by the integralsbn (β ) =

ω
π

Z +π

ω

sin(nωt) sin(ωt)− ωπ(1 − β cos(ωt))3

dt

(10)

and all an vanish due to symmetry. The bn integrals can be analytically evaluated. The first three coefficients
are
1
,
(11)
b1 (β ) =(1 − β 2 )3/2
b2 (β ) =and4
4 − 6β 2,−β 3 β 3 (1 − β 2 )3/2


24 3 3β 4 − 12β 2 + 8
b3 (β ) = 4 −.
ββ 4 (1 − β 2 )3/2

(12)

(13)

The harmonic distortion is characterized by the harmonic distortion coefficients kn , defined as the ratio between
the higher harmonic amplitudes and the fundamental. In our case this amounts tokn (β ) =

bn
b1

(14)

with n ≥ 2. The first two distortion coefficients can be evaluated ask2 (β ) =

3/2 36
4
4
1− 3 + 3 1−β2
= β + β 3 + O(β )5
β β
β
2
4

(15)

3/2 3 2 45 4
36 24 24
= β + β + O(β )6 .
− 4 + 4 1−β2
2
β
β
β
2
16

(16)

and
k3 (β ) = −9 +Finally we use Parseval’s theorem and b1 (β ) to compute the total fluid dynamic harmonic distortion. In ourcase Parseval’s theorem reads as
Z π
∞
ω +ω
2
f
(t)
dt
=
bn (β )2 ,(17)∑
π − ωπn=1
leading to the results
THDSQ (β ) =

s
1
1− ∞
=
k
∑n=1 n (β )21−

4 (1 − β 2 )3/2.
3β 2 + 4(18)

4

COMPARISON OF THE ANALYTICAL RESULTS WITH FEM SIMULATIONS

To assess the analytical model, we numerically solved the Reynolds equation using a 2D finite element method(FEM) for rigid actuator plates. The simulation was implemented as a COMSOL Multiphysics code. Figure 2
shows the damping force for a single time period (cosine-driven), and for different β values calculated with
FEM and with Eq. (8). We observe that both curves (FEM and analytical) in Fig. 2 match each other well.
The model Eq. (8) therefore seems to provide a solid basis for including the effect described by the nonlinear
Reynolds equation into lumped parameter models. With respect to the impact of the squeeze film force on theactuator dynamics, we note that the normalized peak force between β = 0.1 and β = 0.9 increases by a factorof 121 and the peaks of the force amplitude are increasingly concentrating around the center of the period(whiplash effect). This peak shift can be explained by the singular nature of the Reynolds model for the driving
force Eq. (8) at β → 1. A singular damping force, as predicted by the Reynolds equation Eq. (1), will however
hardly occur in reality, since rarefaction will occur (see Knudsen number) and the incompressibility assumptionof the Reynolds equation will be violated beyond a certain amplitude of the motion.
Figure 3a shows a more detailed comparison, based on distortion coefficients kn up-to order 6. Similar
to previous results, we observe overall a good match between the FEM simulation and the analytical curves.
For the highest harmonics and close to β → 1, a minor deviation can be noted for 5th and 6th harmonics at
β = 0.95. Despite this slight deviation close to the singularity, where the accuracy of the FEM simulation results
can be questioned, this supports the suitability of Eq. (8). Accordingly, Fig. 3a suggests that the nonlinear
behavior of the damping force is ruled only by the nondimensional amplitude β , i.e. by the displacement
amplitude normalized by the gap.
Note that the distortion coefficients kn , cf. Eqs. (15) and (16) are continuous and bounded functions overthe entire 0 ≤ β ≤ 1 range. In particular, it can be formally proven that0 ≤ kn (β ) ≤ kn (β = 1) = n .

(19)

The reader is invited to verify this for the examples of Eqs. (15) and (16) .

5

DISCUSSION

In this paper we show that the nonlinearity of the squeeze film damping force, as is relevant to MEMS microspeakers, can be suitably described by a simple lumped parameter force term. A suitable measure of therespective nonlinearity, introduces by the squeeze film force into the dynamics of a MEMS actor, are the distortion coefficients kn as defined in Eqs. (14). Using FEM simulations, we demonstrate that these distortion
coefficients solely depend on the gap-normalized amplitude β . This observation is consistent with the Reynolds
based lumped parameter model for the damping force Eqs. (7) and Eqs. (8). In Fig. 3a we observe that k2
dominates for β ≤ 0.825, while higher harmonics exceed k2 for β > 0.825. In practical applications β ≪ 0.825
and the dominance of k2 can be expected.
However, it should be noted that the situation discussed here contains only the damping force, generated
by a harmonic displacement imposed on the actuator. Inertial and spring restoring forces are neglected. Moresubtle nonlinearities, e.g. originating from Duffing or Coulomb forces, have to be included for more realistic
scenarios [4, 8]. Therefore, in real transducers, β is not necessarily the sole parameter affecting nonlinearity,and the β dependency can significantly change, even if all the other components of the lumped parameter model
are assumed as linear. The simplest lumped parameter model in this regard ism ü(t) +4µw3 Lu̇(t) + k u(t) = Felec (t) .
(u0 − u(t))3(20)

Although the distortion coefficients according to Fig. 3ab allow for a deeper insight into the nonlinear
behaviour of the damping force, they should be not mixed-up with the acoustical harmonic distortion (THD)
of a MEMS microspeaker. Computing the acoustical harmonic distortion requires solving a suitable equation of
motion, such as Eq. (20).

= 0.1= 0.2

FEM
analytical

1.0

1.0
0.5

0.0

0.0

0.0
0.5

0.5

1.0

1.0

1.0

= 0.4

2

1.5

= 0.5

3
1

2

0

0

f(t)

f(t)

0

1

1

2

2

2

4

3

= 0.7

= 0.8

/

20

100

10

50

0

0

10

50

20

100

+/

t/s

= 0.9

f(t)

7.5
5.0
2.5
0.0
2.5
5.0
7.5

f(t)

f(t)

= 0.6
4

2

1
f(t)

f(t)

0.5

0.5

= 0.3

1.5

0.5
f(t)

f(t)

1.0

/

+/

t/s

/

+/

t/s

Figure 2. Time-dependent normalized damping force caused by the squeeze film for different normalized amplitudes β .

5

kn

4

k2 analytical
k3 analytical
k4 analytical
k5 analytical
k6 analytical

1.0

k2 FEM
k3 FEM
k4 FEM
k5 FEM
k6 FEM

2
1

0.4
0.2

0

a

FEM

0.6

3

0.0

analytical

0.8

THDSQ

6

0.2

0.4

0.6

0.8

0.0
0.0

1.0

0.2

0.4

0.6

0.8

1.0

b

Figure 3. a Distortion coefficients kn up-to order 6 calculated using analytical expressions and FEM results. b
The THDSQ calculated using the analytical expression and by FEM.

6

CONCLUSION

In this work we report a lumped parameter model of the damping force based on a simplified Reynolds equation. The model is verified comparing FEM results with closed expressions for the first harmonic coefficients,
the distortion coefficients, and the total fluid dynamic harmonic distortion THDSQ (not to be confused withthe resulting acoustic THD), as provided in this manuscript. The model provides a building block for realistic
lumped parameter models of MEMS microspeakers, including the nonlinearities of squeeze film damping.

References[1] M. S. Ahmed, M. Ghommem, and A. Abdelkefi. Shock response of electrostatically coupled microbeams
under the squeeze-film damping effect. Acta Mechanica, 229(12):5051–5065, 2018.
[2] M. Bao and H. Yang. Squeeze film air damping in MEMS. Sensors and Actuators A: Physical, 136(1):3–27,
2007.
[3] B. Kaiser, S. Langa, L. Ehrig, M. Stolz, H. Schenk, H. Conrad, H. Schenk, K. Schimmanz, and D. Schuffenhauer. Concept and proof for an all-silicon MEMS micro speaker utilizing air chambers. Microsystems& Nanoengineering, 5(1):1–11, 2019.
[4] A. Melnikov, H. A. G. Schenk, J. M. Monsalve, F. Wall, M. Stolz, A. Mrosk, S. Langa, and B. Kaiser.
Coulomb-actuated microbeams revisited: experimental and numerical modal decomposition of the saddlenode bifurcation. Microsystems & Nanoengineering, 7(1), 2021.
[5] A. Melnikov, H. A. G. Schenk, F. Wall, J. M. Monsalve, L. Ehrig, M. Stolz, A. Mrosk, S. Langa, and
B. Kaiser. Squeeze film damping in MEMS loudspeakers: analysis of a clamped-clamped microbeam. In
Microfluidics, BioMEMS, and Medical Microsystems XX, volume 11955, pages 13–19. SPIE.
[6] J. M. Monsalve, A. Melnikov, B. Kaiser, D. Schuffenhauer, M. Stolz, L. Ehrig, H. A. G. Schenk, H. Conrad, and H. Schenk. Large-Signal Equivalent-Circuit Model of Asymmetric Electrostatic Transducers.
IEEE/ASME Transactions on Mechatronics, pages 1–11, 2021.
[7] J. M. Monsalve, A. Melnikov, M. Stolz, A. Mrosk, M. Jongmanns, F. Wall, S. Langa, I. Marica-Bercu,
T. Brändel, M. Kircher, H. A. G. Schenk, B. Kaiser, and H. Schenk. Proof of concept of an air-coupledelectrostatic ultrasonic transducer based on lateral motion. Sensors and Actuators A: Physical, 345:113813.
[8] H. A. Schenk, A. Melnikov, F. Wall, M. Gaudet, M. Stolz, D. Schuffenhauer, and B. Kaiser. Electrically
Actuated Microbeams: An Explicit Calculation of the Coulomb Integral in the Entire Stable and Unstable
Regimes Using a Chebyshev-Edgeworth Approach. Physical Review Applied, 18(1):014059.
[9] H. Yagubizade and M. I. Younis. The Effect of Squeeze-Film Damping on the Shock Response of
Clamped-Clamped Microbeams. Journal of Dynamic Systems, Measurement, and Control, 134(1), 2012.
[10] M. I. Younis. MEMS Linear and Nonlinear Statics and Dynamics, volume 20 of Microsystems. Springer
Science+Business Media LLC, Boston, MA, 2011.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0956

Analysis of nonlinear acoustic propagation from an Atlas V launchvehicleMark C ANDERSON1; Kent L GEE2; Carson F CUNNINGHAM3; Grant W HART4
1-4

Brigham Young University, USA

ABSTRACTHeated, supersonic rocket exhaust is a source of nonlinearly propagating acoustic waves. In this paper,
nonlinear characteristics of shock-containing noise radiated from a United Launch Alliance Atlas V launchvehicle are discussed. Acoustic pressure recordings of a launch out of Vandenberg Space Force Base, CA
were made at several stations at a range of 500-6400 m from the launch pad. The skewness of the pressure
waveform time derivative (“derivative skewness”), a metric useful for quantifying shock content in highamplitude noise, has been calculated. Waveforms, derivatives, skewness, and spectra during the period of
maximum OASPL are shown and discussed. Over these distances, derivative skewness values corresponding
to relatively steep shocks exist during this period of maximum OASPL. Results indicate a strong presence of
nonlinearity and therefore demonstrate the importance of including nonlinearity when modeling launchvehicle liftoff noise.

Keywords: Nonlinear, Launch, Rocket

1.

INTRODUCTION
Over the last ten years, an increase in launch cadence has dramatically increased the total number oforbital launches every year (1). These launches are impressive and can be heard for miles around with lowfrequency acoustic signatures detectable hundreds of kilometers away (2,3). Such launches carry anythingfrom internet satellites, interplanetary exploration probes, telescopes, and weather monitors, greatly
impacting life on Earth.
Despite the positive outcomes of these launches, the acoustics surrounding the launch sites are not wellunderstood. Many models today are still based on what was the state of the art in the 1970s and the primary
document that describes such rocket noise models is NASA SP-8072 (4). Although these models were stateof-the-art when created, an additional 50 years of research indicates a need to consider improved models (1).
One research area for launch vehicle acoustics is the nonlinear acoustic propagation due to the highamplitude pressures created by the exhaust flow. This nonlinear behavior is observable in the form of acoustic
1

anderson.mark.az@gmail.com

2

kentgee@byu.edu

3

carsonfcunningham@gmail.com

4

grant_hart@byu.edu

shocks (5,6). Because such behavior is not limited to launch vehicles, a wealth of research already exists forunderstanding nonlinear acoustic propagation.
This paper demonstrates the nonlinearity observed in rocket launch data and is organized as follows.
Section 2 contains a description of the experimental design used to observe nonlinear acoustic propagationfrom a United Launch Alliance (ULA) Atlas V rocket launch. Section 3 discusses three different methods foranalyzing the shock content in the recordings. These methods are to analyze the waveforms directly, the highfrequency spectral slope, and the skewness of the pressure derivative (“derivative skewness”).

2.

EXPERIMENT DESIGN
The rocket used analyzed in this study is the ULA Atlas V during the Landsat-9 mission. This mission

used a 401 configuration, denoting a first stage with a single RD-180 engine with no solid rocket boosters.
At liftoff, the nominal thrust was approximately 3.9 MN (870,000 lbf). The first stage fires for several minutes
before separating from the upper stage, which carries the payload into orbit. Though evidence of acoustic
signatures from rocket upper stages has been detected in the past (7,8), the noise contribution is considerednegligible. Therefore, it is only the noise generated by the first stage that is analyzed in this paper.
For this analysis, acoustic data were collected at several sites surrounding the launch pad, ranging from500 meters to over 6 kilometers away from the launch pad. Data were collected using the Portable Unit for
Measuring Acoustics (PUMA), consisting of a computer, data acquisition modules, battery, solar charger,
and microphone configuration (9). An example of one such setup is shown in Figure 1. The microphone
configuration used at each PUMA was a custom ground plate design, nicknamed at Brigham Young
University (BYU) the Compact Outdoor Unit for Ground-based Acoustical Recordings (COUGAR). Thisconfiguration consists of an inverted microphone placed above a plastic circular ground plate under a thickdome windscreen (10,11). Several stations used a second-generation design known as the COUGARxt (11).
Depending on location, the microphone used was either a GRAS 46BD or 46AO and data were sampled ateither 51.2 or 102.4 kHz. Recordings began once the computers detected pressure signals greater than a predetermined threshold, and some data in this paper are shown with time referenced to that trigger event.

Figure 1 – An example PUMA setup, complete with a solar charger and a COUGAR microphoneconfiguration. The PUMA setups are designed to be weather-robust, and the COUGAR microphone
configuration is designed to work in windy and rainy conditions.

A map of the measurement locations is shown in Figure 2. Although there were many other BYU
measurement locations around the launch complex, this paper focuses only on those shown in the figure. The
four chosen stations are located at 500, 1600, 3000, and 6400 meters from the launch pad and are roughly ina straight line outwards from the launch pad. This enables nonlinear propagation analysis along a common
radial.

Figure 2 – Measurement locations considered in this paper. Each measurement location is denoted by its
distance to the launch complex.

3.

RESULTS

3.1

Waveforms

As the launch progresses, nonlinear effects can be seen in the recordings during different phases of thelaunch. Figure 3 shows an example of the waveform at a few different points during the launch for the station
located 500 m from the launch pad. Figure 3A is a view of the entire recording up to 80 seconds into thelaunch. The pressure amplitude gradually increases for about 20 seconds and then gradually decreases
afterward. The period of the maximum measured pressure corresponds to the microphone being located near
the peak directivity angle of the exhaust plume (1). The shown data could be coupled with trajectory
information and used to map out a directivity function for the exhaust plume. Figure 3B shows the waveforma few seconds after the trigger. In this waveform segment, there are no visible shocks. Figure 3C shows thewaveform during the period of maximum overall sound pressure level (OASPL). In this waveform segment,
several shocks are clearly visible, and the amplitude of the signal is at least an order of magnitude greaterthan in the other waveform segments. Figure 3D shows a view of the waveform at about 60 seconds into thelaunch where the vehicle is now much further away, and the resulting waveform once again has no visible
shocks. This analysis indicates that there is a finite period during the launch where strong shocks are recordedat the measurement station.
A

B

C

D

Figure 3 – The waveform is shown at different times. (A) The waveform is shown over the first 80 secondsof launch. (B) About 5 seconds after the amplitude trigger. (C) During the period of maximum OASPL.
(D) About 60 seconds into the launch.

The shock content visible in the waveform also changes as a function of distance from the launch pad.
Figure 4 shows the waveform samples during a short portion of the period of maximum OASPL for the four
stations located along the radial. Note that the signals at each station are not anticipated to be coherent because
each station is exposed to the peak directivity angle of the exhaust plume when the rocket is at a different
height. As the distance to the pad increases, the shock strength decreases noticeably. At 500 m from the pad,the shocks are steep and peaked. At 1600 m, the shocks are still present, but the rise times are longer. At
3000 m, the shocks are still present but have even longer rise times. At 6400 m, there is still evidence ofshocks but they become harder to detect visually.

Figure 4 – Waveform samples during the time of max OASPL. Note that the time axes are arbitrary because
each of the waveforms is trimmed to its respective 3-dB down points around the maximum and then a 0.1second interval was chosen within that portion of the recording.
3.2

SpectraThe spectra also indicate the presence of shocks. Figure 5 shows the spectra over the 3 dB-down points
in the recording for the four microphones along the radial. These spectra have all been distance-correctedaccording to:
Distance to Rocket =Distance from Padsin(Peak Directivity Angle)

and,Scaled Spectrum = Measured Spectrum + 20 log10 [Distance to Rocket]
Scaled Distance

The assumed peak directivity angle is 65 degrees, and the scaled distance is 197 m, corresponding to 100
effective (or “equivalent”) nozzle diameters from the rocket (12). Applying this distance correction enables
a more direct comparison of the spectra because they should all have the same amplitudes. For themeasurements closest to the launch pad, the high-frequency spectral slope follows a -10 dB/decade trend,
exactly as expected for shock-containing noise plotted on a one-third octave band basis (1). However, as therecordings are made farther away from the launch pad, the -10 dB/decade slope ultimately gives way to amuch steeper slope. This steeper attenuation at high frequencies is likely due to the atmospheric absorption
overcoming the nonlinear shock formation. A level of 80 dB at several kilohertz at 6400 m clearly
demonstrates nonlinear propagation because the nonlinear propagation moves energy up into higher
frequencies that would have otherwise been absorbed well before reaching the furthest stations.

Figure 5 – Spectral comparisons along the radial. Note that all the spectra have been scaled to a common
distance according to an estimated peak directivity angle of the rocket, but the legend shows the distancesfrom the pad so as to be consistent with the other plots in this paper.
3.3

Derivative Skewness

The final analysis that indicates the presence of shocks is the skewness of the pressure derivative, known
as the “derivative skewness”. This measurement has been shown to correlate well with acoustic shock content
as well as the human perception of “crackle” (13). To aid in understanding this metric, it is useful to consider
an example. Figure 6 shows the acoustic pressure and its derivative during the period of maximum OASPLat one of the measurement stations. Notice that the pressure derivative contains notably more positive values
than negative values, implying a non-gaussian distribution. To further visualize the distributions, Figure 7
shows histograms for the pressure and its derivative. Overlaid on the distributions are gaussian distributions
with the same means and standard deviations. Notice that the distribution for the derivative shows substantial
skewness relative to a normal distribution, and this is what is meant by the derivative skewness. The totalpressure skewness for this waveform segment is 0.1 and the derivative skewness segment is 29.8.

Figure 6 – (Left) The pressure waveform during the period of maximum OASPL at the measurement station
located 500 meters from the launch pad. (Right) The derivative of the pressure waveform during the same
period at the same location. The derivatives in this paper have been normalized by the maximum derivativevalue. Note that waveforms are trimmed, and the time axis indicates the duration of the 3 dB-down portionsof the overall recording.

Figure 7 – (Left) The pressure distribution during the period shown in Figure 6. (Right) The pressure
derivative distribution during the period shown in Figure 6. Note that the derivative skewness remains
constant regardless of whether the data are normalized or not.

Instead of only calculating the derivative skewness over the period of maximum OASPL, it is also useful
to calculate its value during the entire launch, in one-second intervals. Figure 8 shows the derivative skewness
as a function of time for the station located 500 m from the pad. Notice that the derivative skewness increases
and then decreases similarly to the pressure waveform shown in Figure 3A, indicating a correlation between
large pressure amplitudes and high derivative skewness.

Figure 8 – Derivative skewness versus time for the station located 500 m from the pad.

4.

CONCLUSION
Acoustical measurements were made of an Atlas V launch vehicle in the 401 configuration between 500

and 6400 meters from the launch pad. These measurements indicate the presence of acoustic shocks producedby the launch vehicle. Firstly, the shocks are clearly visible in the waveforms during the period of maximum
OASPL. Secondly, the spectra during the 3-dB down points of the recordings have a -10 dB/decade roll-off
at high frequencies, indicating the presence of shocks. Lastly, the pressure derivative skewness – a quantity
known to correlate with the presence of shocks – is positive during the highest-amplitude portions of thelaunch noise.
There are still many questions regarding launch vehicle acoustics that need further study. For example,

datasets like these can be coupled with trajectory data to analyze the sound power and radiation efficiency ofa variety of launch vehicles. Comparative analyses between rocket exhaust plumes and other supersonic jets
should be made to continue to improve physics-based models. There are also questions related to theenvironmental impacts of launch vehicle noise, especially when protected wildlife is found around the launchpads. As resources are devoted to the study of launch vehicle noise, a general understanding of themechanisms behind the noise can be improved and then used to update models and provide more accurate
descriptions of the noise. Such results should be relevant to military operations, civilian spaceflight,
biological studies, and others.
ACKNOWLEDGEMENTS
The authors would like to acknowledge the support of Space Launch Delta 30 at Vandenberg Space Force
Base.

REFERENCES
1. C. P. Lubert, K. L. Gee, and S. Tsutsumi, "Supersonic jet noise from launch vehicles: 50 years since
NASA SP-8072", The Journal of the Acoustical Society of America 151, 752-791 (2022) doi:
10.1121/10.0009160
2. W. L. Donn, E. Posmentier, U. Fehr, and N. K. Balachandran, “Infrasound at Long Range from SaturnV, 1967.” Science, vol. 162, no. 3858, 1968, pp. 1116–20. JSTOR, http://www.jstor.org/stable/1725339.
Accessed 13 Aug. 2022.
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10.1029/2020GL092262
4. K. M. Eldred, “Acoustic loads generated by the propulsion system,” NASA SP-8072, Washington, DC.
(1971)
5. S. A. McInerny, “Launch vehicle acoustics Part 1: Overall levels and spectral characteristics,” J. Aircraft
33(3), 51–517 (1996)
6. S. A. McInerny, “Launch vehicle acoustics Part 2: Statistics of the time domain data,” J. Aircraft 33(3),518–523 (1996)
7. L. T. Mathews, K. L. Gee, G. W. Hart, R. D. Rasband, D. J. Novakovich, F. I. Irrarazabal, A. B. Vaughn,
and P. Nelson, “Comparative analysis of noise from three Falcon 9 launches,” J. Acoust. Soc. Korea 39,
322–330 (2020)
8. D. Cotten and W. L. Donn, “Sound from Apollo Rockets in Space.” Science, vol. 171, no. 3971, 1971,
pp. 565–67. JSTOR, http://www.jstor.org/stable/1730955. Accessed 13 Aug. 2022.
9. K. L. Gee, D. J. Novakovich, L. T. Mathews, M. C. Anderson, R. D. Rasband, “Development of a
Weather-Robust Ground-Based System for Sonic Boom Measurements,” NASA/CR-2020-5001870 (2020)
10. M. James, A. Salton, M. Calton, M. Downing, K. L. Gee, S. A McInerny, “Commercial Space
Operations Noise and Sonic Boom Measurements,” Contractor’s Final Report for ACRP Project 02-81(2020); doi: 10.17226/25834
11. M. C. Anderson, K. L. Gee, D. J. Novakovich, L. T. Mathews, Z. T. Jones, “Comparing two weatherrobust microphone configurations for outdoor measurements,” Proc. Mtgs. Acoust. Vol. 42, 040005 (2022);
doi: 10.1121/2.0001561
12. L. T. Mathews, K. L. Gee, and G. W. Hart. “Characterization of Falcon 9 launch vehicle noise from farfield measurements,” J. Acoust. Soc. Am. 150, 620–633 (2021)
13. K. L. Gee, P. B. Russavage, T. B. Neilsen, S. H. Swift, and A. B. Vaughn, "Subjective rating of the jet
noise crackle percept", The Journal of the Acoustical Society of America 144, EL40-EL45 (2018) doi:
10.1121/1.5046094

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0018

Perfect oblique-incidence ultrasound transmission acrossdissimilar solids
Jeseung LEE1; Minwoo KWEUN2; Woorim LEE1; Chung Il PARK1; Yoon Young KIM1
1
2

Seoul National University, Republic of Korea

Korea Institute of Machinery & Materials, Republic of Korea

ABSTRACT
Perfect ultrasound transmission from one medium to another is critical in many applications, including
medical treatment, imaging, and diagnostics. However, an obliquely incident ultrasonic wave through thesolid-solid interface inevitably generates multiple wave modes (longitudinal waves and transverse waves).
Accordingly, the well-known impedance matching theory considering only single wave mode fails to achieve
the perfect oblique-incidence ultrasound transmission even from one isotropic solid to another. Here, we
present the generalized matching theory that is newly established to enable the perfect transmission ofultrasonic waves across dissimilar solids at an arbitrary incidence angle and frequency. We also design anovel single-phase anisotropic solid-void metamaterial layer that can fulfill all conditions stated by the theory.
It is shown that the insertion of the proposed metamaterial layer between metal (aluminum) and plastic(polyether ether ketone) plates achieves nearly perfect wave transmission over 98% energy efficiency,
without which transmission reduces to 20%-40%. Our findings are validated numerically and experimentally.
Keywords: Ultrasound, Matching layers, Anisotropic metamaterials

1. INTRODUCTION
It is well-known that perfect transmission from one medium to another is possible if an impedance
matching layer is inserted between two dissimilar media [1, 2]. However, the actual application islimited to single-mode wave fields only. Obliquely incident ultrasonic waves in solids (i.e. elastic
waves) inevitably generate both longitudinal (L) and transverse (T) waves at an interface between two
dissimilar solids (Fig. 1, left) [3]. Due to this intrinsic multi-modal property of elastic waves, theperfect transmission of obliquely incident ultrasonic waves across dissimilar solids has been
impossible by inserting the conventional impedance matching layer.

Fig. 1. An overview of perfect oblique-incidence ultrasound transmission across dissimilar media.
Both longitudinal (L) and transverse (T) waves are reflected and transmitted at dissimilar solid-solidinterfaces (left). However, a single-mode wave is fully and solely transmitted without any reflection
by inserting the proposed metamaterial layer at the interface (right). Longitudinal-to-transverse
perfect transmission is illustrated as an example.

Here, we present a novel theory enabling perfect oblique-incidence ultrasound transmission at
solid-solid interfaces and design a new metamaterial layer realizing the theory (Fig. 1, right). This
theory is valid for any form of wave transmission, either mode -preserving (longitudinal-tolongitudinal and transverse-to-transverse) or mode-converting (longitudinal-to-transverse andtransverse-to-longitudinal). Perfect ultrasound transmission across dissimilar media is critical invarious applications such as biomedical ultrasound [4] and nondestructive testing [5]. The dramatictransmission amplification at solid-solid interfaces can be a promising strategy to develop a new kindof powerful medical or industrial ultrasound equipment.

2. RESULTS
2.1 Theory and principle
We present a theory, established in Ref. [6], for perfect transmission of obliquely incident
ultrasonic waves across two dissimilar isotropic solids. There are special conditions that an inserted
metamaterial layer’s material properties (mass density and elasticity tensor) should satisfy to achieve
the perfect transmission. The physical significance of these conditions can be interpreted as the mostgeneralized phase and impedance matching conditions. The detailed equations and derivations are notincluded here due to length limitations. (See Ref. [6] for the theoretical details.) The principle of theperfect transmission represents an interference among four longitudinal -transverse coupled waves(quasi-longitudinal and quasi-transverse waves propagating forward and backward) inside theproposed metamaterial layer.
2.2 Numerical and experimental results
A novel single-phase anisotropic solid-void metamaterial was designed and fabricated to realize
the metamaterial layer satisfying the generalized phase and impedance matching conditions. Without
loss of generality, target frequencies for the longitudinal-to-longitudinal (L-to-L) and longitudinal-totransverse (L-to-T) perfect transmissions were determined as 79.5 kHz and 59.5 kHz. A target
incidence angle for the perfect transmission was determined as 60°. As a case study, an aluminumbased metamaterial layer was designed at the interface between aluminum and polyether ether ketone(PEEK), where the longitudinal and transverse transmittances were only 39.4% and 20.2%,
respectively, due to their very high impedance mismatch. As a unit cell of the metamaterial, we
considered the one shown in Fig. 2 which has two different oblique slit-shaped voids with geometries
characterized by nine design parameters (a, N, l1, r 1, θ1 , l2, r 2, θ 2 and w). The unit cell layouts for the
L-to-L and L-to-T perfect transmissions were optimally designed based on the Method of Moving
Asymptotes algorithm.

Fig. 2. Realization of an anisotropic solid-void metamaterial layer. Geometric dimensions of thedesigned metamaterial layers for the longitudinal-to-longitudinal (L-to-L) and longitudinal-totransverse (L-to-T) perfect transmissions are presented.
Several numerical simulations and experiments performed with the designed metamaterial layers
confirmed the validity of the established theory and our metamaterial design. Numerical simulationsin Fig. 3 reveal that the transmittance from an incident longitudinal wave to a transmitted longitudinal(transverse) wave can be enhanced from 39.4% (20.2%) without the metamaterial layer to 98.6%

(99.6%) with the metamaterial layer at the interface between 5 mm thick aluminum and PEEK plates
under the plane-stress condition. The experimental results also supported nearly perfect transmission;
the measured displacement amplitudes of the transmitted longitudinal (transverse) wave were
amplified by 2.59 (4.55) times when the metamaterial layer was inserted. (See Ref. [6 ] for thenumerical and experimental details.)

Fig. 3. Verification of mode-preserving and mode-converting perfect oblique-incidence
ultrasound transmission. Scattered energy densities without and with the proposed metamaterial
layers were numerically simulated. A longitudinal wave entered from aluminum to PEEK with anincidence angle of 60°. Longitudinal-to-longitudinal (L-to-L) and longitudinal-to-transverse (L-to-T)
perfect transmissions were realized via the designed metamaterial layers at frequencies of 79.5 kHz
and 59.5 kHz, respectively.

3. CONCLUSION
This paper presents a theory that enables both mode-preserving and mode-converting perfecttransmissions of ultrasonic waves across dissimilar isotropic solids at an arbitrary incidence angle andfrequency. Furthermore, the applicability of the theory is practically demonstrated by designing asingle-phase anisotropic solid-void metamaterial, the use of which enabled the perfect obliqueincidence ultrasound transmission. The findings from this paper are expected to open new avenues for
manipulating ultrasonic waves in solids.

ACKNOWLEDGEMENTS
This research was supported by the Global Frontier R&D Program on Center for Wave Energy
Control based on Metamaterials (CAMM-2014M3A6B3063711) contracted through the Institute ofAdvanced Machines and Design at Seoul National University funded by the Korea Ministry of Science,
ICT & Future Planning. This research was also supported by the Na tional Research Foundation of
Korea (NRF-2022R1A2C2008067) funded by the Korea Ministry of Science, ICT & Future Planning.
J.L. acknowledges the Global PhD Fellowship (NRF-2019H1A2A1075829) funded by the Korea
Ministry of Education.

REFERENCES
1. Macleod HA. Thin-film Optical Filters. McGraw-Hill; 1989.
2. Kinsler LE, Frey AR, Coppens AB, Sanders JV. Fundamentals of Acoustics. John Wiley & Sons; 1999.
3. Auld BA. Acoustic Fields and Waves in Solids. RE Krieger; 1990.
4. Clement G, White P, Hynynen K. Enhanced ultrasound transmission through the human skull using shear
mode conversion. J Acoust Soc Am 2004; 115:1356-1364.
5. Piao C, Yang X, Kweun JM, Kim H, Park H, Cho SH, Kim YY. Ultrasonic flow measurement using ahigh-efficient longitudinal-to-shear wave mode-converting meta-slab wedge. Sens Actuator A Phys 2020;
310:112080.
6. Lee J, Kweun M, Park CI, Lee W, Kim YY. Perfect transmission of elastic waves obliquely incident at
solid-solid interfaces. Extreme Mech Lett 2022; 51:101606.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0073

Extraordinary high transmission through ultrasonic barriers usingelastic complementary meta-layer

Chung Il PARK1,3,*; Hyuk LEE2; Chunguang PIAO1,3; Yoon Young KIM1,3
1

Department of Mechanical Engineering, Seoul National University, Korea

2

Department of System Dynamics, Korea Institute of Machinery & Materials, Korea
3

Institute of. Advanced Machines and Design, Seoul National University, Korea*corresponding author, E-mail: picb77@snu.ac.kr (C. I. Park)

ABSTRACT
Because sufficient ultrasound energy should be transmitted to a target system to be inspected for satisfactoryinspection, the existence of a barrier in front of the target system is problematic. Especially when the
impedance of the barrier is far different from that of the target system, sufficient energy transmission throughthe barrier is a big challenge. As this barrier issue occurs in various ultrasonic applications including nondestructive evaluation, underwater detection, and noninvasive surgery, highly efficient wave energy
transmission through a barrier to a target system must be achieved to advance ultrasonic methods. Here, we
introduce a complementary meta-layer (CML) that virtually cancels out from a solid barrier to a fluid barrierso that extraordinary ultrasound transmission through these barriers becomes possible [1]. The concept ofcomplementary media is based on nullification of the physical space of the interfering medium, but the CML
requires realization of negative effective material properties. Our approach is to realize nearly independently
the desired effective negative density and stiffness by spatially separating dipolar and monopolar elastic
resonators in a single meta-atom. Numerical simulations show that the designed CML by our approachrealized extraordinary wave transmission increased by 593%, 1426%, and 3280% respectively for thepolymer, water, and air barriers compared with the nominal barrier-target system configuration. As thedesigned CML requires only a single-cell layer (i.e., mono-layer), unlike conventional complementary
metamaterials with bulky forms, the designed system can be more application-friendly.
Keywords: Elastic metamaterial, Complementary metamaterial, Ultrasound barrier, Transmission
enhancement

1. INTRODUCTION
An idea of complementary media (CM), canceling out a physical space of an aberrating medium
without destruction, has been proposed a few decades ago [2]. It is known that the transformation
mapping by the CM exactly rewinds the phase distortion by the original medium by employing the
double negativity (−  &− or − & −B), and thus pairs of the space cancel out each other. Soon
after the first report, further attempts had been made in an electromagnetic [3] and an acoustic regime[3] with elaborately designed metamaterials. However, a solid CM for an elastic regime has not been
introduced yet. Here we introduce an elastic complementary metamaterial.
A schematic sketch of the envisioned ultrasound penetration using the elastic complementary
metamaterial layer (CML) is illustrated in Fig. 1 [1]. We aim to nullify the barriers of highly
mismatched impedance ranging from solids (waveguides and detecting devices, etc.), to liquids(coupling materials such as gel/oil, human skins, and organs), and even gases (fissures in metal
structures, etc.).

Figure 1, Schematic drawing of wave penetration scheme envisioned by elastic complementarymeta-layer [1].

2. Complementary metamaterial
The key issue of the complementary metamaterial is precisely achieving double negative properties(− & −E) to simultaneously realize impedance matching (Z CML= Z B) and negative phase matching(qCMLdCML=-k Bd B) [1]. Fig. 2(a) illustrates a novel meta-atom structure [1]. A single-phase stainlesssteel with air inclusion consists of our meta-atom whose dimension is all within the subwavelength
regime in which a complicated motion of the atom can be treated as homogenized materials withrepresentative single density and stiffness. Specifically, the width (d=13 mm=0.13λ) and the height(h=18 mm=0.18λ) are far shorter than the wavelength (λ=102.2mm) at the frequency used (55 kHz).
The novelty of the proposition is that our meta-atom virtually independently controls the mass andstiffness by spatially separating the resonance motions. As shown in Fig. 2(a), the atom consists of
M eff-substructure and K eff-substructure [1]. Fig. 2(b) depicts the Eigen motion of the meta-atomnumerically analyzed at the working frequency [1]. M eff-substructure imitates dipolar resonancemotion along the lateral direction. Such a motion dynamically controls the mass (density) of the atom
in an effective medium perspective by manipulating the force-acceleration relation. On the other hand,
Keff -substructure is comprised of two resonators that vertically move in the opposite direction. These
resonances couple to cause the volumetric change mimicking the monopolar resonance that controls
the effective stiffness. We change the geometrical parameters w, l to adjust the effective properties(See Fig. 2(a)).

Figure 2, (a) Proposed meta-atom consisting of spatially separated substructures. (b) dynamic
resonance motion of the meta-atom [1].

3. Effective material properties of meta-atom
Fig. 3 shows the effective material properties (Meff, Keff) of the meta-atom. The M eff contour clearlyshows that M ef is dominantly affected by the w only, not by l. On the other hand, Keff is exclusively
affected by the l. These apparent decoupling properties are a key to independent control of Meff and
Keff , which facilitates the design of the CML meta-atom. See [1] for further details.
To evaluate the transmission spectra, we use the effective impedance Z and introduce

Z r = Z / Z SUS = M eff K eff . Fig. 3(c) shows the Zr contour. The contour consists of nine sections

with respect to the positive and negative signs of Meff and Keff. Although the impedance has either real(Re) or imaginary (Im) corresponding to the combination of the different signs of Meff and Keff, we
exclusively focus on the real Zr rather than the meaningless imaginary values for an evanescentpropagation. In a similar way, Fig. 3(d) illustrates the the Bloch phase contour on the w-l plane. The
center region has the negative phase shift owing to the doubly negative Meff and Keff. We put the zero
value to the complex wavenumber domain for brevity.

Figure 3, Effective material properties: (a) Meff and (b) K eff. (c) Relative impedance Z r and (c)Bloch phase (qd) is calculated using Meff and K eff [1].

4. Wave penetration using CML
We numerically demonstrate the wave penetration using the CML shown in Fig. 4. Fig. 4(a) depicts
the configuration of the simulation. We used commercial FEM software COMSOL Multiphysics 5.3
considering the stable analysis of multi-physical interaction between solid, liquid, and gas. The
thicknesses of the barriers are set to be the anti-Fabry-Perot resonance k BdB=π/2 (d B=5/4λ B) to
consider the most severe situation. We use the results in Fig. 3(c) to (d) to find the desired meta-atomgeometries (w and l) to realize the desired CM functionality. The interesting points (w, l) between the
impedance matching and negative phase matching conditions can then be found.
We conducted simulations with a barrier made of PEEK, water, and air. Fig. 4(b) illustrates thenumerical wave field for the PEEK barrier. A wave generated on the left side hardly propagates to theleft side due to the barrier. Specifically, the impedance of the PEEK is 1/13 of that of the base medium(SUS 304), allowing only a 15.24% transmission coefficient. On the bottom simulation, on the other
hand, we implement the CML adjacent to the barrier. The CML fulfills nullifying the phase deviationof the PEEK barrier. The transmission recovers to 89.58% as if virtually no barrier exists.
Fig. 4(c) and 4(d) respectively present the results when the barrier is water and air, respectively.
The impedance of water is 0.033 times that of the base medium (SUS 304); the transmission is merely6.00%. When CML is installed, the transmission is increased up to 85.55% (improvement by 1426%).
The air barrier permits 0.00016% of transmission while we achieve 0.053% transmission using the
CML installed. Although the air barrier is very tricky even for our CML, the transmission is enhanced
by 3280%.

Figure 4, With (a) a concept drawing of the simulation setup, three barrier cancellation
simulations for (a) PEEK barrier, (b) water barrier, and (c) air barrier are presented [1].

5. Conclusion
We presented the design of a complementary elastic metamaterial. Our meta meta-atom isadvantageous in that its effective material parameters, that is mass (Meff) and stiffness (Keff), can bequite arbitrarily and independently tuned. The designed CML uses nearly independently controlled
dipolar and monopolar resonances so that the effective mass and stiffness can be tuned nearly
independently. The designed elastic CML by our design approach was shown to enhance thetransmission through solid, fluid, and gas barriers by 593%, 1426%, and 3280%, respectively. We
expect that our CML can be useful in practical ultrasound applications including medical treatment
and diagnosis and underwater detection.

ACKNOWLEDGEMENTS
This work was supported by the National Research Foundation of Korea (NRF) Grant [no.
2014M3A6B3063711 (Global Frontier R&D Program on Center for Wave Energy Control based on
Metamaterials)] funded by the Korean Ministry of Science, ICT and Future Planning (MSIP)
contracted through IAMD at Seoul National University. Furthermore, this work was financially
supported by the Main Project of Korea Institute of Machinery and Materials (Project Code: NK228B).

REFERENCES
1. Park CI, Piao C, Lee H, Kim YY, Elastic complementary meta-layer for ultrasound penetration throughsolid/liquid/gas barriers. Int J Mech Sci, 2021, 206: 106619.
2. Pendry J, Ramakrishna SA, Focusing light using negative refraction. J Phys: Condens Matter, 2003,
15(37): 6345.
3. Kobayashi K, Complementary media of electrons. J Phys: Condens Matter, 2006, 18(15): 3703.
4. Shen C, Xu J, Fang NX, Jing Y, Anisotropic complementary acoustic metamaterial for canceling out
aberrating layers. Phys Rev X, 2014, 4(4): 041033.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0131

Metagratings for Near-perfect Mode Conversion and Steering of
Scattered Elastic Waves
Shin Young KIM1*; Yu Bin OH1; Joong Seok LEE2; Yoon Young KIM1
1
2

Seoul National University, Korea

Chungnam National University, Korea

ABSTRACT
When applied in non-destructive evaluating applications, efficient manipulation of propagating modes anddirections of elastic waves is a critical issue. The recent introduction of artificially engineered structural
materials, so-called metamaterials, facilitates extraordinary wave propagation manipulation in various
research fields. However, the simultaneous manipulation of the mode types and directions of elastic waves
in scattered wave fields is still challenging due to complicated motions coupled among multiple modes inelastic media. In this circumstance, we propose a novel metagrating using strip-type members governed by
the diffraction grating theory [1]. The proposed metagratings realize the precise manipulation of modeconverting and wave-steering phenomena simultaneously. A normally incident longitudinal (or shear) wave
mode is near-perfectly converted into a reflected shear (or longitudinal) wave mode with a target reflection
angle from the boundary of an elastic plate. The metagrating unit cell consisting of multiple strip-type
members was designed by the optimization of structural parameters such as the width, length, and distance
between the strips to suppress the undesired scattered diffraction modes effectively. The performance of thedesigned metagratings was verified through numerical simulations and experiments. The proposed
metagratings are expected to be applied in ultrasonic non-destructive evaluating applications.
Keywords: Elastic metagratings, Mode conversion, Wave steering

1. INTRODUCTION
To detect small defects inside the structure composed of a solid elastic medium, ultrasonic elastic
waves are widely applied in the field, for example, in structural health monitoring [2] and ultrasonicnon-destructive testing [3]. It is common sense that a precise manipulation of the mode and directionin the scattered wave field is essential for achieving high performance in such applications. However,simultaneous control of the mode and direction of scattered elastic waves is still challenging becauseof the intrinsic nature of elastic waves - coupling motions by various modes.
In this circumstance, we recently proposed novel elastic metagratings consisting of rectangular
strip-type members for realizing anomalous mode-converting reflections [1]. With delicately designed
metagratings, periodically repeated structures governed by the diffraction grating theory [4] , werealized the near-perfect mode-converting and wave-steering precisely. We converted the normally
incident longitudinal (or shear) wave mode into a reflected shear (or longitudinal) wave mode withthe desired target angle from the boundary of an elastic plate. For the design of the unit cell, we
numerically analyzed the scattered wave field by applying FFT in the spatial domain and searched thegeometric values of the strip-type members with a gradient-based optimization algorithm. The
proposed strip-type metagratings are expected to be applied for non-destructive structural healthmonitoring in the near future.

2. DESCRIPTIONS OF SCATTERED ELASTIC WAVES
Fig. 1 presents a schematic illustration of the scattered phenomenon induced by the strip-type* Corresponding Author.
E-mail: mental1019@snu.ac.kr (S. Y. Kim)

elastic metagratings. When longitudinal or shear wave impinges on the periodically structured
boundary, several longitudinal and shear waves are scattered from the boundary. To analyze thescattered field of elastic waves, the diffraction grating theory for the longitudinal and shear waves canbe applied as follows:

k p sin  pn,ref  kinc sin inc 

2 n
,
a

(1a)

ks sin  sm,ref  kinc sin inc 

2 m
,
a

(1b)

where inc denotes an incident angle and  p,ref and  s , ref denote reflective angles of n, m-th ordernmlongitudinal and shear waves, respectively. In addition, a denotes the periodicity of the unit cells
installed along the boundary. The symbol k denotes the wave number of the longitudinal or shear
wave.

Figure 1: A schematic illustration of the strip-type elastic metagratings for near-perfect mode conversion
and steering [1].

3. DESIGN OF ELASTIC METAGRATINGS USING STRIP-TYPE MEMBERS
The motions of the elastic waves in an elastic media are numerically analyzed using the finite
element-based software COMSOL Multiphysics ® . For the FEM simulation, the solid mechanics
module is used for the modeling and the plane stress condition is applied to represent a thin elastic
plate. The Floquet periodic boundary conditions are applied for the periodicity of the unit cell in the
metagratings. Throughout the research [1], the working frequency is set to 120 kHz. At this frequency,
the wavelengths of the longitudinal and shear waves in the one mm-thick aluminum plate arecalculated to be 0.045 m and 0.026 m, respectively.
For analyzing the diffraction phenomenon, the amplitudes of the entire diffraction modes in the
scattered wave fields should be evaluated quantitatively. We applied the fast Fourier transform (FFT)in the displacement fields obtained from the finite element analyses.
Along with the numerical analysis, optimization for the design problem should be accomplished
for the actual realization of the near-perfect mode conversion and steering. An optimization algorithm
determines the appropriate geometric dimensions of the strip-type members in the unit cell for theselective control of the scattered diffraction modes. As design variables, relative positions, lengths,
and widths of the members were chosen. The method of moving asymptotes was applied in this work
as a gradient-based optimization algorithm. More details can be found in [1,5].
As examples of near-perfect mode conversion and steering, we showed an example when theincident and reflected angle is 0 o and 45 o , respectively. For the target reflection angle, we
determined the periodicity of the unit cell, which is calculated from Eq. (1). The actual configurationof the strip-type metagratings and the near-perfect mode conversion and steering induced by the
structures are shown in Fig. 2.

Figure 2: The near-perfect mode conversion and steering with the target reflection angle of s 1  45 ousing the proposed strip-type metagrating [1].

4. APPLICATION OF NEAR-PERFECT MODE CONVERSION AND STEERINGThe strip-type elastic metagratings are directly applicable to the field of non-destructive wavebased evaluation. Fig. 3 shows the reflection-type mode converter. The piezoelectric transducer is
attached to the one side of the reflection-type mode converter. In contrast, the periodic structuresconsisting of strip-type metagratings are attached to the other side of the mode-converter. Using themode converter, the shear wave propagating through the plate is possible with high efficiency. In Fig.
3, the configuration of the reflection-type mode converter and schematic illustration of thephenomenon is shown in the left figure, and simulated results are presented in the right figure.

Figure 3: Application of the proposed metagratings: reflection-type mode converter for nondestructive
testing [1].

5. Conclusions
We proposed the simultaneous realization of the near-perfect mode conversion and wave steering
simultaneously using elaborately designed metagratings. The metagratings were realized inrectangular strip-type metagratings, which are simple and easy to manipulate the geometric parameters.
The specific geometries of the metagratings were determined by solving an optimization problem tosuppress undesired high-order diffraction modes. We showed that the designed metagratings could
realize the desired simultaneous near-perfect mode conversion and wave steering. The designed
metagratings in non-destructing testing demonstrated the engineering practicality of the designed
novel metagratings.

ACKNOWLEDGEMENTS
This work was supported by the National Research Foundation of Korea (NRF) grant (No.
2014M3A6B3063711, Global Frontier R&D Program on Center for Wave Energy Control based on
Metamaterials) funded by the Korea Ministry of Science, ICT & Future Planning, contracted throughthe Institute of Advanced Machines and Design at Seoul National University in Korea (to Y. Y. Kim).

This work was also supported by the National Research Foundation of Korea (NRF) grant (No.
2021R1F1A1063190) funded by the Korea Ministry of Science and ICT), contracted through
Chungnam National University in Korea (to J. S. Lee).

REFERENCES
1. Kim SY, Oh YB, Lee JS, Kim YY. Anomalous mode-converting reflection of elastic waves using striptype metagratings. 2022; under review.
2. Farrar CR, Worden K. An introduction to structural health monitoring. Phil Trans R Soc A. 2007; 365:303315.
3. Burrascano P, Callegari S, Montisci A, Ricci M, Versaci M. Ultrasonic nondestructive evaluation systems:
Industrial application issues. Springer International Publishing, New York, USA; 2015.
4. Ra'di Y, Sounas DL, Alù A. Metagratings: beyond the limits of graded metasurfaces for wave front control.
Phys Rev Lett. 2017;119:067404.
5. Kim SY, Lee W, Lee JS, Kim YY. Longitudinal wave steering using beam-type elastic metagratings. Mech
Syst Signal Proc. 2021;151:107688.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0186

The Bragg and locally resonant band gaps of in-plane waves along
thickness in periodically laminated piezoelectric composite
structures under electrical boundaries
Qiang-qiang LI 1; Yong-qiang GUO 2
1,2

College of Civil Engineering and Mechanics, Lanzhou University; and Key Laboratory of Mechanics on
Disaster and Environment in Western China, Ministry of Education, Lanzhou 730000, P. R. China

ABSTRACTThe periodically laminated piezoelectric composite structures composed of alternately arranged piezoelectric
layers and elastic layers have band gaps resulting from the Bragg scattering mechanism within which theelastic waves cannot propagate. The locally resonant band gaps are usually deemed to occurr only in elastic
metamaterials containing resonators. However, this paper reveals that locally resonant band gaps can also
appear in periodically laminated piezoelectric composite structures without any resonators but with external
electrical boundaries. Based on the transfer matrix method and Floquet-Bloch theorem, the active tuning of
the Bragg band gaps as well as the transition between the Bragg band gaps and the locally resonant band gaps
are realized by the applied electrical capacitance or voltage on the piezoelectric layer.
Keywords: Periodically laminated piezoelectric composite structures, Bragg band gaps, Locally resonantband gaps, Electrical boundaries

1. INTRODUCTION
The Bragg band gap is common in general periodic structures due to the Bragg scattering
mechanism (1). The Bragg scattering condition requires the band gap wavelength to be of the sameorder of magnitude as the lattice constant. To generate the low frequency Bragg band gap, periodic
structures with large lattice constants are required, which limits the application of the Bragg band gap
within low frequencies. The locally resonant band gap is resulted from the interaction between theresonance of the scatters and the elastic wave in the matrix. It is commonly seen in atomic chains (2–6), rods (7–10), beams (8,11–17), plates (8,18,19), and two- and three-dimensional periodic structurescontaining resonators (20–25). These structures with locally resonant band gaps have the following
two characteristics. On the one hand, the broadband and low-frequency locally resonant band gaps canbe realized by adjusting the connection stiffness and mass of the resonator without increasing th e
structure size. On the other hand, the periodic structure with resonators also has the Bragg band gap,
and the mutual transformation between the Bragg band gap and the local resonance band gap as well as
the fusion of the adjacent locally resonant band gap can be realized by adjusting the connection
stiffness and mass of the resonator.
The periodically laminated piezoelectric composite structure generally does not have a local
resonant band gap due to the lack of the necessary resonator. However, when the periodicallylaminated piezoelectric composite structure is attached with electrical boundary conditions, thelocally resonant band gap will appear. Kutsenko et al. (26) realized the mutual transformation
between the Bragg band gap and the locally resonant band gap by connecting negative capacitor onto
the piezoelectric layer. Inspired by this idea, this paper studies the effects of the negative capacitanceand the voltage of the piezoelectric layer in the periodically laminated piezoelectric composite
structure on the formation of Bragg and locally resonant band gaps. It is found that the voltage and thenegative capacitance have the same effect on the in-plane P-wave band gap, which can not only realize
the mutual transformation of the Bragg band gap and the locally resonant band gap but also promote12

liqq2013@lzu.edu.cn
Correspondence.guoyq@lzu.edu.cn

the merging of adjacent band gaps.

2. BASIC MODEL
Consider the propagation of in-plane waves in periodically laminated piezoelectric composite structures
whose unit cell as shown in Figure 1(a), contains m piezoelectric, electrode and elastic layers arranged
repeatedly with total thickness h   j 1 h j and N interfaces. The electrical boundaries includingmelectric-open, applied capacitance, electric-short and applied feedback control, are applied to thepiezoelectric layer through the external circuit. The relationship between electrical quantities, including thecharge Q , capacitance C , voltage V , and gain coefficient K g under different boundary conditions is
also shown in Figure 1. To facilitate the description of the global physical quantities of the unit cell and the
local physical quantities of each layer j , the global coordinate system OXYZ and the local coordinate
system ox j y j z j are established as shown in Figure 1(a) and (b). For the typical layer j shown in Figure1(b), the displacements and stresses at the position z along the three coordinate axes are represented by
u j ( z) , v j ( z) as well as w j ( z ) and  zxj ( z) ,  zyj ( z) and  z ( z ) , respectively. Since only the propagationof in-plane waves is considered here, so we only concern with the displacements and stresses associatedwith in-plane waves. The displacement column vector vˆ u  [uˆ, wˆ ]T and stress column vectorvˆ   [ˆzx , ˆ z ]T are combined to form the state vector vˆ  [( vˆ u )T ,( vˆ  )T ]T . q and  denote the Floquet
wavenumber and the circular frequency, respectively.

h1

h3

h2

X
1
2
3
4
5







hi 1
hi
hj

O

Y

1
2
3
4




h

q




i 1
i
j
k




J
K









Q0
Q  CVV 0V  K g   w(h j )  w(0) m2

hm1 m  1

m

(a)



N 1
N

q
Z

The unit cell and its description in the global coordinate system of periodically laminatedpiezoelectric composite structures.

Ju j , zxj

o
xj

hj j

yj

zj

v j , zyj

w j ,  zj

Q0Q  CVV 0V  K g   w( h j )  w(0) K(b)

The typical piezoelectric layer j and its description in the local coordinate system of
periodically laminated piezoelectric composite structures.
Figure 1 – The basic model of the periodically laminated piezoelectric composite structures.

3. STATE SPACE FORMALISM
3.1 Derivation of the State Equation of a Layer
For the typical piezoelectric layer j shown in Figure 1(b), the governing equation of in-plane
wave propagation is determined by the geometry equations, the constitutive equations, and thedifferential equations of motion. For the convenience of derivation, the subscript
quantities indicating the layer is omitted.
ε  LT u , E   ;σ  cε  e E , D  eε  βE ;Tj of the physical

(1)
(2)

Lσ  u ,  D  0 ;(3)where u  [u, w] , ε  [ x ,  z ,  zx ] , and σ  [ x ,  z , zx ] are the displacement, strain, and stress vectors,
respectively.  , E  [ Ex , Ez ]T , and D  [ Dx , Dz ]T are the electric potential scalar, the electric field, andthe electric displacement vectors, respectively.  is the material density. c , e and β are the elastic
stiffness matrix, the piezoelectric coefficient matrix, and the dielectric coefficient matrix written as c11 c13 c15 
e
e 
13 
e

c  c13 c33 c35  , e   11 13 15  , β   11
(4)
.
e
e
e


31
33
35 
13
33 


c15 c35 c55 
T

T

T

L and  are the differential operator matrix and vector

 x
L
0


0

z



 x 
z  ,
  .
 z x 

(5)Suppose that all the mechanical quantities, including the displacements, strains, stresses, andelectric quantities, including the electric potential, electric fields, and electric displacements, havethe form of a harmonic solution(6)
( x, z, t )  ˆ (k; z; )ei( kx t ) , (  u, ε, σ,  , E, D) ,where i  1 is the imaginary unit. k denotes the wavenumber along the x axis. The superscript“” means the corresponding to physical quantity in the k   region.
It is worth noting that only the z direction relates to the piezoelectric effect with accumulation of
charge and potential difference. From the second formula of the equation (1), one can obtainEˆ x  0 , Eˆ z   d ˆ d z .
(7)
Besides, only considering the change rate of electric displacement in the z direction, then the second
formula of the equation (3) becomes d Dˆ z d z  0 , which means that Dˆ z is a constant and can bedecided by the charge per unit areaDˆ z  Qˆ S .
(8)
Substituting the second formula of equation (3) into the second formula of equation (2) and using equations(6), (7), and (8), one can obtaind vˆ ( z ) 1 Qˆ
d ˆ
1
1

Fvˆ u ( z ) 
F3 u

,
(9)
d z 33
33
dz
33 S
where F  i kF1 , Fi  e3i1 ,e3i 3  ( i  1,3 ). Then, through making use of the first formula of equations (1),(2), and (3) and eliminating the electric potential ̂ through equation (9) and the stress ˆ x , one canobtain the inhomogeneous state equation of the piezoelectric layer written as
 d vˆ u 
T-1F3T
T-1   vˆ u  1 Qˆ 
d vˆ ( z )  d z   T-1S



,
(10)



T -1 TT
T -1
T -1 
dz
 d vˆ   M  S T S S T   vˆ    33 S  S T F3  F  d z 
whereS  i kG 31 

1

33

F3T F , T  G 33 

1

33

F3T F3 , M  k 2 G11   2  I 2 c
G kl   1k1l
c3k1l1

33

FT F ;

c1k 3l .
c3k 3l 

Furthermore, taking advantage of the relationship between voltage and potential

(11)

d ˆ
Vˆ  
dz,
dz
0
h

(12)

and the electric boundary conditions shown in Figure 1, we can obtaind vˆ ( z ) Avˆ ( z )  B  vˆ (h)  vˆ (0)  ,
dz

(13)


T-1F3T R
T-1F3T U

T-1S
T-1 
1 
A
,
B


.
T -1
T -1 
T -1 TT
T -1 TT
33 S    S T F3  F  R   S T F3  F  U 
(M  S T S) S T 

(14)

where

According to the four different electric boundaries, the form of the 1 2 matrices R and U arelisted in Table 1.
Table 1 – The expressions of matrices R and U
Electric boundaries
Electric-open

Electrical relationships
Qˆ  0U

00 FM S  F 1Applied capacitance

R

1

1

T

33

3

Qˆ
Vˆ 
C

33

 1
 h 1 1FM 1F T  1



 33
 33 S C

 FM S  F 

1

1

 33

3

Electric-shortVˆ  033

h 1 1FM 1F T  1
 33 S   33
KApplied feedback controlVˆ   K g  w(h)  w(0)  FM S  F 

1

1

 1
 h 1 1FM 1F T  1



 33
 33 S C
1

T

T

3

33


h 1 1FM 1F T  1
 33 S   33


FM 1

FM 1


h 1 1FM 1F T  1
 33 S   33

1

 33

FM 1


h 1 1FM 1F T  1
 33 S   33
where K  [0, K g ] .
For the electrode layer and the elastic layer, the state equation can be obtained by setting theepiezoelectric constants as zero.
3.2 Traveling Wave Solution of the State Equation for a Layer
The solution of the inhomogeneous state equation can be written asvˆ  Φ 44  e Λz  P 4 4

 vˆ   (Φ ) w   u    u 24   e Λz  P  w ,4 4
 vˆ   (Φ ) 24 (15)

where Φ and Λ are eigenvector and eigenvalue matrices of coefficient matrix A , respectively. The
eigenvector matrix can be divided into Φu and Φ according to their association with displacement andstress, respectively. The eigenvalue matrix Λ is a diagonal matrix. The matrix P  Λ-1Φ-1BΦ(eΛh  I 4 )with the 4 th order unit matrix I 4 . w means the pending column vector.
3.3 Transfer Matrix Method
Consider arbitrary layer j and take advantage of the solution of the state equation, the state vector on
its top surface and bottom surface can be written asΛ hvˆ j (h j )  Φ j e  Pj w j , vˆ j (0)  Φ j  I 4  P j  w j .
(16)



j j



One can get the transfer relation of the state vectors between the top surface and the bottom surface by
combining the two formulas of equation (16)vˆ j (h j )  Tj vˆ j (0) ,(17)
where Tj is referred to as the transfer matrix of layer jTj  Φ j e j j  P j  I 4  P j  Φ -1j .
Λ h

-1

(18)

Considering further the relation of the state vectors at interface K between layer j and j  1 , wehavevˆ j 1 (0)  TK vˆ j (hj ) ,(19)
K
where T  I 4 is referred to as the transfer matrix of interface K .
Using the above transfer relations repeatedly from top surface to bottom surface of the unit cell, thetransfer relation of the state vectors between the top surface of layer 1 and bottom surface of layer mcan be obtained asvˆ m (hm )  Tm Tm1 T2T1 vˆ 1 (0)  Tvˆ 1 (0) ,(20)
where T is called the transfer matrix of the unit cellT  TT
Tm1Tm .
(21)
1 2
In addition, according to the Floquet-Bloch theorem, the state vectors between the top surface andbottom surface of the unit cell satisfy the periodic conditionvˆ m (hm )  eiqh vˆ 1 (0) .
(22)
As mentioned before, q stands for the Floquet wavenumber along thickness, h is the total thickness of
the unit cell. Combining equation (20) and equation (22), one can get
Tvˆ 1 (0)  eiqh vˆ 1 (0) or (T  eiqh ) vˆ 1 (0)  0 ,(23)
For vˆ 1 (0) has nontrivial solution, then one has
eiqh  Eigenvalues(T) or det(T  eiqh )  0 .

(24)
The above equation is called the dispersion equation that involves the wavenumber q and the frequency
 . The dispersion curves reflecting the relation between wavenumber q and frequency  are obtain bysolving the dispersion equation numerically.

4. NUMERICAL EXAMPLES
Consider the in-plane waves propagating in a vertical direction (that is kh   0 ) in the periodicallylaminated piezoelectric composite structures composed of Glass-Brass-PZT-5H-Brass multilayers, where
Glass (10mm), PZT-5H (10mm) and Brass (0.025mm) are the elastic layer, piezoelectric layer andelectrode layer. The physical parameters are the same as those in Reference (27).
Figure 2 gives the frequency-real wavenumber spectra in subfigures (a), (b) and thefrequency-imaginary wavenumber spectra in subfigures (c), (d) under different capacitance and voltage.
Electrical open circuit and electrical short circuit are plotted in each subplot as comparison benchmarks.
The electrical boundary conditions only affect the P-wave and do not affect the S-wave when the elastic
wave travels along thickness through the periodically laminated piezoelectric material described above (27).
Both changes in negative capacitance and voltage can convert the Bragg band gap into the local resonanceband gap, and vice versa. Specifically, during the change of negative capacitance C S from
0.75pF mm to 0.80pF mm , the lower boundaries of the first and second P-wave Bragg band gaps
(around 150kHz and 280kHz) remain unchanged, while the upper boundaries gradually shift to higher
frequencies and the stopbands become into locally resonant band gaps as the capacitanceC S  0.78pF mm2 . With the further decrease of negative capacitance C S , the two local resonanceband gaps separate and their anti-resonant frequencies shift to the lower and higher frequencies,
respectively. The effect of voltage (applied feedback control K g ) on the band gap of the P-wave is similar
to that of negative capacitance. The first band gap of the P-wave (around 200kHz) is the local resonanceband gap. The change in voltage only changes the magnitude of the anti-resonance frequency but does notchange the band gap width. In addition, when K g  0.5 1011 V m , the third P-wave band gap (around
500kHz) is the locally resonant band gap with the two anti-resonance frequencies. In case of applied
feedback control K g increases from 1.4 1011 V m to 5.0 1011 V m , the third P-wave band gap (around
530kHz) is the locally resonant band gap with a constant lower bound frequency, and the fourth P-wave
band gap (around 650kHz) is the Bragg band gap with a constant upper bound frequency. As the applied
feedback control increases, the third and fourth P-wave band gaps merge into a locally resonant band gap at
2

2

K g  2.3 1011 V m . Thereafter, the third band gap is transformed into a Bragg band gap with a smaller
upper bound frequency, while the fourth band gap is transformed into a locally resonant band gap with asmaller lower bound frequency.
C/S=-0.75pF/mm2
C/S=-0.78pF/mm2

C/S=-0.76pF/mm2
C/S=-0.79pF/mm2

C/S=-0.77pF/mm2
C/S=-0.80pF/mm2
1

qR h/

qR h/

Electric-open (C/S=0)
Electric-open (C/S=¥)
1

0

-1

Electric-open

Kg=0.51011V/m2

Kg=1.41011V/m2

Kg=2.31011V/m2

Electric-short (Kg=0)

Kg=3.21011V/m2

Kg=4.11011V/m2

Kg=5.01011V/m2

0

-1
0

100

200

300

400

500

600

700

800

0

100

200

Frequency (kHz)

300

(a)
C/S=-0.75pF/mm2
C/S=-0.78pF/mm2

500

600

700

800

(b)
C/S=-0.76pF/mm2
C/S=-0.79pF/mm2

C/S=-0.77pF/mm2
C/S=-0.80pF/mm2
3

2

2

1

1

qI h/

qI h/

Electric-open (C/S=0)
Electric-open (C/S=¥)
3400

Frequency (kHz)0

Kg=0.51011V/m2

Kg=1.41011V/m2

Kg=2.31011V/m2

Electric-short (Kg=0)

Kg=3.21011V/m2

Kg=4.11011V/m2

Kg=5.01011V/m2

0

-1

-1

-2

-2

-3

Electric-open

-3
0

100

200

300

400

500

Frequency (kHz)

600

700

800

0

100

200

300

400

500

600

700

Frequency (kHz)(c)(d)
Figure 2 – Frequency-real wavenumber spectra (a), (b) and frequency-imaginary wavenumber
spectra (c), (d) under different capacitance and voltage.

5. CONCLUSIONS
The influence of electrical boundary conditions (including negative capacitance and voltage) on
the band gap of the vertical incident in-plane wave in periodically laminated piezoelectric composite
structures is studied. Firstly, the basic model of the periodically laminated piezoelectric composite
structure is given, and then the inhomogeneous state differential equation determining the in -plane
state vector of a material layer is derived by using the elastic dynamics principle. Secondly, thedispersion equation describing the propagation characteristics of in-plane waves is derived by
combining the transfer matrix method with the Floquet-Bloch theorem. Finally, the frequency
wavenumber spectra under different negative capacitance and voltage conditions are obtained bynumerical examples. It is found that the negative capacitance and voltage alteration can not onlychange the band gap width of the in-plane wave (mainly for the P wave), but also can results in themutual transformation between the Bragg band gap and the local resonant band gap. The widthchange may due to the alteration of Bragg or locally resonant band gap or may due to that theadjacent band gaps fuse into wider locally resonant band gaps.

ACKNOWLEDGEMENTS
This work was funded by the National Key Research and Development Program of China (No.:2019YFC1511005).

800

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ABS-0348

A micromorphic description of Willis coupling
Samuel P. Wallen(1) , Benjamin M. Goldsberry(2) , Michael R. Haberman(3)(1) Applied Research Laboratories, The University of Texas at Austin, USA, sam.wallen@utexas.edu(2) Applied Research Laboratories, The University of Texas at Austin, USA, bgoldsberry@utexas.edu(3) Applied Research Laboratories and Walker Department of Mechanical Engineering, The University of Texas at Austin, USA,haberman@utexas.edu

ABSTRACT

Acoustic and elastic metamaterials (AEMM) are artificial materials whose performance is engineered to
exceed that of naturally-occurring materials and conventional composites, often leveraging complex microstructural geometry and strong dispersion. Generalized continuum (GC) theories, such as micromorphic elasticity, are extensions of classical elasticity that capture the effects of microstructure via the inclusion of extra degrees of freedom and/or nonlocal effects, and have received renewed interest as potential reduced-order models for AEMM. An alternative AEMM modeling framework is the elastodynamic
homogenization theory of Willis, which yields an effective medium whose constitutive relations containstress-acceleration and momentum-strain-rate couplings. Despite the recent popularity of the GC and Willis
methods, a comprehensive description of their relationship remains lacking. In the present work, we derive
an explicit connection between the Willis and GC models. Specifically, we augment the micromorphic
theory of Mindlin to account for microscale asymmetry of mass density and demonstrate that the resulting
boundary value problem may be expressed in the form of a Willis material. This approach helps to elucidate the physical origins of Willis coupling using an energy-based variational formulation, and may offer
insights about how other emergent material properties arise from microstructural constituents.
Keywords:

1

INTRODUCTION

Acoustic and elastic metamaterials (AEMM) are artificial materials whose performance is engineered to exceedthat of naturally-occurring materials and conventional composites, often leveraging complex microstructural geometry and strong dispersion [1, 2]. Generalized continuum (GC) theories, such as micromorphic elasticity, areextensions of classical elasticity that capture the effects of microstructure via the inclusion of extra degrees of
freedom and/or nonlocal effects [3, 4, 5], and have received renewed interest as potential low-dimensional models for AEMM [6, 7, 8, 9]. Generalized continuum theories are usually viewed as a top-down approach, i.e., a
GC model is postulated a priori and its parameters must be fit to a particular microstructure.
An alternative AEMM modeling framework is the elastodynamic homogenization theory of Willis, whichyields an effective medium whose constitutive relations contain stress-acceleration and momentum-strain-rate
couplings (i.e., the so-called Willis coupling) [11, 12, 13, 14]. This homogenization framework represents abottom-up approach, since the effective medium is derived from a particular microstructure. Since the seminal
work of Willis [11], many authors have noted the emergence of Willis coupling in their own homogenization
schemes, though its physical origin and mathematical properties vary across models.
Despite the recent popularity of the GC and Willis methods, a comprehensive description of their relationshipremains lacking. In the present work, we derive an explicit connection between the Willis and GC models.
Specifically, we augment the micromorphic theory of Mindlin [15] to account for microscale asymmetry ofmass density and demonstrate that the resulting boundary value problem may be expressed in the form of aWillis material. This approach helps to elucidate the physical origins of Willis coupling using a time-domain,energy-based, variational formulation, and may offer insights about how other emergent material properties arise
from microstructural constituents.

2

AUGMENTED MICROMORPHIC SOLID

2.1 Notation
In the following derivations, we express vector and tensor quantities using indicial notation with components
relative to an orthonormal, Cartesian basis. Einstein summation convention is assumed throughout. Time andspace derivatives are denoted by overdots and subscripted commas, respectively (e.g., ȧi, j denotes the time
derivative of the gradient of the vector with components ai , or ȧi, j ⇔ ∇ (∂~a/∂t)).
2.2 Kinematics, Deformation, and Internal Energy
The GC formulation developed herein is based on the linear, elastic theory of Mindlin [15]. To briefly summarize, Mindlin’s theory extends classical elasticity by endowing each material point with a so-called microdeformation, which is a second-rank tensor field with components ψi j . Physically, the microdeformation represents
the deformation gradient of the material in the neighborhood of each material point. Thus, in three dimensions, Mindlin’s theory contains twelve independent variables (three components of the displacement ui andnine components of the microdeformation ψi j ). To increase the applicability of the theory to AEMM, which
are often composed of architected units that exhibit translational local resonances (e.g., the seminal work ofRef. [16]), we additionally define the microtranslation vi , which allows the microstructure to translate relative
to the macroscopic coordinate points. Macro- and microscopic deformations of a body in our GC framework
are shown schematically in Fig. 1.

Figure 1. Schematic of deformation in the augmented micromorphic continuum model. Each material point X
undergoes the macroscopic deformation χ(X,t), while the microstructure experiences linear deformation gradientI + ψ(x,t) and translation v(X,t).
In the spirit of Ref. [15], we proceed by developing an energy-based variational formulation based on

Hamilton’s Principle [17]. To this end, we define the following measures of deformation:ui, j + u j,i,2
γi j = ψi j − ui, j ,(1b)

κi jk = ψi j,k .

(1c)

εi j =(1a)Here, εi j is a macroscopic, symmetric strain, analogous to a classical elastic solid; γi j is the deformation of
the microstructure, relative to the macroscale; and κi jk is the macroscopic gradient of the microdeformation.
The dynamic behavior of the medium is determined by the manner in which its kinetic energy density T and
deformation energy density W vary with changes in the kinematic variables. The kinetic energy density may
be expressed as1
1(2)
T = ρ(u̇ j + v̇ j )2 + ρ̄k ψ̇k j (u̇ j + v̇ j ) + ψ̇k j ρ̄¯ kl ψ̇l j ,22
where ρ is the mass per unit macro-volume, and ρ̄k and ρ̄¯ kl are the first and second moments of mass density
about the centroid of the micro-volume, respectively. We note that our expression differs from that of Ref. [15]due the addition of the microtranslation vi and the removal of the assumption that the center of mass of the
micro-volume coincides with its centroid. We postulate the deformation energy as a general, quadratic functionof the deformation measures defined by Eqs. (1) [15], with an additional term generated by the microtranslation
vi :
1
1
1
1
W = vi Ki j v j + εi jCi jkl εkl + γi j Ai jkl γkl + κi jk Bi jklmn κlmn
2
2
2
2
(3)
+ εi j Di jkl γkl + εi j Ei jklm κklm + γi j Fi jklm κklm ,where Ki j , Ci jkl , Ai jkl , etc. are constants that depend on the material and geometric properties of the microstructure.

3

EQUATIONS OF MOTION

The equations of motion of the body are defined by Hamilton’s Principle [17], which seeks to minimize theaction defined by the space-time integral of the Lagrangian density T − W , i.e.,
Z T Z 
T (u̇i , v̇i , ψ̇i j ) − W vi , ψi j , εi j , γi j , κi jk dV + Wnc dt = 0,
(4)
δ
0

V

where the symbol δ (·) denotes the first variation of the quantity (·), T is an arbitrary, temporal boundary, V
is the domain of integration with infinitessimal volumetric element dV , and Wnc is the work done by nonconservative forces throughout the body. The non-conservative work is defined by its variation,δ Wnc =Z
V

Z

( fi δ ui + gi δ vi + Fi j δ ψi j ) dV +(ti δ ui + hi δ vi + Mi j δ ψi j ) dA,

(5)

∂V

where fi , gi , and Fi j are generalized body forces that are energetically conjugate to the displacement, microtranslation, and microdeformation, respectively; ti , hi , and Mi j are analogous surface forces; and ∂V is theboundary of the volume V , with infinitessimal area element dA. Substituting Eqs. (2), (3) and (5) into Eq. (4)and following the typical procedure from calculus of variations, yields the following partial differential equationsof motion of the body:ṗi − (τi j − σi j ), j = fi ,(6a)q̇i + φi = gi ,(6b)

L̇i j + σi j − µi jk,k = Mi j ,(6c)where we have defined the following generalized momenta:
∂T
,
∂ u̇i
∂T
,
qi =
∂ v̇i
∂T
Li j =,∂ ψ̇i j
pi =

(7a)
(7b)
(7c)

and the following generalized stresses:∂W,∂ εi j∂Wσi j =,∂ γi j
∂W
φi =
,
∂ vi
∂W
.
µi jk =
∂ κi jk
τi j =

(8a)
(8b)
(8c)
(8d)

The variational formulation also yields the following boundary conditions:(τi j − σi j )n j = ti
µi jk nk = Mi jorδ ui = 0,

(9a)

or

δ ψi j = 0,(9b)where ni is the outward-facing unit normal vector. The complete boundary value problem for the body comprises the equations of motion, Eqs. (6); the strain-displacement relations, Eqs. (1); the stress constitutive relations, Eqs. (8); the momentum constitutive relations, Eqs. (7); and the boundary conditions, Eqs. (9).

4

WILLIS MATERIAL ANALOGY

4.1 Long-wavelength, Low-frequency limit
For problems with sufficiently low frequencies and long wavelengths, the internal degrees of freedom ψi j andvi may be approximated as linearly related to the macro-displacement gradient and acceleration as follows:
ψi j ≈ Qi jkl uk,l ,(10a)vi ≈ −Ri j ü j ,(10b)where Qi jkl and Ri j are constants that depend on the material and geometric properties of the microstructure.
We note that Eq. (10a) is equivalent to a reduction to strain gradient elasticity, as in Ref. [15], while Eq. (10b)is a quasi-equilibrium, internal deformation caused by a constant acceleration.
4.2 Displacement Form
The approximations given by Eqs. (10) may be used to eliminate the internal degrees of freedom, ψi j and vi ,yielding equations of motion in terms of the macroscopic displacement ui only. To this end, we incorporate
these approximations into the variational formulation as constraints with corresponding Lagrange multipliers λi jand ηi . This yields Hamilton’s Principle in a modified form:Z T Z δT (u̇i , v̇i , ψ̇i j ) − W vi , ψi j , εi j , γi j , κi jk + λi j (ψi j − Qi jkl uk,l ) + ηi (vi + Ri j ü j ) dV + Wnc dT = 0. (11)0V

Carrying out the variation leads to the following equation of motion:Π̇i − Σi j, j = fi − R ji g̈ j − Qlki j Mlk, j ,(12)and the following boundary conditions:Σi j n j = ti + Qkli j Mkl n jorδ ui = 0,(13a)µi jk nk = Mi jorδ ui, j = 0,(13b)where we have defined the effective macroscopic stress Σi j and effective macroscopic momentum Πi , which are
given by
(14)
Σi j = τi j − σi j + Qkli j L̇kl + σkl − µklm,m ,Πi = pi − R ji q̈ j + φ˙j .
(15)
The terms in parentheses in Eqs. (14)-(15) are expressions of the balance of momentum for the microstructure, which are passed to the macroscale as effective stress and momentum quantities. Finally, by substituting
Eqs. (1), (7), (8) and (10) and Eqs. (2) and (3) into Eqs. (14) and (15), we may express the effective macroscopic stress and momentum in the familiar, Willis-like form:
Σi j = Ĉi jkl uk,l + Ŝi jk ük ,(16)

Πi = ρ̃i j u̇ j + S̃i jk u̇ j,k ,(17)

where the effective material properties are linear, differential operators of the form
(0)

(1)

(2)

(18a)

(0)

(2)

(odd)

(18b)

Ĉi jkl = Ĉi jkl + Ĉi jklm ∂m + Ĉi jklmn ∂m ∂n ,Ŝi jk = Ŝi jk + Ŝi jk ∂t2 + Ŝi jkl ∂l ,
(2)

(4)

ρ̃i j = ρ̃ (0) δi j + ρ̃i j ∂t2 + ρ̃i j ∂t4 ,
(0)

(18c)

(2)

S̃i jk = S̃i jk + S̃i jk ∂t2 ,

(18d)
(0)

(2)

δi j is the Kronecker symbol, and the various coefficients on the right-hand side of Eqs. (18) (e.g., Ĉi jkl , S̃i jk ,etc.) are constants that depend on the microstructure.

5

CONCLUSION

In this work, we have demonstrated that the linear, micromorphic elasticity theory of Mindlin [15] may be
re-cast in a form that is analogous to that obtained via the homogenization theory of Willis [11]. Specifically, we have introduced a time-domain, energy-based, variational formulation in which generalized stressand momentum-like quantities, which arise naturally through variations of the kinetic and deformation energy
densities, combine to yield effective constitutive relations that include Willis coupling.
These results present many opportunities for future study, such as: development of an energy-based homogenization scheme to model realizable materials within our framework; conversion of our model into the frequency
domain, in which the internal degrees of freedom could be eliminated without approximation; derivation ofsymmetries and physical bounds on the constitutive tensors; and the extension of the variational formulation to
include other physical effects, such as dissipation, nonlinearity, piezoelectricity, and electromomentum coupling[18, 19, 20].

ACKNOWLEDGEMENTS
Research was sponsored by the Defense Advance Research Project Agency (DARPA) and the Army Research
Office and was accomplished under Grant Number W911NF-20-1-0349. The views and conclusions containedin this document are those of the authors and should not be interpreted as representing the official policies,
either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government isauthorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation
herein.

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10.1115/1.4026911.
[2] Haberman MR, Norris AN. Acoustic Metamaterials. Acoustics Today. 2016;12:31-9. Available from:http://www.rci.rutgers.edu/~norris/papers/2016_Acoustics_Today_Metamaterials.pdf.
[3] Maugin GA, Metrikine AV, editors. Mechanics of Generalized Continua. vol. 21. Springer New York;2010. Available from: http://link.springer.com/10.1007/978-1-4419-5695-8.
[4] Madeo A. Generalized Continuum Mechanics and Engineering Applications. ISTE Press Ltd and Elsevier
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[5] Srinivasa AR, Reddy JN. An overview of theories of continuum mechanics with nonlocal elastic response
and a general framework for conservative and dissipative systems. Applied Mechanics Reviews. 2017 5;69.
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69/3/030802/421537/An-Overview-of-Theories-of-Continuum-Mechanics.
[6] Ha CS, Plesha ME, Lakes RS. Chiral three-dimensional isotropic lattices with negative Poisson’s ratio.
physica status solidi (b). 2016 7;253:1243-51. Available from: https://onlinelibrary.wiley.com/
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International Journal of Solids and Structures. 2017 2;106-107:91-105.
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2019 2;123:119-37.
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[11] Willis JR. Variational principles for dynamic problems for inhomogeneous elastic media. Wave
Motion. 1981 1;3:1-11. Available from: https://www.sciencedirect.com/science/article/pii/
0165212581900081.
[12] Norris AN, Shuvalov AL, Kutsenko AA. Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems. Proceedings of the Royal Society A: Mathematical, Physical andEngineering Sciences. 2012 6;468:1629-51. Available from: https://royalsocietypublishing.org/
doi/10.1098/rspa.2011.0698.
[13] Nassar H, He QC, Auffray N. Willis elastodynamic homogenization theory revisited for periodic media. Journal of the Mechanics and Physics of Solids. 2015 4;77:158-78. Available from: https://www.sciencedirect.com/science/article/pii/S0022509614002579.
[14] Muhlestein MB, Sieck CF, Alù A, Haberman MR. Reciprocity, passivity and causality in Willis
materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science.
2016 10;472:20160604. Available from: http://rspa.royalsocietypublishing.org/lookup/doi/
10.1098/rspa.2016.0604.

[15] Mindlin RD. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis. 1964
1;16:51-78.
[16] Sheng P, Zhang XX, Liu Z, Chan CT. Locally resonant sonic materials. Physica B: Condensed Matter.
2003 9;338:201-5. Available from: http://www.sciencemag.org/cgi/doi/10.1126/science.289.
5485.1734.
[17] Bedford A. Hamilton’s principle in continuum mechanics. vol. 139. Springer; 1985.
[18] Pernas-Salomón R, Shmuel G. Symmetry breaking creates electro-momentum coupling in piezoelectric
metamaterials. Journal of the Mechanics and Physics of Solids. 2020 1;134:103770.
[19] Pernas-Salomón R, Shmuel G. Fundamental Principles for Generalized Willis Metamaterials. PhysicalReview Applied. 2020;14:064005.
[20] Pernas-Salomón R, Haberman MR, Norris AN, Shmuel G. The electromomentum effect in piezoelectric
Willis scatterers. Wave Motion. 2021 11;106:102797.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0394

Theoretical study on the elastic wave propagation and optomechanical couplings in a sub-wavelength phoxonic waveguidebased on the perturbation method
Xue-Yuan Xing1; Xiao-Xing Su1,2; Jixing Qin2
1
2

School of Physical Science and Engineering, Beijing Jiaotong University, Beijing 100044, China

State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China

ABSTRACT
Based on the perturbation method, a theoretical study on the elastic wave propagation and opto-mechanicalcouplings in a sub-wavelength phoxonic waveguide is conducted. The considered waveguide material is the
single-crystalline silicon, which is optically isotropic, but elastically and photo-elastically anisotropic. First,
under a weakly viscoelastic assumption, the attenuation of the guided elastic wave is studied. Then, for the
waveguide with slightly misaligned material and structural axes, we study the effects of such misalignmenton the elastic eigenmodes and opto-mechanical coupling strengths. Theoretical analysis and numerical results
show the applicability and effectiveness of the perturbation method. The non-degenerate first-order
perturbation approximation is found to be able to accurately estimate the imaginary parts of the elastic
eigenfrequencies due to the weak viscoelasticity, while the perturbations on the elastic eigenfrequencies andopto-mechanical coupling coefficients, which are introduced in a slightly misaligned waveguide, can be
accurately captured by the second-order approximation. Furthermore, some analytical insights into theunderlying physics suggestive of potential practical applications can be gained.
Keywords: Brillouin opto-mechanics, Phoxonic micro-structures, Perturbation method

1. INTRODUCTION
In recent years, the studies on the sub-wavelength/nano opto-mechanics and phoxonics have been
paid much attention around the world. The stimulated Brillouin scattering (SBS) (1-3) is the mostsignificant acousto-optical interaction phenomenon in a waveguide structure, which is a third -ordernonlinearity optical process induced by the collective interaction among the pump, Stokes photons
and phonons. It has many applications such as Brillouin lasers and sensors. In the past, it has been
well established that the SBS is generally mediated by the photoelastic (PE) and electrostrictive (ES)effects. However, at the sub-wavelength/nano scale, the much enhanced surface/interface physicaleffects: the moving interface (MI) effect and optical radiation pressure (RP), may also become relevant
coupling mechanisms of SBS. Therefore, the SBS at the sub-wavelength scale may exhibit quite
radical behaviors, such as the strong dependence on the waveguide geometry, giant enhancement of
the SBS gain [especially in the case of forward SBS (FSBS) which is generally weak in a traditional
waveguide] and coherent interference between different coupling mechanisms (4).
The strengths of the coupling effects underlying SBS can be quantitatively characterized by the
opto-mechanical coupling coefficients (OMCCs), which are generally expressed as the overlap
integrals of the involved optical and elastic wave fields over the waveguide cross -section. The
derivation of the OMCCs and SBS gain is an important part of the theoretical research on the sub wavelength SBS. Based on a systematical coupled-mode formulation, Wolff et al. (5) derived the
explicit integral formulae of the OMCCs due to all the known coupling mechanisms in SBS. In
reference (6), such integral formulae are extended from the optically isotropic case to the general
optically anisotropic case. Based on the extended formulae, the authors of reference (7) calculated the
SBS gain spectra of the waveguide made of the strongly birefringent material rutile (titanium dioxide),
1

xxsu@bjtu.edu.cn

and found that a small misalignment between the material and structural axes of the waveguide maylead to a rapid emergence of many new resonance peaks in the SBS gain spectra. This finding
demonstrates the sensitivity of the opto-mechanical couplings and SBS gain to a structural symmetry
breaking at sub-wavelength scale, and may be utilized for some tuning and control purposes in thepractical applications of SBS.
To the authors' knowledge, the perturbation method is seldom used in the theoretical studies of
sub-wavelength SBSs. Being one of the most widely used approximation techniques, the perturbation
method is generally based on the idea of expressing the small changes by asymptotic expansions, andcan thus provide some analytical insights into a complex system. Based on the perturbation method,
we first study the attenuation of elastic waves in a sub-wavelength silicon waveguide under thecondition of weak viscoelasticity in this paper. Then, considering the photoelastic and mechanicalanisotropy of silicon, the effects of the small misalignment between the material and structural axeson the elastic eigenfrequencies and OMCCs of the waveguides are studied. Th e obtained numericalresults show the effectiveness of the perturbation method, and some perturbative -expansion based
discussions are provided.

2. FUNDAMENTALS OF THE WORK
2.1 Theory of Sub-wavelength Stimulated Brillouin Scatterings
We consider a waveguide which is longitudinally invariant in the z-direction, as shown in Figure
1. The doublets (𝜔, 𝛽) and (𝛺,q) represent, respectively, the angular frequencies and wave numbersof the light and elastic waves involved in a SBS process. They should meet the phase -matching
relation as shown on the right-hand part of Figure 1. Here, subscripts s or p, label the Stokes and pumplights, respectively.

Figure 1- Schematic sketch of the considered waveguide structure and the phase-matching relation
In a steady-state, the growth of the Stokes light along the z-direction can be described by thefollowing equation:𝑑𝑃s(1)= 𝐺B 𝑃p 𝑃s − 𝛼s 𝑃s ,𝑑𝑧
where P and α are, respectively, the guided power and linear loss-factor of the optical wave corresponding
to the subscript, and 𝐺B the SBS gain.
According to reference (4), the SBS gain can be expressed as a function of 𝛺：
𝐺B (𝛺) = ∑ 𝐺B,𝑚𝑚

(𝛤𝑚 /2)2,(𝛺 − 𝛺𝑚 )2 + (𝛤𝑚 /2)2(2)

where 𝐺B,𝑚 and 𝛺𝑚 are the SBS gain and eigenfrequency of the 𝑚𝑡ℎ elastic eigenmode,
respectively. Therefore, the peaks in a SBS gain spectrum exhibit a Lorenz shape, and the total SBS
gain can be understood as the weighted sum of individual SBS gains. If considering mechanical loss,
𝛺𝑚 becomes 𝛺𝑚 − i𝛤𝑚 /2, with i = √−1. The SBS gain 𝐺B,𝑚 can be expressed as:

𝐺B,𝑚 =

𝜔s 𝑄𝑚 |𝐶𝑚 |2,
4𝑃p 𝑃s 𝜀𝑚

(3)

2

𝛺
𝛺
̃ 𝑚 ) 𝑑 2 𝐫 , 𝑄𝑚 = 𝑚 and 𝜀𝑚 = 𝑚 ∫ 𝜌|𝐮
̃ 𝑚 |2 𝑑 2 𝐫 represent the OMCC, thewhere 𝐶𝑚 = ∫(𝐟̃∗ ⋅ 𝐮𝛤𝑚2mechanical Q-factor and the energy per unit length of the 𝑚𝑡ℎ elastic eigenmode, respectively. Here,the hat "~" means the operation to take the (x, y)-dependent complex amplitude of a physical field,
the integration is performed over the whole cross-section of the waveguide, 𝐟 is the total optical force
density generated by the interaction of the pump and Stokes lights, 𝐮𝑚 is the displacement field of
the 𝑚𝑡ℎ elastic eigenmode, and 𝜌 is the mass density.
As per the integral definition of the OMCC given above, the OMCC essentially represents the work
per unit length which is done by the optical force acting on the displacement of the elastic wave. Thus,
the OMCC can be further expressed as the algebraic sum of the two parts corresponding to theelectrostrictive force and optical radiation pressure, respectively. Alternatively, according to reference(5), if the irreversible coupling effects can be ignored and the optical loss is small enough, the OMCC
can also be evaluated via the PE and MI effects, and we have 𝐶ES = (𝐶PE )∗ and 𝐶RP = (𝐶MI )∗ . (Here,
we have ignored the subscript m labelling the number of the elastic wave modes). The integral
expressions of the PE and MI OMCCs and their detailed derivations can be found in reference (6).
2.2 Material and Structure
Silicon is ubiquitous in integrated photonics. With no extraordinary material axis, singlecrystalline silicon belongs to the point group m3m of the cubic crystal system (8). It is opticallyisotropic, but photo-elastically and mechanically anisotropic. The material parameters used in thispaper, including the refractive index, photoelastic coefficients, elastic modulus and visco sity
coefficients, are all taken from references (3, 9). All optical parameters correspond to the free space
wavelength of 1550 nm.
In the present work, we only consider the circular or elliptical waveguides, and the optical andelastic eigenmodes are calculated by the commercial finite element solver COMSOL Multiphysics.
Figure 2 presents the definitions of the structural and material coordinate systems x-y-z and ξ-ζ-η in aconsidered waveguide. As shown here, we use the Euler angles to characterize the rotation between
such two sets of axes. In the present work, we assume that only the Euler angle θ may be nonzero.

Figure 2- Material and structural coordinate systems of the waveguide and definition of the Euler angles. Theline of nodes is denoted by L.

3. THEORETICAL DERIVATION AND NUMERICAL RESULTS
3.1 Basic Derivation
The elastic wave equation in the frequency domain is written as:𝜌𝛺2 𝐮 + ∇ ∙ (𝐜: 𝐬) = 0 .

(4)Here, 𝐜 is the stiffness tensor, and 𝐬 is the strain field. Following the general procedure of theperturbation method (10, 11), we first suppose that:(5)𝐜 = 𝐜 (0) + 𝜖∆𝐜 ,where 𝜖 is a small parameter characterizing the perturbation. Then, it can be further supposed that:𝛺 = 𝛺 (0) + 𝜖𝛺(1) + 𝜖 2 𝛺(2) +∙∙∙ ,

(6a)

̃=𝐮
̃ (0) + 𝜖𝐮
̃ (1) + 𝜖 2 𝐮̃ (2) +∙∙∙ ,𝐮(6b)𝐬̃ = 𝐬̃ (0) + 𝜖𝐬̃ (1) + 𝜖 2 𝐬̃ (2) +∙∙∙ .
In the non-degenerate case, we can obtain that:(6c)̃ 𝑚 (1) = ∑𝐮̃ 𝑛 (0) ,
𝑎𝑛 𝐮

(7)

̃ 𝑚 (2) = ∑𝐮̃ 𝑘 (0) .
𝑏𝑘 𝐮

(8)

𝑛≠𝑚

𝑘≠𝑚

Here,
𝑎𝑛 = −𝑏𝑘 =< 𝐬̃𝑛 (0) , (∆𝐜: 𝐬̃𝑚 (0) ) >
[(𝛺𝑛

(0) 2

) − (𝛺𝑚

(0) 2

̃𝑛
) ]<𝐮

(0)

̃𝑛
, 𝜌𝐮

(0)

>

(9)

,

̃ 𝑘 (0) , 𝜌𝐮
̃ 𝑘 (0) >− ∑𝑛≠𝑚 𝑎𝑛 < 𝐬̃𝑘 (0) , (∆𝐜: 𝐬̃𝑛 (0) ) > +2𝛺𝑚 (0) 𝛺𝑚 (1) 𝑎𝑘 < 𝐮
[(𝛺𝑘

(0) 2

) − (𝛺𝑚(0) 2

̃ 𝑘 (0) , 𝜌𝐮
̃ 𝑘 (0) >
) ]<𝐮

,

(10)

where <A, B> is the inner product between two vector fields A and B defined as an overlap integral
over the waveguide cross-section:< 𝐀, 𝐁 >= ∫ 𝐀∗ ∙ 𝐁 𝑑 2 𝐫 .

(11)

And for the frequency corrections, we have:𝛺𝑚 (1) =𝛺𝑚 (2) =< 𝐬̃𝑚 (0) , (∆𝐜: 𝐬̃𝑚 (0) ) >̃ 𝑚 (0) , 𝜌𝐮
̃ 𝑚 (0) >2𝛺𝑚 (0) < 𝐮∑𝑛≠𝑚 𝑎𝑛 < 𝐬̃𝑚 (0) , (∆𝐜: 𝐬̃𝑛 (0) ) > − [𝛺𝑚 (1) ]2 < 𝐮̃ 𝑚 (0) , 𝜌𝐮
̃ 𝑚 (0) >̃ 𝑚 (0) , 𝜌𝐮
̃ 𝑚 (0) >2𝛺𝑚 (0) < 𝐮

(12)

,

.

(13)

3.2 First-order Correction of Elastic Eigenfrequency due to Viscoelasticity
Silicon is generally a material with weak viscoelasticity, so the perturbation method is used here
to calculate the frequency correction due to the viscoelasticity for the considered waveguides.
According to the linear elasticity theory, we can assume that ∆𝐜 = −i𝛺𝛈 , with 𝛈 being theviscoelasticity tensor. Then, the substitution of such expression of ∆𝐜 into Eq. (12) yields theimaginary first-order correction of the 𝑚𝑡ℎ elastic eigenfrequency:𝛺𝑚 (1) = −< 𝐬̃𝑚 (0) , (𝛈: 𝐬̃𝑚 (0) ) >
̃𝑚
2<𝐮

(0)

̃𝑚
, 𝜌𝐮

(0)

>

i.

(14)

Based on the definition of 𝛤𝑚 given above, we essentially have 𝛺𝑚 (1) = −i𝛤𝑚 /2. Let us take acircular waveguide with a radius of 111.3 nm as an example to verify Eq. ( 14). Table 1 lists thecalculated eigenfrequencies of two non-degenerate modes and two doubly degenerate modes whichcorrespond to the elastic wavenumber q = −3.5752 × 107 𝑚−1 . For each eigenfrequency, the results
calculated by COMSOL with and without the inclusion of viscoelasticity, as well as the imaginary
corrections obtained by Eq. (14), are all given. Obviously, the first-order perturbation theory canaccurately predict the frequency correction due to viscoelasticity. As the viscoelasticity tensor and thestiffness tensor possess exactly the same symmetry, the introduction of the visco elasticity tensor will
not break the original symmetry of the system, resulting in no split of the original degeneracy as well
as no significant change of the elastic field. Therefore, the first -order perturbation can make a good
approximation here.

Table 1- The eigenfrequencies obtained with and without considering viscoelasticity as well as thecorresponding imaginary corrections obtained by the first-order perturbative approximation. (in units of Hz)
Imaginary correction
Eigenfrequencies calculated without

Eigenfrequencies calculated with theobtained by first-

the inclusion of viscoelasticity

inclusion of viscoelasticity

order perturbative
approximation

Non-degenerate

2.83153 × 1010

2.83154 × 1010 − 3.22301 × 107 i−3.22309 × 107 icase

4.46410 × 1010

4.46419 × 1010 − 8.60848 × 107 i−8.61033 × 107 i

Degenerate

8.94285 × 1010

8.94297 × 1010 − 3.61241 × 108 i−3.61018 × 108 i

case

8.94285 × 1010

8.94297 × 1010 − 3.61241 × 108 i−3.61018 × 108 i

3.3 Correction of Elastic Eigenfrequency and Change of Opto-mechanical Coupling
Coefficient due to Small Misalignment between Material and Structural Axes
If a small angular misalignment between the material and structural axes of the waveguide isintroduced (due to fabrication defect or in order to realize some tuning and control purposes), theguided physical fields may change significantly as a result of symmetry breaking. Then, to accurately
capture the effects of such small axes misalignment, we will use the second -order perturbative
approximation in the following calculations. The small changes of the stiffness and photoelastic
tensors ( ∆𝐜 and ∆𝐩 ) corresponding to any given Euler angle θ are calculated based on thetransformation matrix between the two sets of coordinate systems (7). Substituting the obtained
second-order perturbative approximation of the displacement and strain fields into the expression of
the OMCCs, we may have:(0)

𝐶𝑚 = 𝐶𝑚 + ∑

(0)

𝑛≠𝑚

(𝑎𝑛 + 𝑏𝑛 )∗ ∙ 𝐶𝑛

−𝜀0 ∫(𝐞̃s )∗ {[𝛆r ∙ (∆𝐩: 𝐬̃𝑚

(15)

(0) ∗

) ∙ 𝛆r ] ∙ 𝐞̃p } 𝑑 2 𝐫,

where e denotes the electric field of the involved optical wave. Additionally, the unit length elastic
energy 𝜀𝑚 in Eq. (3) is corrected as:
𝜀𝑚 =

(𝛺𝑚

(0)

+ 𝛺𝑚 (1) + 𝛺𝑚 (2) )2
(𝛺𝑚

+∑

(0) 2

)

(𝛺𝑚
|𝑎𝑛 + 𝑏𝑛 |2

(0)

(0)

∙ 𝜀𝑚

+ 𝛺𝑚 (1) + 𝛺𝑚 (2) )2
(0)

(16)
(0)
∙ 𝜀𝑛 .

(𝛺𝑛 )2On the right-hand side of Eq. (15), the second term is the contribution of mechanical anisotropy,
while the third term is the contribution of photoelastic anisotropy. We consider an elliptical waveguideto verify the accuracy of the perturbative approximation obtained above. The semi -major and semiminor axes of the waveguide’s elliptical cross-section are 342.8 nm and 220.4 nm, respectively. And
we assume that the elastic wavenumber q = 0. First, the elastic eigenmodes are calculated with
COMSOL for θ = 0˚, 1˚ and 2 ˚, respectively. Then, for each mode, the accurate corrections of theeigenfrequency due to the introduction of the two nonzero misalignment angles are compared withthe corresponding perturbative predictions. We select the obtained results for two elastic eigenmodes
and present them in Tables 2 and 3, in which the presented perturbative predictions are corresponding
to θ = 1˚ and 2 ˚, respectively. The accuracy of the sum of the perturbative corrections of the first twoorders is shown by the relative errors between them and the corresponding accurate results, which are
all smaller than 4%.
𝑛≠𝑚Table 2-The obtained results for showing the accuracy of the perturbative corrections of the elastic
eigenfrequencies of the considered elliptical waveguide due to the introduction of a nonzero misalignment
angle θ. (The nonzero misalignment angle θ are assumed to be 1˚. The eigenfrequencies corresponding to θ(0)

= 0˚ and the assumed nonzero value are denoted by 𝑓𝑚

and 𝑓𝑚 , respectively. And the first and second order

perturbative corrections of the eigenfrequencies are denoted by 𝑓𝑚 (1) and 𝑓𝑚 (2) , respectively. The
frequencies along with their corrections are all in units of Hz.)(0)5.06453
× 10

(0)

𝑓𝑚

𝑓𝑚

9

5.06450× 10

91.30691

1.30685

10

10

× 10

× 10

𝑓𝑚 − 𝑓𝑚

𝑓𝑚 (1)

𝑓𝑚 (2)

𝑓𝑚 (1) + 𝑓𝑚 (2)−2.7860−2.2933−0.3891−2.6824× 10

4−5.7559
× 10

5

× 10

4

−3.9043
× 10

5

× 10

4

−1.7408
× 10

5

× 104
−5.6451
× 105

Relative
error
3.72%
1.93%

Table 3- The obtained results for showing the accuracy of the perturbative corrections of the elastic
eigenfrequencies of the considered elliptical waveguide due to the introduction of a nonzero misalignment
angle θ. (The nonzero misalignment angle θ are assumed to be 2˚. See the caption of Table 2 for more detailsof what is presented here.)
(0)

(0)

𝑓𝑚

𝑓𝑚

𝑓𝑚 − 𝑓𝑚

𝑓𝑚 (1)

𝑓𝑚 (2)

𝑓𝑚 (1) + 𝑓𝑚 (2)5.06453

5.06442−1.1138−9.1622−1.5580−1.0720

× 109

× 109

× 105

× 104

× 104

× 105

1.30691

1.30668−2.3009−1.5598−0.6955−2.2553

× 1010× 1010

× 106

× 106

× 106

× 106

Relative
error
3.75%
1.98%

In Figure 3 are presented the FSBS gain spectra of the considered elliptical waveguide calculated
for the misalignment angle θ = 0° and 2°. It can be seen that after the nonzero misalignment angle isintroduced, three obvious resonance peaks appear (marked with the symbol 'o' in the figure). Using
the derived approximate formula (15), we also calculate the OMCCs corresponding to these threeresonant peaks, which are then compared with the corresponding accurate results. By detailed
comparisons, the accuracy of the perturbatively approximate formula (15) is verified. The detailed
results are presented in Table 4. (Note that in the case that θ = 0°, the OMCCs essentially vanish
because of the absence of resonance peaks.) In addition, it is also found here that, due to the weak
photoelastic anisotropy of silicon, the magnitude of third term on the right-hand side of Eq. (15) can
essentially be neglected. Therefore, in a sub-wavelength silicon waveguide with slightly-misaligned
material and structural axes, the misalignment-sensitive behaviors of the OMCCs and SBS gains areprimarily attributed to the elastic anisotropy of the material, suggesting the possible utilization ofelastic anisotropy to realize some tuning purposes in silicon photonics.

Figure 3- FSBS gain spectra of the considered elliptical waveguides with different misalignment angle θ
Table 4- The perturbatively approximated versus accurate values of the OMCCs (in units of N) obtained forthe elastic eigenmodes of the considered elliptical waveguide with different order numbers m (whichcorrespond to the three misalignment-sensitive resonance peaks shown in Figure 3).
m

12

36

76

Approximate value of Cm

7.0044 × 10−9

1.3375 × 10−92.3151 × 10−9

Accurate value of Cm

7.0119 × 10−9

1.3429 × 10−92.3031 × 10−9

4. CONCLUSION
In this paper, the perturbation method used for the theoretical study of the elastic wave propagationand SBS in a sub-wavelength phoxonic waveguide is developed. Numerical simulations about thecircular or elliptical silicon waveguides verify the applicability of the developed method. The
corrections of the elastic eigenfrequencies due to the weak viscoelasticity as well as the OMCCs dueto the introduction of the small misalignment between the material and structural axes of the
waveguide can be perturbatively approximated with high accuracy. The obtained results may
contribute to deepen our understanding of the elastic wave propagation and opto -mechanical coupling
characteristics in phoxonic microstructures, and may provide some theoretical guidance for thepractical application of SBS.

ACKNOWLEDGEMENT
This work is supported by the State Key Laboratory of Acoustics and the Youth Innovation
Promotion Association CAS.

REFERENCES
1. Eggleton BJ, Poulton CG, Rakich PT, Steel MJ, Bahl G. Brillouin integrated photonics. Nature Photonics.
2019;13(10):664-77.
2. Wiederhecker GS, Dainese P, Mayer Alegre TP. Brillouin optomechanics in nanophotonic structures. APL
Photonics. 2019;4(7).
3. Wolff C, Smith MJA, Stiller B, Poulton CG. Brillouin scattering—theory and experiment: tutorial. Journalof the Optical Society of America B. 2021;38(4).
4. Rakich PT, Reinke C, Camacho R, Davids P, Wang Z. Giant Enhancement of Stimulated Brillouin
Scattering in the Subwavelength Limit. Physical Review X. 2012;2(1).
5. Wolff C, Steel MJ, Eggleton BJ, Poulton CG. Stimulated Brillouin scattering in integrated photonic
waveguides: Forces, scattering mechanisms, and coupled-mode analysis. Physical Review A. 2015;92(1).
6. Su X-X, Li X-S, Wang Y-S, Lee HP. Theoretical study on the stimulated Brillouin scattering in a sub-

wavelength anisotropic waveguide: acousto-optical coupling coefficients and effects of transverse
anisotropies. Journal of the Optical Society of America B. 2017;34(12).
7. Su X-X, Dou Z-L, Lee HP. Stimulated Brillouin scattering in a sub-wavelength anisotropic waveguidewith slightly-misaligned material and structural axes: misalignment-sensitive behaviors and underlying
physics. Journal of Optics. 2022;24(4).
8. Auld BA. Acoustic fields and waves in solids. Weinheim,Germany: Wiley.Ltd; 1973.
9. Weber MJ. Handbook of optical materials. Boca Raton,USA: CRC Press; 2002.
10. Zeng J-Y. Quantum mechanics. Beijing,China: Science Press; 2007.(in Chinese)
11. Landau LD, Lifshitz EM. Quantum mechanics. Oxford,UK: Pergamon Press; 1977.

ABS-0586

An analysis of transformation functions for flexural cloaking
Aidin Nojavan,(1) , Fiona Gottschalk(2) , Sabine Christine Langer(3)(1) Institute for Acoustics, TU Braunschweig, Germany, a.nojavan@tu-braunschweig.de(2) Institute for Acoustics, TU Braunschweig, Germany, f.gottschalk@tu-braunschweig.de(3) Institute for Acoustics, TU Braunschweig, Germany, s.langer@tu-braunschweig.de

ABSTRACT
Cloaking is a special case of wave guidance that aims at steering of waves around an object or space. Cloaking can be beneficial in different future applications to reduce wave scattering or to isolate a space from thesurrounding waves. In order to transform wave propagation within the cloak, geometrical transformations are
introduced. Performing a suitable coordinate transformation of the wave equation yields the required material
distribution for steering the waves in the desired path. Hence, the cloak’s performance is directly and deeply
dependent on the chosen coordinate transformation. While in most of the related literature a “linear” transformation is used to design a cloak, few studies claim that a “quadratic” function results in a more effective cloak.
Based on this observation, a more extensive study is conducted in this paper, where the linear and the quadratic
approaches are generalized to a coordinate transformation of arbitrary polynomial degree. The study explores
the influence of the explicit choice of the underlying transformation on the quality of the cloaking effect bothanalytically and numerically for several of the aforementioned polynomial degrees. The numerical investigation
is based on the finite element method.

Keywords: Elastic Metamaterials, Flexural Cloaking, Wave Guidance

1

MOTIVATION

The developments in the field of metamaterials in the recent decades have encouraged scientists to pursue
inspirations, which previously appeared as science fiction fantasy. In cloaking, the waves are guided smoothly
around a space by engineering the material properties around it. The space in the middle is called the core andthe neighboring region around it with engineered material matrix is called the cloak. Usually, a circular shapeis assumed for the core and cloak to make the concept independent of the direction of the incident wave. In
the right panel of Figure 1, the schematic of the cross section of a cylindrical cloak is shown, which is usedas the reference for the discussions in this paper. The core and cloak in the figure have the radii R1 and R2 ,respectively. The core typically represents an obstacle or a discontinuity in the structure or space, which causes
the incident waves to be scattered. Moreover, the obstacle causes a shadow region directly behind it. Smoothguidance of the waves in the cloak region around the core reduces the wave scattering and shadow region.
The guidance of the waves in the cloak makes the space appear as if there was no obstacle in the way of thephysical field. It transforms the obstacle to an infinitesimally small point, which is schematically shown in theleft panel of Figure 1. It is feasible that a cloak could be used, for example, to guide the waves around thepillars in buildings or around the cabin in automobiles and airplanes.
The idea of cloaking was firstly introduced in electromagnetism [10], but was soon extended for applicationsin other fields such as acoustics [4] and elastodynamics [7]. Regarding the widespread usage of thin plates inthe engineering field, the concept of cloaking was also adopted and introduced for manipulating flexural waves
in thin plates [6], which is the mainly considered type of mechanical waves in thin plates. In this regard,
normally the term flexural cloaking is used to refer specifically to the cloaking of flexural waves in thin plates.
The compactness and light weight of thin plates make the flexural cloaking concept more attractive for a large
variety of applications and helps to integrate cloaks. Flexural cloaking could be used to reduce wave scatteringR2transformation

R1 7→ 0

R1
core
cloak

Figure 1. Schematic of a cloak. Left: the virtual space. Right: the physical space. The waves are bent aroundthe core space by a transformation in the cloak region, which is realized by an appropriate material matrix. The
transformation identifies the core space in the physical space with a single point in the virtual space (after [8]).

caused by a hole in a plate or on positions, where other structures are jointed to a plate. It could also be used
for isolating sensitive devices or as previously mentioned in automotive and aerospace industries.
Contrary to the usual engineering problems, where the distribution of a physical field is solved for specificknown material properties, a suitable material matrix in a cloak is searched for in order to obtain the desired
physical field distribution. More precisely, the waves are desired to propagate in specific routes around the core;and based on that, the required material properties are calculated and engineered in the cloak region. One of
the few existing methods to design a cloak inversely is the transformation method [3, p. 197]. In this method,
firstly the physical field and material properties are linked together in the space. This is normally describedby the governing equations of the physical field. Afterwards, a suitable coordinate transformation is used todescribe the desired distribution of the physical field in the space, for example the routes around the core
shown in Figure 1. At the end, the governing equations are transformed according to the chosen coordinatetransformation in order to transform the physical field and calculate the required transformed material matrix.
The underlying coordinate transformation is not limited to one specific choice. In the literature, mostly apolynomial function of order n = 1, i.e., an affine function, is used to describe the distortion of the space, seefor example [6]. Notice that this function differs from a linear one only by an additive constant, and consequently, even though it is technically not linear, it is referred to as such for simplicity. Zareei et al. [12] were
the first who proposed using a non-affine, and as such a truly nonlinear transformation, for flexural cloaking.
They claimed that the "nonlinear cloak has much less scattering in a broader range of frequencies” [12, p. 4].
Later, Darabi et al. [5] confirmed experimentally that a cloak designed based on the nonlinear transformation
introduced by Zareei et al. [12] shows greater potential in cloaking at higher frequencies in comparison to
the linear transformation introduced by Farhat et al. [6]. Since the cloak, namely the distortion of the waves
in the space, is designed based on the coordinate transformation, the choice of the coordinate transformation
has a fundamental impact on the performance of the cloak. Therefore, different polynomial transformations are
considered in this study and the performance of the corresponding cloaks is compared numerically.

2

POLYNOMIAL TRANSFORMATIONIn the transformation technique, the coordinate mapping is used to transform a physical field, which represents
the flexural vibrations in a cylindrical cloak in a thin plate for the present study. The free vibrations in a thinplate are described by the Kirchhoff equation, which is written as follows in polar form:
D0 ∇2 ∇2 w(r, θ ) + ρ0 hω 2 w(r, θ ) = 0 ,(1)where D0 , w, ρ0 , h, and ω refer to the flexural rigidity, deflection, mass density, thickness, and angular frequency of the plate, respectively, and ∇2 is the Laplace operator in polar coordinates.

For the construction of a transformation function, inspiration is taken from the transformation functions usedin [2] and [12]. These are generalized to a map Fn : (0, R2 ) × (0, 2π) → Bk·k2 (0, R2 )\Bk·k2 (0, R1 ) defined by


1

n
n
n
sin
θ
R −R
 , r ∈ (0, R2 ) , θ ∈ (0, 2π) ,(2)Fn (r, θ ) := Rn1 + 2 n 1 rn 
R2cos θthat maps the a disk of radius R2 onto an annulus of radii R1 and R2 . Since one is interested in expressing thegoverning differential equations in terms of new coordinates (r̃, θ̃ ) rather than (r, θ ), one considers the inverse,
i.e., the coordinate transformation Φn : (R1 , R2 ) × (0, 2π) → Bk·k2 (0, R2 ) defined by


1

n
n
sin
θ̃
1
R2
 , r̃ ∈ (R1 , R2 ) , θ̃ ∈ (0, 2π) .
Φn (r̃, θ̃ ) :=(r̃n − Rn1 ) n (3)
Rn2 − Rn1
cos θ̃
Notice that for n = 1 and n = 2 the function Fn coincides with the (affine-) linear and nonlinear maps used in
the literature (see [2] and [12]). The transformation acts on horizontal lines by bending them away from thecore region as displayed in Figure 2 for some
 explicit choices of n.
The family of coordinate transformations Φn | n1 , n ∈ N gives rise to an infinite number of coordinate trans-

n=0.5
n=1
n=2
n=3
n=4

Figure 2. Images of horizontal lines under Fn for all n ∈1

2 , 1, 2, 3, 4

, R1 = 1, and R2 = 2.

formations, where each transformation relates the old and the new radial variables, r and r̃, respectively, by theequationr̃n −Rn2 − Rn1 n
r = Rn1 ,Rn2(4)

which is a polynomial equation of degree n. Therefore, it is convenient to refer to the cases n ∈ {1, 2, 3, 4} aslinear, quadratic, cubic, and quartic, respectively, facilitating the distinction of different degrees of nonlinearity.
Regarding Figure 2, one observation can be made instantly: The transformation behaves monotonously withrespect to the parameter n in the sense that the larger one chooses n, the less strongly the horizontal lines
are bent away from the core region. This causes the equally spaced horizontal lines both to be mapped more
densely and to curve more drastically in the vicinity of the inner boundary of the cloak for larger n. On theother hand, this also results in a smoother transition into the cloak at the outer boundary at radius r̃ = R2 forlarger n.

The formalism necessary to transform the differential equation describing the considered problem is provided
by differential geometry. It is sufficient to consider the transformation of the Laplace operator,  
p∂1
 r̃   ,∇2 = p∂xe  det gΦn g−1
(5)∑
Φn
det gΦn xe∈{r̃,θ̃ }∂θ̃xewhere gΦn is the metric induced by the coordinate transformation Φn defined by gΦn := JΦTn JΦn and JΦn is
the Jacobian of Φn . Carrying out the transformation given by equation (5) with respect to Φn defined in (3)and applying the operator to itself to extend the transformation to the operator ∇2 ∇2 , tedious calculations andsimplifications finally yield for every coordinate transformation Φn , n, 1n ∈ N the transformed Kirchhoff equation.
More explicitly, equation (1) writes as follows in the transformed coordinates:
4 "
 n
4
4
4
R2 − Rn1 n (r̃n − Rn1 )1− n∂ w (r̃n − Rn1 )2− n∂ 2 w 2(r̃n − Rn1 )3− n n
∂ 3w
f
(r̃)
f
(r̃)
+
(r̃ + 3(n − 1)Rn1 ) 3
+
D0
1
2
n
4n−1
4n−2
2
4n−3
R2
r̃
∂ r̃
r̃
∂ r̃
r̃
∂ r̃
4
 n

4
4
r̃ − Rn1 4− n ∂ 4 w 2(r̃n − Rn1 )1− n n2(r̃n − Rn1 )2− n ∂ 4 w
∂ 3w
n
−
(r̃
−
(n
−
1)R
)
+
+
1
r̃n
∂ r̃4
r̃2n−1r̃2n−2∂ θ̃ 2 ∂ r̃
∂ θ̃ 2 ∂ r̃2

2
4
4 ∂ w
n
n − 4n ∂ w+4(r̃n − Rn1 )− n
+
(r̃
−
R
)
+ ρ0 hω 2 w = 0 ,(6)1∂ θ̃ 2∂ θ̃ 4wheren
3n
f1 (r̃) := r̃3n + (n2 + n − 1)(n − 1)Rn1 r̃2n + (−6n2 + n + 1)(n − 1)R2n
1 r̃ + (3n − 1)(2n − 1)(n − 1)R1

(7)

f2 (r̃) := −r̃2n − 2 (2n − 1) (n − 1)Rn1 r̃n + (11n − 7)(n − 1)R2n
1 .

(8)

and

A cylindrical cloak can be realized by arranging rings of different materials beside each other, which iscalled a multilayered cloak and is shown schematically in the left panel of Figure 3. The realization of a cloak
is explained in detail in Section 3. However, first the required material properties need to be calculated. Inorder to derive the required material properties for the field transformation in a multilayered circular cloak, thetransformed terms from equation (6) are compared with the original Kirchhoff equation in the undeformed space.
Notice that equation (1) is formulated assuming a homogeneous isotropic material for the plate. Therefore, thegeneral Kirchhoff equation including the polar orthotropy and radial inhomogeneity of a multilayered circularcloak is the correct choice for comparing the original and transformed equations. The corresponding extended
equation in the cylindrical coordinates is written as



 3
∂ 4 w Dθ ∂ 4 w 2Drθ ∂ 4 w
Dr ∂ Dr ∂ 3 w 2 ∂ Drθ Drθ
∂ w
Dr 4 + 4
+
+
2
+
+
−
4
2
2
2
3
2
∂r
r ∂θ
r ∂r ∂θ
r
∂r
∂r
r
∂r
r
∂ r∂ θ 2
 2
 2


∂ Dr 1 ∂ (Dr νθ ) Dθ 2 ∂ Dr ∂ w
2Drθ
2 ∂ Drθ 2Dθ
1 ∂ Dθ
1 ∂ 2 (Dr νθ ) ∂ 2 w
+
− 2 +
+
− 3
+ 4 − 3
+ 2
+
∂ r2
r
∂r
r
r ∂r
∂ r2
r4
r ∂r
r
r ∂r
r
∂ r2
∂θ2


Dθ
1 ∂ Dθ 1 ∂ 2 (Dr νθ ) ∂ w+−++ ρhω 2 w = 0 ,(9)r3
r2 ∂ r
r
∂ r2
∂r
where

Gh3
.
(10)
6
More details on the derivation of (9) and (10) can be found in [12].
More precisely, the comparison of the equations succeeds the following idea: One regards the physicalspace with an inhomogeneous material distribution as the deformation of a virtual space with a homogeneousDrθ = Dr νθ + 2Dk = Dr νθ +Ẽθ̃ (r)/Ehost
104
host

j

Ẽθ̃ /Ehost

102

core

s

lay

h
er
lay

er

n

100

1

10−2

Ẽr̃ (r)/Ehost

10−4jẼr̃ /Ehost
1

2

3
r̃/R1

4

5

Figure 3. Left: schematic of the layers in a multilayered cloak to discretize the material properties distribution.
Right: continuous and discretized material properties distribution in a cloak [8].

material distribution. The respective coordinates are matched by a suitable coordinate transformation T . Expressing the plate equation with homogeneous material parameters in virtual space in terms of the coordinatesin physical space yields transformed differential operators with additional factors according to the chain rule asdemonstrated by the exemplary calculation


 −1

∂w
∂
∂T
∂ (w ◦ T )−1
D0◦ T = D0
(w ◦ T ◦ T ) ◦ T = D0
◦T
,
(11)
∂r
∂r
∂r
∂ r̃
|
{z
}
transformed material properties

where r and r̃ refer to the radial coordinates in virtual and physical space, respectively, and D0 refers to homogeneous material properties in virtual space. In other words, the left-hand side of equation (11) exemplarily
depicts part of the Kirchhoff equation with homogeneous material parameters transformed via the chosen coordinate transformation T . The resulting expression on the right-hand side of (11) is a differential operator withrespect to the transformed coordinates in physical space with an additional factor stemming from the underlying
transformation due to the chain rule. The right-hand side of equation (11) further illustrates that this additional
factor can be interpreted as part of the respective material parameter of an inhomogeneous material. Consequently, it is matched with the respective unknown inhomogeneous parameters of equation (9), thus defining theinhomogeneous material matrix necessary to bend the waves in the initially desired way.
Choosing the following relations, the terms referring to the 4th order derivatives of the transformed equation(6) and original equation (9) match together perfectly:
 n
r̃ − Rn1 2
D̃r̃ = J
D0 ,
r̃n
2

D̃θ̃ = J

2



r̃n
n
r̃ − Rn1

where

J=

Rn2 − Rn1
Rn2

2


D0 ,

ρ̃ =

r̃n
n
r̃ − Rn1

 2(n−2)
n

ρ0 ,

(12)

2

n

.

(13)

It is noticed that for all n ∈ N the constant J 2 converges to unity in the limit of large ratios of R2 /R1 .
For polynomials with n ≥ 2, the rate of convergence is sufficiently fast in order for J 2 to be neglected forR2 /R1 ≥ 4. In this case, the transformed properties converge to the host properties (D0 , ρ0 ) at the outer edgeof the cloak around R2 , which is essential to avoid undesired large discrepancies between the cloak and thesurroundings. For the linear transformation, a very large ratio of R2 /R1 is required to reach J 2 ≈ 1 and avoid
large differences with the surroundings. For the transformation with respect to n = 0.5 the situation is worseand extremely large values of R2 /R1 are necessary to fulfill J 2 ≈ 1. Therefore, in this study, the constant J 2
is simply disregarded for all the transformations to avoid the reflection of the waves on the outer boundariesof the cloak around R2 . Moreover, the adjustment of the flexural rigidity is challenging, as it depends onthe elasticity modulus, Poisson’s ratio, and thickness of the plate. Instead, the transformation relations (12)are assigned only to the elasticity modulus. The experiments performed by Stenger et al. [11] proved that aflexural cloak designed merely by the transformation of the elasticity modulus guides the waves around the core.
Therefore, considering the above explanations, the following relations are used in this study to design flexural
cloak configurations based on transformations with different polynomial orders: nr̃ − Rn1 2
E0 ,Ẽr̃ =r̃nẼθ̃ =r̃nr̃n − Rn1

2


E0 ,

ρ̃ =

r̃n
r̃n − Rn1

 2(n−2)
n

ρ0 ,

(14)

where E0 and ρ0 refer to the original properties of the host. It should be mentioned that the transformed
elasticity moduli for a linear cloak based on relation (14) differ from those achieved by Farhat et al. [6],which are used in most of the literature of flexural cloaking. Farhat et al. calculated the transformed material
properties by applying the linear transformation on a simplified version of the Kirchhoff equation for vibrationof thin plates. The relations in the present work are, on the other hand, calculated by applying the same
coordinate mapping on equation (1). In this regard, the approach in the present work, and accordingly theachieved relations, are in accordance with the study of Brun et al. [2].

3

MODELING AND SIMULATION

The modeling of the cloak in this study is based on the multilayer method, which is a technique to realize thecomplex anisotropic inhomogeneous material matrix of a cloak. In the multilayer technique, continuously varying material properties of a cloak, e.g., the elasticity modulus, are discretized. An example of the continuous
elasticity distribution in a flexural cloak with a polynomial transformation of order n = 2 and the corresponding
discretization is shown in the right panel of Figure 3. The distribution of the elasticity modulus is shown in radial and azimuthal directions with solid and dashed lines, respectively. More exactly, the continuous distribution
in the radial and azimuthal directions is shown in ( ) and ( ), respectively. The corresponding discretizedvalues are depicted with ( ) in the radial (r̃), and ( ) in the azimuthal (θ̃ ) directions. The values shownin the figure correspond to a cloak with R1 = 15 mm and R2 = 70 mm. Therefore, the ratios of the elasticity
moduli are equal to unity for r̃/R1 > 4.67, since this refers to the region outside the cloak. The discretization
takes place by arranging several layers beside each other, each of which consists of a mixture of two or moredifferent materials. The schematic of a multilayered cloak is shown in the left panel of Figure 3.
The use of the multilayer technique for the realization of flexural cloaks was firstly introduced by Farhat etal. [6]. The multilayer method uses the effective medium theory to mix different materials in each layer andreach the required orthotropic properties. Therefore, the multilayer method helps, on the one hand, to approximate the continuously varying properties by discretization, and, on the other hand, to realize inhomogeneous
orthotropic properties through using the effective properties of a mixture of different materials in each layer.
Based on the effective medium theory, if the size of a microstructure is much smaller than the wavelength, themacro-structural behavior of the mixture can be interpreted by the resulting effective properties and without theneed for micro-structural analysis. Normally, a softer material (s) and a harder one (h) are used in each layer ofa cloak. In a circular cloak, which is the main focus of this study, normally the mixed materials are arranged
beside each other as rings, which is also illustrated in the left panel of Figure 3. The corresponding effective
elasticity moduli in the radial and azimuthal directions are calculated by the rule of mixture:!
jj
Eh + γ j Es
1
1
1
γj
j
j
,
Ẽ
=
=
+
,
γ j = dsj /dh ,
(15)
j
j
j
j
j
θ̃
1
+
γ
1
+
γ
Ẽr̃
E
Es
h

where d and j refer to the width of each ring or layer and the layer number, respectively. For the rings in eachlayer, the calculated mass density according to equation (14) is assigned.
In order to design a multilayered cloak, a reference point r̃ j is assumed for each layer. Afterwards, thetransformed elasticity moduli in the radial and azimuthal directions are calculated for each layer based on its
reference r̃ j and using equation (14). At the end, the volume fraction of the constituents in each layer iscalculated according to the rule of mixture in equation (15). The volume fraction of each material can be
adjusted by different methods. One of the methods for a circular cloak is to adjust the width of the rings ineach layer. In this work, however, identical rings are used, which means that γ j = 1 is supposed. Accordingly,
the respective elasticity moduli of the softer and harder rings in each layer are calculated using equation (16),which is a reformulation of equation (15) for γ j = 1:
r 
r 
2
j 2
j j
j
j
j
j
(16)
Ẽ
Ẽ j − Ẽr̃j Ẽ j .
− Ẽr̃ Ẽ ,Es = Ẽ −E = Ẽ +
h

θ̃

θ̃

θ̃

θ̃

θ̃

θ̃

The calculated elasticity moduli and mass densities are then set in the model regardless of the availability ofmaterials with the calculated properties. In other words, the cloak configurations in this study are designed byartificial materials. The use of artificial materials helps to concentrate on the influences of the different appliedtransformations and exclude the limitations of using available and conventional materials in the design. Nevertheless, it is important to investigate the realization of cloaks with innovative solutions and available materials in
future studies. Regarding the above explanations on the multilayer method, the modeling process is elucidated
more exactly in the following lines.
The modeling and simulation in this study is performed using the commercial finite element analysis software Abaqus 2018. The host plate has dimensions of 1000 × 850 × 1 mm and an elasticity modulus of Ehost =0.5 GPa. It is shown schematically in Figure 4. The plate is modeled using thin shell elements S8R5, which
ignores the shear stresses and is in accordance with Kirchhoff’s hypotheses. The rings of the cloak are modeledas circular element sets, to which the respective calculated transformed elasticity modulus and mass density values are assigned. For all materials in all configurations a Poisson’s ratio of 0.45 is assumed. The cloaks havedimensions of R1 = 15 mm and R2 = 75 mm. Each cloak consists of 20 layers.

excitation
surfacewave
propagation

2R2

cloak
(radius R2 )2R2

clamped core(radius R1 )2R2

simplysupported

correlation
region2R2

assessment circle(radius 1.2 R2 )dashpots in rows withincreasing constants

Figure 4. Schematic of the modeled cloak [9].

In order to investigate the cloaking effect, plane waves are propagated in the plate and the scattering of the
waves is observed and evaluated. For this aim, the plate is simply supported in the deflection’s direction on
two edges in accordance with Figure 4. The excitation is generated by a pressure on a small surface on oneedge of the plate as shown in the same figure. The nodes on the excitation surface are tied to a reference nodeusing the multipoint constraint in Abaqus, which “makes the global displacements and rotations as well as allother active degrees of freedom equal” for the tied nodes [1]. As a result, the nodes on the excitation surface
move in phase and propagate plane waves. Though propagation of plane waves facilitates the observation of the
cloaking effect, it is not sufficient. It is necessary to avoid the reflection of the waves by the edges of the plate.
For this, engineering dashpots presented by Abaqus are arranged in rows on the edges of the plate as shown in
the figure. The damping coefficients of the dashpots are increased linearly to avoid the reflection of the waves
due to an impedance jump. The increase of the damping coefficients is illustrated in Figure 4 by using darker
colors for larger coefficients and lighter colors for smaller coefficients. Moreover, a small damping factor of 1%
is assigned to all materials to observe the cloaking effect more clearly.
The model is meshed with a geometrical global size of 0.014 based on a convergence study at 800 Hz. The
total simulation time is equal to 0.03 s, which is adequate for part of the waves to pass the cloak and reach the
opposite edge of the plate where they are dissipated. From the viewpoint of time, the model is discretized witha total number of time steps equal to the frequency. As a result, the model is discretized finer in time domain
as frequency increases.

4

RESULTS AND DISCUSSION

In order to assess and compare the performance of the cloaks with polynomial transformations of different
orders, the out-of-plane velocity of the nodes on a circle around the cloak is collected. When the incident planewaves meet an obstacle, they are scattered in different directions. The obstacle also causes a shadow region justbehind itself. These interactions are more intense in the close neighborhood of the obstacle. For that reason,
the performance of a cloak can be assessed more appropriately by focusing on a close neighborhood of it. In
this regard, a radius of 1.2 R2 is chosen for the assessment circle, as depicted in Figure 4.
Since the simulations are performed in time domain, the velocity of the nodes on the assessment circle fora single time step as well as the average behavior over a specific time interval is investigated. The velocities
of nodes on the assessment circle are evaluated for each cloak configuration with regard to the reference plate,which has no obstacles and allows the plane waves to propagate without disturbance. More exactly, the scatteredvelocity vscatt =| v − vref | is calculated for each node on the assessment circle, where v and vref refer to the node
velocities of the cloak configuration and the reference plate, respectively. For the average behavior over time,
the root mean square (rms) values of v and vref are calculated over the last 50 time steps of the simulations.
In the case of single time step assessment, the values are selected from one single time step to calculate thecorresponding scattered velocity vscatt .
However, the compared cloak configurations possess different material distributions and are therefore regarded as different systems. Hence, it is not reasonable to compare the same time steps of the different systems
together, as the wave propagation is different in them. In order to overcome this challenge, a correlation is used
to select the best representative state of each cloak configuration, which indicates the best performance of the
cloak in guiding the waves. The used correlation is Bravais-Pearson and is calculated on the correlation region
shown in Figure 4. The correlation is computed for the last 15 time steps of each cloak configuration and withreference to the last time step of the reference plate. The time step with the best correlation is selected as therepresentative state of the cloak configuration.
The scattered velocity field on the assessment circle for single time steps is shown in Figure 5 for cloak
configurations with polynomial transformations of different orders n. The directivity of the plots is caused by thescattering of the waves and their interference with the surrounding waves. It is noticed that the transformationsof order n = 0.5 and n = 1 show the poorest performance at all investigated frequencies. The plots show larger
velocity differences for the mentioned configurations than for other configurations, especially in the post-cloak
region, i.e., the left semicircle in the plots. The quadratic, cubic, and quartic cloaks show a very similar
behavior and very small scattered velocity values. However, the quadratic and cubic cloaks imply a slightlybetter performance and smaller scattered velocities than the quartic cloak. It is indicated by a close observationof the plots that the quadratic cloak shows generally the best performance among all, though its response isvery close to the cubic cloak.
n = 0.554200 Hz

12

21

n= 1

29

n = 2

n = 3

vscatt .10−5
dB
3

3

400 Hz

8

14

19

n = 4

vscatt .10−5
dB
2

2

6 9

13

vscatt .10−5
dB

600 HzFigure 5. Comparison of the single time step scattered velocity field for polynomial transformations of different
orders n. The velocities are calculated on the assessment line shown in Figure 4.

The average scattered velocity field of the cloaks over time is shown in Figure 6. The plots show thedifference between rms velocities of the cloak configurations and the reference plate. The rms plots approvegenerally the observations from Figure 5. However, in these plots, the cloak with order n = 0.5 shows by far
the weakest performance at all investigated frequencies, especially in the post-cloak region. The linear cloak
exhibits a weaker performance than the quadratic, cubic, and quartic cloaks at frequencies of 200 and 400 Hz.
At frequency 600 Hz, however, it shows a better performance in the shadow region. The plots in Figure 5 andFigure 6 indicate that the cloak with polynomial order n = 1 performs better at higher frequencies. It should berecalled that the transformed properties for the linear cloak investigated in this study are different from those
introduced by Farhat et al. [6] and used in most of the literature. The study of Farhat et al. [6] is based on asimplified version of the Kirchhoff theory for thin plates. The transformed relations and also the performanceof the corresponding linear cloak in the present study are in agreement to the study of Brun et al. [2].
The quadratic, cubic, and quartic cloaks show a very similar behavior. Nevertheless, the quadratic cloak
performs slightly better than the others, which is especially recognizable at frequency 600 Hz. We believe theweaker performance of the cubic and quartic cloaks is due to their abrupt and dense transformations in the close
neighborhood of the core, which requires extremely fine discretization to be properly realized. In fact, the finediscretization used in this study by rings of 1.5 mm is still not adequate to realize the higher compression of thetransformed space in the vicinity of the core in the cubic and quartic cloaks. As the frequency increases, theratio of the microstructure, namely the rings of the investigated cloaks, to the wavelength increases. As a result,
the reliability of the effective properties decreases. This is one of the main reasons for the weaker performanceof the cloaks at higher frequencies. The situation is aggravated in the cubic and quartic cloaks due to the more
densely compressed transformed space around the core in these cloaks and the necessity for finer discretization.
In addition to the above plots, the correlations of the cloak configurations at their representative time step
are also used as a quantitative measure for comparing the performance of the cloaks. The correlations are notedin Table 1. The correlations are in agreement with the previously discussed plots. The cloaks with ordersn=0.5

5

10

15

n=1

n=2

vscatt .10−5
dB
20
2 4

200 Hz

n=3

6

8 10

n=4

vscatt .10−5
dB

1

400 Hz34

567

vscatt .10−5
dB

600 HzFigure 6. Comparison of the root mean square (rms) scattered velocity field for polynomial transformations ofdifferent orders n. The velocities are calculated on the assessment line shown in Figure 4.

n = 0.5 and n = 1 have noticeably lower correlations than the other cloak configurations. The quadratic, cubic,
and quartic configurations possess similar correlations. Similar to the observations in Figure 6, the quadratic
cloak shows slightly better correlations, especially at frequency 600 Hz.
Table 1. Correlation of cloaks with polynomial transformations of different orders n.

5Frequency (Hz)n = 0.5

n = 1

n = 2

n = 3

n = 4

200

44%

51%

87%

87%

83%

400

48%

58%

89%

87%

82%

600

43%

51%

81%

75%

72%

CONCLUSION

The coordinate transformation is the basis of the design procedure of cloaks and has therefore a great influenceon the performance of the resulting cloak. A deformation of space for designing a cloak can be realized anddescribed by different functions, i.e., the underlying coordinate transformation is not limited to one specificchoice. Despite the importance of the topic, it is not discussed sufficiently in the literature. In most of theliterature, the cloak is designed based on a polynomial function of order n = 1 and only few studies proposedusing a nonlinear transformation [12, 5]. In this study, cloak configurations are designed based on polynomialtransformations of orders 0.5, 1, 2, 3, and 4 and their performance is compared numerically.
The investigations in this study show a remarkably weaker performance for cloaks with polynomials of ordern = 0.5 and n = 1 in comparison to the other investigated cloaks. For the quadratic, cubic, and quartic cloaks,
a very similar behavior is observed, though the quadratic one indicates a slightly better performance, especiallyat higher frequencies. To the opinion of the authors, the weaker performance of the cubic and quartic cloaks ispossibly due to the transformed space being more densely compressed in the close neighborhood of the core inthese cloaks, which demands a very fine discretization and microstructure. We believe based on the results ofthis study that a cloak with higher polynomial order performs better. A further study with finer discretization canassess this claim. However, the realization challenges also increase as the order of the polynomial transformation

increases, as finer microstructure is demanded. A higher order polynomial transformation can be an option
for specific applications, where extreme performance is required. Alternatively, it can be used at very low
frequencies, where a coarser microstructure suffices.
It should also be mentioned that this study is based on different assumptions and simplifications. Additional
studies can be pursued for a deeper knowledge on the topic, e.g., with modified transformations, different assessment criteria, or other theoretical and practical improvements. Concretely, this study disregards the factor J 2
for transformed properties to avoid abrupt impedance change between the cloak and the surroundings. However,a closer focus can be granted in future studies on the interactions at the outer boundaries of the cloak. Anotherassumption for the present study is the use of artificial materials regardless of their availability in the nature.
The use of artificial materials helps to concentrate on the influence of the underlying transformations beyond thelimitations caused by material availability. The realization aspects of cloaks with innovative fabrication solutions
and with use of available materials need to be investigated in further studies.

A

APPENDIX

Since for certain choices of the polynomial order n the transformed differential operator from (6) further simplifies, it may be interesting to take an explicit look at these specific cases. For n = 21 one obtains

 1
8 
 1
1
1
1
2 − 3a2
1 21
3
2
2
2
2r̃
r̃
2
r̃ − 4 a ∂ w
R −R
2
∂ 3w
r̃2
∂ 4w
 r̃ + 8 a ∂ w∇2 ∇2 w =  2 1 1   
−
+ 
+




7
4
6
5
2
3
1
1
1
1
1
1
1
1
∂ r̃
∂ r̃
∂ r̃
∂ r̃4
R22
r̃ 2 − a 2r̃ 2 − a 2r̃ 2 − a 2r̃ 2 − a 2

 1

1
2
2
2 r̃ + a∂ 3w∂ 2w∂ 4w 
2r̃
∂ 4w
4
1
−
+
+
+
 . (17)




7
8
8
6
2
2
2
2
4
1
1
1
1
1
1
1
1
∂ θ̃ ∂ r̃∂ θ̃ ∂ r̃
r̃ 2 − a 2r̃ 2 − a 2 ∂ θ̃r̃ 2 − a 2 ∂ θ̃r̃ 2 − a 2
As for n = 1, one has 
R2 − R1 4
2 ∂ 3w ∂ 4w
1
∂w
∂ 2w
1
2 2
∇ ∇ w=++ 4−
R2(r̃ − R1 )3 ∂ r̃ (r̃ − R1 )2 ∂ r̃2 r̃ − R1 ∂ r̃3
∂ r̃

4
1
2
∂ 3w
2
∂ 4w
∂ 2w
∂ 4w
+
+
.
−
+
(r̃ − R1 )3 ∂ θ̃ 2 ∂ r̃ (r̃ − R1 )2 ∂ θ̃ 2 ∂ r̃2 (r̃ − R1 )4 ∂ θ̃ 2 (r̃ − R1 )4 ∂ θ̃ 4

(18)

For n = 2, the transformed differential operator assumes the following form: 2
2 " 4(r̃2 − R21 ) r̃2 + 3R21 ∂ 3 w
r̃ + 6R21 r̃2 − 15R61 ∂ w r̃4 + 6R21 r̃2 − 15R41 ∂ 2 w
R2 − R21
2 2∇ ∇ w=
−
+2
r̃7
∂ r̃
r̃6
∂ r̃2
r̃5
∂ r̃3
R22
+


(r̃2 − R21 )2 ∂ 4 w 2 ∂ 3 w
2 ∂ 4w
4
∂ 2w
1
∂ 4w
−
+
+
+
. (19)r̃4∂ r̃4 r̃3 ∂ θ̃ 2 ∂ r̃ r̃2 ∂ θ̃ 2 ∂ r̃2 (r̃2 − R21 )2 ∂ θ̃ 2 (r̃2 − R21 )2 ∂ θ̃ 4

For n = 3, one obtains
22∇ ∇ w=

R32 − R31
R32

 2
 43 " 92 r̃ + 22R31 r̃6 − 100R61 r̃3 + 80R91 ∂ w (r̃3 − R31 ) 3 6
3 3
6 ∂ w
−
r̃
+
20R
r̃
−
48R
1
1
1
∂ r̃
r̃10
∂ r̃2
(r̃3 − R31 ) 3 r̃11
5
82(r̃3 − R31 ) 3 r̃3 + 6R31 ∂ 3 w (r̃3 − R31 ) 3 ∂ 4 w 2 r̃3 − 2R31 ∂ 3 w
+
+
−
r̃9
∂ r̃3
r̃8
∂ r̃4 (r̃3 − R3 ) 13 r̃5 ∂ θ̃ 2 ∂ r̃
1
#
2
3
3
4
2
4
3
(r̃ − R1 )
∂ w
4
∂ w
1
∂ w
.
+2
+
+
r̃4
∂ θ̃ 2 ∂ r̃2 (r̃3 − R3 ) 43 ∂ θ̃ 2 (r̃3 − R3 ) 43 ∂ θ̃ 4
1

1

(20)

Finally, the case n = 4 gives" 44  r̃ 12 + 57R4 r̃ 8 − 273R8 r̃ 4 + 231R12 ∂ w(r̃4 − R41 ) −r̃8 − 42R41 r̃4 + 111R81 ∂ 2 w
R
−
R
2 2
1
1
1
2
1
+
∇ ∇ w=
r̃15
∂ r̃
r̃14
∂ r̃2
R42


2(r̃4 − R41 )2 r̃4 + 9R41 ∂ 3 w (r̃4 − R41 )3 ∂ 4 w 2 r̃4 − 3R41 ∂ 3 w
+
+
−
r̃13
∂ r̃3
r̃12
∂ r̃4
r̃5
∂ θ̃ 2 ∂ r̃

4
4
4
2
4
2(r̃ − R1 ) ∂ w
∂ w
∂ w
4
1
+
+
+
.
(21)
r̃6
∂ θ̃ 2 ∂ r̃2 (r̃4 − R41 ) ∂ θ̃ 2 (r̃4 − R41 ) ∂ θ̃ 4

REFERENCES[1] General multi-point constraints, 2016-09-21T00:05:58.000Z.
SIMACAECSTRefMap/simacst-c-mpc.htm.

https://abaqus-docs.mit.edu/2017/English/[2] M. Brun, D. J. Colquitt, I. S. Jones, A. B. Movchan, and N. V. Movchan. Transformation cloaking and radial approximations for flexural waves in elastic plates. Journal of Physics D: Applied Physics, 16(9):093020,
2014.
[3] R. V. Craster, editor. Acoustic metamaterials: Negative refraction, imaging, lensing and cloaking, volume
166 of Springer Series in Materials Science. Springer, Dordrecht, 2013.
[4] S. A. Cummer and D. Schurig. One path to acoustic cloaking. Mathematical Research Letters, 9(3):45,
2007.
[5] A. Darabi, A. Zareei, M.-R. Alam, and M. J. Leamy. Experimental demonstration of an ultrabroadband
nonlinear cloak for flexural waves. Physical Review Letters, 121(17):174301, 2018.
[6] M. Farhat, S. Guenneau, and S. Enoch. Ultrabroadband elastic cloaking in thin plates. Physical Review
Letters, 103(2):024301, 2009.
[7] G. W. Milton, M. Briane, and J. R. Willis. On cloaking for elasticity and physical equations with atransformation invariant form. New Journal of Physics, 8(10):248, 2006.
[8] A. Nojavan and S. Langer. Vibration isolation in plates using flexural cloaking - in press. In Inter-noise.
Glasgow, Scotland, 2022.
[9] A. Nojavan and S. C. Langer. Frequency dependency of multilayered flexural cloaks. In Fortschritte der
Akustik - DAGA 2022, pages 236–239. Stuttgart, 2022.
[10] J. B. Pendry, D. Schurig, and D. R. Smith. Controlling electromagnetic fields. Science, 312(5781):1780–
1782, 2006.
[11] N. Stenger, M. Wilhelm, and M. Wegener. Experiments on elastic cloaking in thin plates. Physical Review
Letters, 108(1):014301, 2012.
[12] A. Zareei and M.-R. Alam. Broadband cloaking of flexural waves. Physical review. E, 95(6-1):063002,
2017.

ABS-0597

Design rules of negative stiffness absorber and experimental verification fordifferent buckling conditions
Pascal FOSSAT(1) , Mohamed ICHCHOU(1) , Olivier BAREILLE(1)(1) Vibroacoustics and Complex Media Research Group VIAME, LTDS - CNRS UMR 5513, Centre Lyonnais d’Acoustique

CeLyA, École Centrale de Lyon, Université de Lyon, France, pascal.fossat@ec-lyon.fr

ABSTRACTThis work deals with the identification of optimal working conditions of a negative stiffness absorber consisting of a two-spring assembly. Design rules are derived from the energy stored and the force generated by eachspring on the center mass, and location of stable equilibrium points. Optimal sets of mechanical and geometricalparameters will be implemented on a finite element model to predict either static or dynamic behavior. Experiments are performed on a mock-up in which spring like components are 3D printed. Buckling condition is
introduced on the springs to evaluate its influence on the negative stiffness effect. The efficiency of the negative
stiffness resonator on vibration isolation is evaluated for different conditions of buckling force.

Keywords: Vibration isolation, negative stiffness, high static low dynamic isolator

1

INTRODUCTION

It is known from studies on dynamic vibration absorbers that stiffness determines the range of efficiency of an
isolator. However, one limitation of the classical linear description is that stiffness should be reduced as a larger
frequency range of performance is expected. This would result in a large static displacement that would be unsatisfactory for some applications. To overcome this limitation, negative stiffness was introduced to reach low
frequency ranges of efficiency. Negative stiffness springs were introduced in the form of curved beam by [10]and it was shown that such nonlinear springs provide low frequency performance and while they support thestatic load of the equipment to be isolated. The mechanism has been further developed as industrial vibration
isolator for sensitive equipment such as microscopes [8], or installed as seat suspension in vehicles [5], however
the specific nonlinear behavior relying on the buckling instability motivated numerous fundamentals studies. One
may cite the pioneering papers of Lakes [4, 3], in which negative stiffness was introduced to bring additional
damping in composites. Similar behavior of pre-stressed negative stiffness springs was previously investigatedin [8]. The negative stiffness assembly is usually presented a two horizontal springs in series, whose connecting
point is also attached to a vertical spring. In this way, the two horizontal springs provide the negative stiffness
while the vertical spring brings the positive stiffness. This assembly is largely documented to identify its highstatic low dynamic features, as well as the conditions for which the negative stiffness compensates or overcome the positive one, leading to specific nonlinear behaviors [7] followed by [2], and [13]. Additional ways
to produce quasi-zero stiffness are also presented in the comprehensive book of [6]. As examples of recent
developments, the negative stiffness component can be inserted as magnets, as in [1], and that such devices arealso suitable shock performance in [9].
The objective of this work is to present some design rules and equations associated to the description of anegative stiffness resonator prototype. The design under study is largely inspired from [10] and [12]. These arecomplemented by preliminary finite element comparisons performed on the different components. Experimental
measurements are finally conducted on the mock-up and demonstrates the tunable aspect of the resonator withrespect to different buckling loads.

Slider

Support springs

Center mass

Curved beams

Frame

Figure 1. Picture of experimental setup with components of the resonator, aluminium rigid elements, and 3D
printed soft components, and instrumentation ; accelerometer on mass, and impact hammer

2

THEORETICAL INSIGHT

This section gives an overview of preliminary theoretical calculations that can be made to assess the static
behavior and dynamic performance of the resonator depicted in Fig. 1. It consists of an aluminum frame
assumed to be rigid inside which there are several moving parts. The center mass is made of aluminium and is
supported on its top and bottom by four folded beams that provide positive stiffness in vertical direction. The
center mass is also attached on its left and right sides to two pairs of curved beam that provide the negative
stiffness. As for the four folded beams, the two pairs of curved beam are 3D printed. Two aluminium sliders
are designed to move horizontally in opposite directions to induce local buckling of the curved beams.
2.1 Negative stiffness from buckled Euler beams
drawn in Fig. 2 The first element considered is the beam with initial imperfection w = Q0 , length L, depth band thickness h. As for the case of a straight beam, the beam will not deform until the applied load reaches thecritical buckling load Pe , but in case of initial imperfection, a corrected expression can be found between theend shortening δ and the axial force P, as derived in [11]. Assuming w < 0.2L, the force displacement relation
is :


2
 !−1/2

2
 !

πQ
δ
πQ
1
δ
πQ
1
0
0
0
 1+
P = Pe 1 −
+4
+
(1)
L
L
L
8
L
2 L
with Pe = nEI(π/L)2 the Euler critical load of the beam with n = 4 in case it has both ends clamped, EI
is the bending stiffness, L is initial the length of the beam. Geometric parameters and mechanical properties ofcurved beam are indicated in Table 1.
Table 1. Geometrical parameters and mechanical properties of curved beams
Dimensions [m]Young’s Modulus [GPa]Density [kg/m3 ]L = 0.14, Q0 = 0.004, b = 0.06, h = 0.004

2.5

1008

Curved beam:

w

P
δ

L
Initial state:m

Buckled state:

δ

Fn

δ

Figure 2. Compressed Euler beam clamped at both ends, physical configuration of the negative stiffness resonator at rest and working state with initial displacement
Two pairs of curved beams are then connected on both sides of the center mass, as illustrated in Fig. 2, andan initial axial displacement δ is applied in opposite direction. Increasing the initial displacement makes thebeams to buckle and provides a restoring force F. Assuming δ < 0.1L as recommended by [11], and assuming
the two pairs of curved beams and mass are aligned, the vertical reaction force F is formulated as :

 x
−1/2
F(x) = −4Pe 1 − sa1
(2)
a2
γ
p
with s = πQ0 /L, γ = L − δ )2 + x2 , a1 = s2 + 4 − 4γ/L, and a2 = 3/2 + s2 /8 − γ/2L. Differentiating (2) versus
the displacement gives the effective stiffness provided by the element as  γ 2 − x2 
a3 
−3/2
−1/2
−1/2
k(x) = −2s a3 a2 a1−
1 − s a1(3)
+ a2 1 − b a12
γ3with a3 = x2 /(Lγ 2 ). The resulting expression of effective stiffness (3) exhibits the unconventional nature of theresulting force with respect to initial axial displacement δ . This aspect is closely reported in [10, 2].

3

Numerical modeling of the negative stiffness component

Numerical simulation were conducted at different scale to emphasize the relevancy of theoretical approach introduced above. These deal with the computation of the restoring force with respect to initial deflection 3.1,
3.1 Force-displacement relations
The force-displacement relation is obtained numerically using commercial finite element software. The reaction
force is plotted in Fig. 3 as a function of the vertical imposed displacement for 2 mm and 4 mm. Twocomments can be formulated, First, for same values of displacements, the tension restored by the thick beam islarger than that generated by the thin one. Second is the flattening of the curve and especially the widening of
the area where the stiffness is negative, which appears as a fundamental feature for the design of the resonator.
In the next section, buckling aspect is implemented in order to demonstrate the softening of the structure ascompressive load is introduced.
3.2 Eigenfrequency versus initial buckling force
The previous study is extended to the assembly of a pair of curved beams, as represented in Fig. 2. Fixedboundary conditions are specified on the outer boundary of the structure while compressive load is applied
towards the mass to create the buckling, and the first eigenfrequency is successively computed and reported in

300

40

200

Restoring force [N]Restoring force [N]

60

20
0
-20
-40

100
0
-100
-200

-60
-5

0

-300

5

Displacement at center [m]

(a)

-5

10-3

0

Displacement at center [m]

5
10-3

(b)

Figure 3. Vertical reaction force versus vertical displacement at the middle of the assembly, for (a) 2mm and(b) 4mm thick curved beams, and increasing values of initial displacement δ [0;0.1L] from light to dark blue
Fig. 4. It can be seen that as the load increases, the fundamental frequency tends to zero, as the system is madegradually softer. As a limit behavior, this curve crosses the load axis at the critical buckling load obtained froma preliminary buckling analysis.
In the next section, mass is introduced in between the two pairs of curved beam, which results in the ashifting of eigenfrequencies to lower frequencies. Support beams are also modeled, as they prevent the curved
beam to naturally bend when the middle mass is introduced.
3.3 Eigenfrequency study
From the geometrical parameters and mechanical properties indicated above, a preliminary eigenfrequency computation is performed on a three dimensional finite element model of the assembly. Fixed boundary conditions
are specified on left, right, top, and bottom boundary of the assembly to mimic clamped conditions with theframe. This results in the modal shapes illustrated in Fig. 5. The modes of interest are the ones having in-planekinematics, and particularly the first one associated with symmetrical bending of curved beams on each sideof the mass, resulting in its vertical motion. For this specific kinematic to occur, it is outlined that negative
stiffness introduced by curved beam should overcome positive stiffness produced by the four supports. The
second mode mode presenting rotation of the mass is not of interest in this contribution, although it is in some
applications related to structure stabilization, an damping in joints of reticulated structures.

4

EXPERIMENTAL VALIDATION

4.1 Measurement setup
This is the mockup of the system shown in Fig. 1. The components governing the behavior are i) the four
folded beams that provide positive stiffness in vertical direction, ii) the center mass is also attached on its left
and right sides to iii) two pairs of curved beam that provide the negative stiffness. As for the four folded
beams, the two pairs of curved beam are 3D printed. Two aluminium sliders are designed to move horizontallyin opposite directions to induce local buckling of the curved beams. The position of the sliders is adjusted
using two screws and determines the initial force experienced by the curved beams. In the following, the same
movement will be applied to the sliders so as not to introduce any asymmetry into the system.
The aluminium frame with sliders and mass have mass density ρ = 2700 kg/m3 and Young’s modulus E = 69
GPa. After weighing of the 3D printed springs components and modal testing, the mass density of ABS parts

First eigenfrequency [Hz]

100

80

60

40

20

0
0

500

1000

1500

2000

Force [N]Figure 4. First eigenfrequency versus compressive load applied towards the mass, obtained with two pairs of 4mm thick curved beams.

(a)

(b)

(c)

(d)

Figure 5. Modal shapes associated with the first four eigenfrequencies, (a) symmetrical bending mode withvertical motion of mass at 44 Hz, (b) asymmetrical bending mode with rotational motion of mass 217 Hz , (c)symmetrical bending mode with inverted bending of curved beams 238 Hz, (d) symmetrical bending mode withparallel bending of curved beams 300 Hz

FRF Acc/Force [dB]

10
0
-10
-20
-30

COH

1

20

40

60

80

100

20

40

60

80

100

0.5
0

Frequency [Hz]Figure 6. Structural response (inertance) from experimental setup delta = 1.5mm (−), delta = 3.5mm (−), anddelta = 5mm (−), and frequency shift evolving respectively to 30 Hz, 26 Hz, 20Hz
is calculated as ρs = 1008 kg/m3 , and the Young’s modulus is estimated as Es = 2.5 GPa.
4.2 Response of the resonator
The influence of initial force on the response is also verified from experimental measurements. These wereobtained by exciting the structure using an shock hammer, and measuring the resulting acceleration. Both theexcitation and sensor are located on the mass, where the transverse motion is the quantity of interest. The
structural mobility is shown in Fig. 6 for different values of initial force. It shows the frequency shift due
to the softening effect as a consequence of increasing initial load, from 30 Hz down to 20 Hz, as well as
the remarkable gradual decrease in amplitudes. The trend associated with decrease of frequency as force isapplied is confirmed experimentally, at least for values of initial displacement δ from 0 to 5 mm. For large
values of force, the structural integrity of the 3D printed curved beam could also be discussed. From thispreliminary mock-up, it appears that setting the overall stiffness to zero and overcome the positive stiffness
at equilibrium point is challenging because of the uncertainties when manufacturing the frame and junctions
between the components.

5

CONCLUSIONSThis paper theoretical formulations describing the relation between restoring force with respect to initial deflection, as well as simplified model of flexural behavior are introduced. In this way, the key parameters arecorrelated to negative stiffness mechanism as well as overall bending kinematics governing the motion of theresonator. Numerical calculations confirmed trends identified analytically and static studies together with dynamic analysis confirmed the high static low dynamic behavior of the system. According to the design procedure
presented, a variable passive negative stiffness resonator is manufactured and tested. Although the correlation
between numerical and experimental data is very sensitive to manufacturing conditions, the main trends are
identified and the weakening of the system is evidenced experimentally.

ACKNOWLEDGEMENTS
This work was supported by the LabEx CeLyA (Centre Lyonnais d’Acoustique, ANR-10-LABX-0060) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French
National Research Agency (ANR).

REFERENCES[1] A. Carrella, M. Brennan, T. Waters, and K. Shin. On the design of a high-static–low-dynamic stiffness
isolator using linear mechanical springs and magnets. Journal of Sound and Vibration, 315(3):712–720,
2008. EUROMECH colloquium 483, Geometrically non-linear vibrations of structures.
[2] X. Huang, X. Liu, J. Sun, Z. Zhang, and H. Hua. Vibration isolation characteristics of a nonlinear isolator
using euler buckled beam as negative stiffness corrector: A theoretical and experimental study. Journal ofSound and Vibration, 333(4):1132–1148, 2014.
[3] R. Lakes. Extreme damping in compliant composites with a negative-stiffness phase. Philosophical Magazine Letters, 81(2):95–100, 2001.
[4] R. S. Lakes, T. Lee, A. Bersie, and Y.-C. Wang. Extreme damping in composite materials with negativestiffness inclusions. Nature, 410(6828):565–567, 2001.
[5] C.-M. Lee, V. Goverdovskiy, and A. Temnikov. Design of springs with “negative” stiffness to improvevehicle driver vibration isolation. Journal of Sound and Vibration, 302(4):865–874, 2007.
[6] C.-M. Lee and V. N. Goverdovskiy. Vibration Protection Systems: Negative and Quasi-zero Stiffness. Cambridge University Press, 2021.
[7] X. Liu, X. Huang, and H. Hua. On the characteristics of a quasi-zero stiffness isolator using euler buckled
beam as negative stiffness corrector. Journal of Sound and Vibration, 332(14):3359–3376, 2013.
[8] D. L. Platus. Negative-stiffness-mechanism vibration isolation systems. In Vibration control in microelectronics, optics, and metrology, volume 1619, pages 44–54. SPIE, 1992.
[9] B. Tang and M. Brennan. On the shock performance of a nonlinear vibration isolator with high-static-lowdynamic-stiffness. International Journal of Mechanical Sciences, 81:207–214, 2014.
[10] S. A. Tobias. Design of small isolator units for the suppression of low-frequency vibration. Journal of
Mechanical Engineering Science, 1(3):280–292, 1959.
[11] L. Virgin and R. Davis. Vibration isolation using buckled struts. Journal of Sound Vibration, 260:965–973,
02 2003.
[12] Q. Wu, C. Droz, P. Fossat, M. Ichchou, and S. Xie. Negative stiffness mechanisms for the broadeningof low frequency bandgaps performance of euler-bernoulli resonators. In Proc. of the 29th International
Conference on Noise and Vibration engineering, 2020.
[13] Q. Wu, G. Huang, C. Liu, S. Xie, and M. Xu. Low-frequency multi-mode vibration suppressionof a metastructure beam with two-stage high-static-low-dynamic stiffness oscillators. Acta Mechanica,
230(12):4341–4356, 2019.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0599

Transformable Metamaterial with Programmable Control of Elastic
Wave Polarization and Refraction
Zhou Hu1; Zhibo Wei2; Kun Wang1; Yan Chen2,3*; Guoliang Huang4; Gengkai Hu1; Rui Zhu1*
1

School of Aerospace Engineering, Beijing Institute of Technology, China
2

3

School of Mechanical Engineering, Tianjin University, China

Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University,
China4

Department of Mechanical and Aerospace Engineering, University of Missouri, USA

Mechanical metamaterials [1,2] are artificial materials with special microstructures, which exhibit
unconventional mechanical properties, such as negative Poisson ’s ratio, negative mass density,
negative modulus, etc., and also bring various and novel wave phenomena, such as negative refraction,
wave cloaking etc. Recently, rational design with flexible microstructures has become an emerging
trend shaping the future of mechanical metamaterials with multiple exotic functionalities [3-6]. However,most flexible metamaterial designs are only engaged in a quantitative enhancement of certain property.
A qualitative transformation of the metamaterials’ properties or/and functionalities could be extremely
important for the prospects of new generation metamaterials, yet tremendously challenging indesigning.
In this presentation, a class of unprecedented 2D extremal metamaterials that can transform froma null-mode (solid state) to a tri-mode (near-gaseous state) will be mainly focused. First, we will
present a modular-origami-based kinematic design methodology for the transformable metamaterial,
where a four-bar mechanism with weakly connections is construted, as shown in Fig. 1.

Fig.1. Design concept for the unit cell of the transformable metamaterial.
Then, considering the four-bar mechanism as the basic building block, four different configurations
that can be transformed among each other can be constructed by periodically arranging the same unit
cell with different configurations (Fig. 2). To quantitatively investigate the mechanical behaviors of
the four configurations, we calculate their effective properties including effective densities andeffective elastic tensors based on the homogenization theory. Mechanically speaking, if one any
eigenvalue of the elastic tensor disappears or much smaller than the others, the corresponding
deformation mode will cost little to no elastic energy and this deformation mode is a zero mode [7].
Based on the number of zero modes, a class of 2D extremal metamaterials is determined, which covers
nullmode, unimode, bimode, and trimode metamaterials related to the four metamaterials, respectively.

12

yan_chen@tju.edu.cn
ruizhu@bit.edu.cn

Fig. 2. Four transformable metamaterials designed based on modular-origami
Finally, wave properties of the transformable metamaterials are systematically analyzed, and theirunique wave control capabilities, such as programmable elastic wave polarization filtering andmanipulation of positive and negative refractions, will be numerically demonstrated. The 2D
transformable mechanical metamaterial proposed here can be easily extended to 3D systems, andopens up vistas for 3D architected materials with transformable material states and functions.

REFERENCES
1. Zadpoor AA. Mechanical meta-materials. Mater. Horiz. 2016; 3, 371-381.
2. Kadic M, Milton GW, van Hecke M, Wegener M. 3D metamaterials. Nat. Rev. Phys. 2019; 1, 198-210.
3. Bertoldi K, Vitelli V, Christensen J, van Hecke M. Flexible mechanical metamaterials. Nat. Rev. Mater.
2017; 2, 17066.
4. Coulais C, Sabbadini A, Vink F, van Hecke M. Multi-step self-guided pathways for shape-changing
metamaterials. Nature 2018; 561, 512-515.
5. Li Y, Zhang Q, Hong Y, Yin J. 3D Transformable Modular Kirigami Based Programmable Metamaterials.
Adv. Funct. Mater. 2021; 31, 2105641.
6. Zhang MK, Yang JK, Zhu R. Origami-Based Bistable Metastructures for Low-Frequency Vibration
Control. J. Appl. Mech. 2021; 88, 051009.
7. Milton GW, Cherkaev AV. Which Elasticity Tensors are Realizable? J. Eng. Mater. Technol. 1995; 117,483-493

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstract

ABS–0643

Phononic crystals, elastic metamaterials, and bio-inspiration
MINIACI MARCO1
1

CNRS, Centrale Lille, ISEN, Univ. Lille, Univ. Valenciennes, UMR 8520 - IEMN, F-59000 Lille, France

ABSTRACT
Nature has always represented a fundamental source of inspiration to solve mankind’s scientific challenges
and engineering tasks. For instance, it has been shown that a hierarchical organization over multiple lengthscales allows enhanced quasi-static mechanical properties, while the relative orientation of adjacent chiral
centers may affect the physical properties of a polymer, and internal heterogeneous architectures may resultin shape changing systems, to cite a few examples.
In this paper, we discuss some examples of how bio-inspiration may be used to enhance the potential ofphononic crystals and acoustic metamaterials in terms of vibration attenuation and control of wave
propagation.
Keywords: Phononic Crystals, Metamaterials, Sound Insulation, Bio-inspiration, Chirality.

1. INTRODUCTION
Phononic crystals and acoustic metamaterials can be considered as composite materials exhibiting
ad-hoc designed architectures made of periodic, quasi-periodic or even randomly disposed building
blocks (unit cells) characterized by extraordinary dynamic properties, such as frequency-dependent
directionality and band gap (BG) behavior [1]. Since their introduction a few decades ago, researchers
have tried to explore more and more configurations capable of (i) achieving low and broadband
frequency effects (without recurring to unpractical increases of the unit cell size or decrease s of the
global structural stiffness) and (ii) practical solutions to exhibit a tunable frequency response once thestructures are fabricated.
In this context, Nature has always represented a formidable source of inspiration to solve
mankind’s scientific challenges and engineering tasks. For instance, it has been shown that ahierarchical organization over multiple length scales allows enhanced quasi -static mechanicalproperties, while the relative orientation of adjacent chiral centers strongly affects the physicalproperties of a polymer, and internal heterogeneous architectures may result in shape changing
systems, to cite a few examples [2].
In this research, we discuss how bio-inspiration may be used to enhance the potential of phononic
crystals and acoustic metamaterials [3]. Specifically, a comparison of the dynamic behavior of
conventional and bio-inspired phononic crystals / metamaterials will is presented through theevaluation of the corresponding dispersion diagrams and / or transmission properties.

2. METHODS
From the modeling point of view, we consider small-amplitude waves. In the case of a linear elastic
medium, in the absence of external forces, the wave field is described by standard wave equations for
two-dimensional displacements, and the analysis is restricted to harmonic waves. Due to the structural
periodicity of the phononic crystals / metamaterials, the displacements are also periodic in space andcan be thus represented using the Floquet-Bloch expansion theorem. This allows to restrict theanalysis of an infinite periodic medium to that of a representative unit cell with Bloch periodic
conditions at the unit cell boundaries.

1

Marco.miniaci@gmail.com, marco.miniaci@univ-lille.fr.

3. CASE STUDIES
3.1 Structural hierarchy
Structural hierarchy is here understood in the sense that a representative (metamaterial) unit cellcomprises multiple arrangements of inhomogeneities at various size scales. If the same arrangement
occurs at every scale, the pattern is called self-similar. We consider single-phase metamaterials formedby self-similar unit cells with different hierarchical levels, as schematically shown in Fig. 1A.
The dispersion diagrams for the hierarchical cross-like metamaterial with regular and externalhierarchical organization is shown in Figures 1A. It clearly appears that the wide band gap ranging
from approximately 42 kHz to 96 kHz is almost preserved. Pass bands located inside the latter band
gap are represented by mostly flat lines corresponding to localized modes. These modes do not couple
with the external field in realistic structures, which always have a certain level of energy dissipation.
Also, the introduction of hierarchy allows to shift the band gap corresponding to the ordinary unit cellto an approximately 3 times lower frequency. Several other band gaps appear at the band gap
frequencies of the regular structure [4]. The BG shift can be seen as an alteration of the ratio stiffness
over mass (decreasing the first more than the second), where the unit cell assembly strategy has been
driven by a simple translation of the unit cell.

Figure 1: (A) Hierarchical structures found in natural diatom cell walls and the corresponding bio -inspired hierarchicalmetamaterial unit cells, along with their dispersion diagrams. The hierarchical organization allows to preserve the BG originatingfrom a smaller constitutive geometry and opening frequency band gaps at lower frequencies [4]. (B) Schematic representation of
the concept of tacticity inspiring the design of two periodic diadic arrays of translational-rotational resonators in isotactic andsyndiotactic configuration, along with their band diagrams and mode shapes. A large BG between 374 and 1816 Hz in the
syndiotactic arrangement is visible [5].

3.2 Tacticity
All the classical amino acids present in nature are chiral (except glycine). A molecule is calledchiral if it cannot be superimposed on its mirror image by any combination of rotations andtranslations. While the concept of chirality itself has been largely exploited in the field of phononic
crystals and acoustic metamaterials in recent years, the effects of the concatenation of such non centrosymmetric unit cells (tacticity from polymer science), has not been explored, yet. Inspire d bythis concept, we show its effects on the dynamic behavior of chiral structured periodic media, bycontrasting the dispersion properties of iso- and a syndiotactic crystal.
Our numerical results show that the syndiotactic phononic crystal arrangement n ucleates lowfrequency full bandgaps, while the isotactic variant exhibits wave modes in the same frequency range,
responsible for a transmissive behavior (see Fig. 1B). Nonetheless, the two variants are characterized
by the same density and quasistatic stiffness [5].

4. CONCLUSIONSIn conclusion, we have shown that as the dynamic properties of phononic crystals / metamterials
can be enhanced by taking inspiration from biology-related concepts. Specifically, we have provedthat a hierarchical organization of self-similar unit cells may lead to multiple BG opening at different
frequency scales, as well as to the lowering of the existing ones (whose frequency width is partiallypreserved). Similarly, concatenation of noncentrosymmetric unit cells leads to a BG frequency
lowering, keeping the overall mass and stiffness unchanged.

ACKNOWLEDGEMENTS
This project has received funding from the European Union Horizon 2020 research and innovation
programme under grant agreement No. 863179.

REFERENCES
1. Hussein MI, Leamy MJ, Ruzzene M. Dynamics of Phononic Materials and Structures: Historical Origins,
Recent Progress, and Future Outlook. Applied Mechanics Reviews. 2014; 66(4):040802.
2. Meyers M, Chen P, Lin A. Seki Y. Biological materials: Structure and mechanical properties. Progress inMaterials Science. 2008; 53:1-206.
3. Miniaci M, Krushynska A, Movchan AB, Bosia F, Pugno NM. Spider web-inspired acoustic metamaterials.
Applied Physics Letters. 2016; 109:071905.
4. Miniaci M, Krushynska A, Gliozzi AS, Kherraz N, Bosia F, Pugno NM. Design and Fabrication of
Bioinspired Hierarchical Dissipative Elastic Metamaterials. Physical Review Applied. 2018; 10:024012.
5. Bergamini A, Miniaci M, Delpero T, Tallarico D, Van Damme B, Hannema G, Leibacher I, Zemp A.
Tacticity in chiral phononic crystals. Nature Communications. 2019; 10:4525.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0916

Collimation and redirection of flexural waves in thin elastic plates
José SÁNCHEZ-DEHESA 1; Penglin GAO 2, Francisco CERVERA1; Alberto BROATCH 3, JorgeGARCÍA-TÍSCAR3; Andrés FELGUEROSO3
1

Wave Phenomena Group, Department of Electronic Engineering, Universitat Politècnica de València, ES-46022
Valencia, Spain.
2

State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, 200240 Shanghai,
China
3

CMT-Motores Térmicos, Universitat Politècnica de València, ES-46022 Valencia, Spain.

ABSTRACTWe report the feasibility of several structures specifically designed for the control of flexural waves
propagating in thin perforated plates. First, a collimator of flexural waves is designed to generate a collimatedbeam from an omnidirectional source of flexural waves. The practical demonstration is performed here using
an aluminum thin plate containing 50 x 30 free holes. In addition, we have designed a redirector device thatsplits the collimated beam into two beams propagating sideways. The amount of the redirected energy isimproved by adding a structure consisting of an array of two-beam resonators, which is designed as a barrier
to reflect the transmitted energy through the redirector. Good agreement is found between the experimental
data and the theoretical predictions.
Keywords: Sound collimation, sound redirection, flexural waves, thin plates

1. INTRODUCTION

Arrays of holes and/or lattices of resonant structures are particularly useful to design devices controlling theflow of waves along the plate. Thus, the collimation and redirection of flexural waves [1] were inspired bysimilar devices introduced in acoustic for the control of acoustic waves using lattices of sound scatterers.
Acoustic collimation was achieved by phononic crystals slabs [2-3] and by stacking holey Plexiglas plates[4]. On the other hand, the redirection of acoustic waves in the air was demonstrated by using lattices of
perforated cylindrical shells [5-6]. For the case of water background, the redirection has been predicted byusing lattices of elastic shells [7].
In what follows, we describe both the collimator and the redirector of flexural waves that havebeen designed and experimentally tested. Firstly, in Section 2, we present the numerical simulations
predicting the expected behavior. Later, Sect. 3 reports the experimental setup employed tocharacterize the structures. In Section 4, we show that measurements support the designed structures,
and, finally, Sect. 5 summarizes the results of this work.

2. THEORY

The structures presented here have been obtained using the biharmonic equation, which describes
well the interaction of flexural waves with lattices of holes and inclusions [8]. The scattering
properties of the waves interacting with the structures under study have been obtained in theframework of the multiple scattering formalism reported in Refs. [9] and [10]. We point out that theaccuracy of this simplified approach has been validated by its comparison with three-dimensional(3D) numerical simulations based on finite elements using a commercial package (COMSOL
jsdehesa@upv.es
jplhit@gmail.com
3
abroatch@mot.upv.es
12

multiphysics) where the 3D full elastic equations were taken into account.
The structures are designed by considering that the arrays of holes and resonators are drilled in a
homogeneous aluminum plate with the following parameters: Young’s modulus E= 70 GPa, Poisson’s
ratio ν= 0.33, mass density ρ =2.7×10 3 kg/m3, and plate thickness h=1 mm.
2.1

Collimation by an array of free holes

For designing the collimator, we calculate the band structure of a square lattice of free holes drilledin the plate. After fixing 15 kHz as the operational frequency of our collimator, we have optimized
the lattice period (ac ) and the hole radius (rc ) to obtain a beam along the direction of interest. Figure
1 shows the performance of the designed collimator, which consists of 20×30 free holes with radius
Rc=5 mm and lattice constant a c= 12 mm. The principal beam, which is defined by the blue arrow, isobtained thanks to the interaction of an omnidirectional source of flexural waves located at the middleof the first right-hand side layer. Notice that two additional beams (red arrows) also appear on theleft-hand side of the cluster. These unavoidable sidelobes are also generated by the properties of theisofrequency contour resulting from the band structure. In order to avoid interferences with the main
beam, in the experiment, they are suppressed by adding two absorbing stripes.

Figure 1 – Snapshot of the amplitude of the out-of-plane displacement created by a punctual omnidirectional source located at the position y=0 in the first column in the cluster free holes (white circles).
2.2

Redirection of flexural waves

The redirection of flexural waves was predicted in Ref. [11]. This work follows a similar approach
but uses as impinging beam the one generated by the collimator shown in Fig. 1. The parameters of
the new design are obtained by an optimization procedure in which the features of the impinging beam
are taken into account. The number of holes in the lateral side of the slab is determined by optimizing
the energy ratio of the redirected waves to the incident ones. The cluster of holes resulting from theoptimization process contains 3×7 holes with radius R r =6.5 mm and (square) lattice period ar =24mm. To further increase the sideways transmission, we add a barrier at the rear of the redirector device
with the goal of totally reflecting the waves partially transmitted across the slab. This barrier is madeof three layers of two-beam resonators, where the resonance-induced bandgap can be tuned to theworking frequency of the redirector [8].
Figure 2 shows the predicted performance of the redirector device. Notice how the principal beamof the collimator impinges the redirector device and a strong deviation to the lateral sides is obtainedthanks to the reflecting structure added to the rear side. The pattern of the Z-displacement field clearlyshows that the designed structure produces two sideways-transmitted beams. This phenomenon
represents the elastic analog of the Poisson-like effect described in [7] for acoustic waves but here iscaused by a different resonant mechanism. The physical mechanism explaining the redirection of
flexural waves can be understood in terms of the coupling of the incident wave to the fundamental
symmetric S0 mode of the crystal slab made of three layers of free holes. The S0 mode is a mixed
mode containing standing waves modes in the two orthogonal directions along the slab [11]. Also, it
is noticeable that the sidelobes generated by the collimator are completely suppressed by the absorbing
stripes. Let us stress that, in comparison with the result obtained without the reflecting barrier, thesideways transmitted waves have been strongly enhanced and show a perfect symmetry with respect
to y-axis. The integration of the energy flux over the boundaries of the finite slab indicates that 81%of the impinging energy is transmitted sideways.

Figure 2 – Snapshot of the amplitude of the out-of-plane displacement created by the collimated beam
impinging a redirector device, which consists of a rectangular array of free holes with a barrier of flexural
waves added at its rear surface.

3. EXPERIMENTAL SETUP

The experimental setup shown in Fig. 3 serves to validate the expected performance of thecollimator and redirector devices described in the previous section. Aluminum (Al-5745) plates witha thickness of 1 mm and dimensions of 800 x 600 mm2 were used to make the holey structures.
Different plates have been laser-cut with the holes and resonators necessary for the produced designs.
The accuracy of the laser-cutting method used is greater than 0.1mm.
The sample incorporating the collimator and redirector is shown in Fig. 3. To imitate the anechoic
boundary conditions employed in the numerical simulations, a mastic tape is applied to the plate's
exterior edges to prevent wave reflections there. We specifically used a 3M Scotch film ElectricalInsulation Putty that is 38.1 mm wide and 3.175 mm thick.

Figure 3 – Experimental setup employed to capture the out-of-plane displacement of the flexural waves
propagating in the aluminum thin plate with holey structures.
A laptop's audio card is used to create a sinusoidal signal for the measurements with an amplitudeof 2 Vpp, 24-bit resolution and 48 Hz sampling frequency, which is then amplified by a 200W Samson
signal amplifier. An electromagnetic transducer (EMAT) is fed by the amplified electrical signal. This
device is positioned next to the plate without contacting it and is composed of a coil encircling a 12mm diameter cylindrical magnetic core. Without any physical contact between the transducer and theplate, the EMAT excite flexural waves at the target frequency in the plate.
The propagation of the flexural waves generated by the EMAT is examined using a Polytec PSV500 scanning laser Doppler vibrometer (LDV). The scanning head features a 633 nm He-Ne laser andalso includes a number of motorized beam splitters and mirrors that can accurately position the emitted
beam with an angular resolution of greater than 0.001º. When compared to a reference beam, theDoppler effect caused by the target surface's vibration in the backscattered beam creates aninterference pattern that is correlated with the frequency of the surface vibration. The LDV may alsocount the number of dark and bright fringes and report displacements (on the order of pm) rather thanvelocities using interpolation and demodulation algorithms.
With this approach, no addition of mass is needed (avoiding thus undesired influences) and acomplete 2D map of the Z-displacement field in the entire area of the plate can be obtained. The LDV
head is positioned at a distance of about 2 meters from the perpendicular surface plate, which is
discretized into a scanning grid of the order of 23000 points, excluding the areas inhabited by the
drilled structures and the insulating strips. At least seven measurements points per target wavelength
are chosen for the scan spatial resolution, which is more than 3.5 x 3.5 mm2 .
By using a total of 3201 FFT lines, the vibrometer can examine frequencies between 10 and 20
kHz with a resolution of 3.125 Hz. In addition, when the laser beam is redirected to the subsequentscanning position, we maintain the phase reference introducing the audio signal in the LDV front-end
too. In order to reconstruct the time development of the waves across the entire plate, the LDV
captures both the amplitude and the phase of each measurement point.
3.1 Equipment validation
Prior to perform the measurements, a preliminary test was conducted to assess the operability ofthe experimental setup. To do so, a smaller aluminum plate (600 x 250 mm2 ) also 1 mm thick with thesame characteristics as those in the tests was used to determine the adequation of the LDV head tothis study by analyzing the group velocity propagation in the plate.
The excitation of plate was carried out by means of a Piezoelectric Transducer (PZT) instead of
the EMAT, but the source was also the amplified 24-bit audio card. The surface of the plate wasdiscretized into 3589 points, with a point, approximately, every 5 mm, from which eight of them,aligned with the location of the PZT, were used to determine the group velocity propagation. Figure

4 shows the discretization of the plate surface, highlighting the location of the PZT and the points
used to derive the velocity propagation.

y coord[m]

0.1
Origin (PZT)

0.05
0

Velocity Sample Points

-0.05
-0.1
-0.1

0

0.1

0.2

x coord

0.3

0.4

0.5

[m]

Figure 4 – Sketch of the preliminary test with the discretized surface, showing the location of the PZTtransducer and the samples used to derive the group velocity
To determine the group velocity, a 15 kHz burst was emitted through the PZT. This signal had aduration of 0.5 ms and enough time is left before sending the next burst to ensure the plate vibration
had been completely dissipated between one pulse and the following. As previously explained, thesignal was also fed to the LDV system to maintain the phase reference. In this case, it did not only
work as a reference signal, but also as a trigger to the system to begin a 10 ms measurement.
The acquisition settings were slightly different to accomplish the specific purpose of capturing thegroup velocity. In order to characterize every point of the discretized surface, the LDV was triggeredwith the analog output from the audio card, recording a pre-trigger sample of 2 ms (20% of the sampletime) and a band pass filter between 12.5 and 17.5 kHz was employed to insulate the burst from otherpossible sources. The sample frequency during the acquisition was 250 kHz, providing a temporal
resolution of 4 µs.
In order to determine the group velocity, 9 sample points were used: the origin (point 0) where thePZT was placed and another 8 points aligned with the transducer (points 1 to 8). The 10 ms signals
from each point were then analyzed and compared to each other. Thus, point 0 was compared against
point 1, point 2… until point 8. Then, point 1 was compared against points 2 to 8, point 2 against
points 3 to 8… so a total of 36 comparison samples were obtained. The comparison was carried out
by computing the cross-correlation between both signals and determining the lag between them attheir maximum peak. The left panels in Fig. 5 show the time signals from point 0 (Origin), point 3
and point 7. It is possible to see the burst signal in the origin and how the vibration losses intensity as
it gets further from the PZT. On the right side of the figure, the normalized cross-correlation from thesignals of point 3 and point 7 is displayed, indicating the maximum value.
101

Origin

-3

1

[-]

[m/s]

xcorr(v
0

10

1

[m/s]

2

4

6

8

Point 3

-4

V

z

0
-1
0
5

[m/s]

2
10

4

6

8

10

6

8

10

Point 7

-5

V

z

0
-5
0

2

4

Time

7

)

10

Norm. cross-correlation

0

,v

Max

0.5

z

V

-1

3

0

-0.5

-1
-10

-5

[ms]

0

Lag

5

10

[ms]

Figure 5 – (left panels) Vibration signals of the origin and two velocity sample points. (Right panel) crosscorrelation of the two velocity sample points

Finally, since the distance between sample points was known, the group velocity of the Lambwaves could be easily determined with the computed lag. The outliers were discarded, selecting 30
samples of the comparison. The mean group velocity from these samples was 764.24 m/s, with astandard deviation of ±8.25%. The theoretical value for Lamb waves propagation in the A0 mode underthese circumstances (1 mm plate with an excitation of 15 kHz) is 747.5 m/s, presenting a discrepancy
of only 2.23% with the actual measured data. Figure 6 shows the individual velocity samples in theupper part, with the mean and the theoretical values and the bottom of the figure represents the actual
experimental measured data in the theoretical A 0 lamb wave velocity curve.
1500
Sample velocity

[m/s]

1300

Mean sample velocity
Theroretical velocity

1100

Group velocity

900
700
500
300
100
0

5

10

15

20

Sample

25

30

[-]

3000

Group velocity

[m/s]

2500
2000
Lamb theoretical A

1500

0

mode

Experimental data (15 kHz · 1 mm)

1000
500
0

0

100

200

Frequency-thickness

300

400

500

[mm kHz]

Figure 6 – Group velocity of each sample with the mean and the theoretical value (top) and the value of theexperimental data in the lamb theoretical A0 mode curve (bottom)
Besides, another interesting purpose to fulfill prior to the final measurements was the performance
evaluation of the insulation putty absorbing the reflected waves from the edge of the plate at theworking frequency of 15 kHz. It was observed that for excitations of low frequency, up to 8 – 10 kHz,the mastic tape was not able to completely absorb the reflections from the edges, being theperformance the worse the lower the frequencies were. With frequencies above 12 kHz it was seenthat the tape dampened the vibrations satisfactorily, proving to be adequate for this case of study.

4. EXPERIMENTAL CHARACTERIZATIONThe setup shown in Fig. 3 has been employed to validate the predicted performance of thecollimator and redirector designs. Figure 7 shows a snapshot of the Z-displacement filed created by
the EMAT, which is now the excitation source of omnidirectional flexural waves.

Figure 7 – Measured snapshot of the amplitude of the out-of-plane displacement field created in thealuminum plate containing the collimator device. The EMAT employed as the source of omnidirectional
flexural waves is approximately located (behind the plate) at the middle position in the first column ofdrilled holes.

Figure 8 – Measured snapshot of the amplitude of the out-of-plane displacement field created in thealuminum plate containing the collimator and the redirector.

5. CONCLUSION

In summary, we have demonstrated the feasibility of two interesting devices for controlling theflexural waves propagating in thin elastic plates. First, we have introduced a structure of holes, here
denominated as collimator, which has been specifically designed to generate a collimated beam along
a given selected direction. In addition, another device, here denominated redirector, has been
specifically designed to produce the splitting and the transmission sideways of the vibrational energy
in a similar manner to that described in acoustics. The measurements corroborated the performanceobtained from numerical simulations performed with a 2D modeling (thin plate approach) as well aswith a full 3D elastic treatment [1]. The structures shown here can have potential applications for thecontrol and guiding of flexural waves propagating in different environments. For example, inaeronautic and astronautic industries, these structures can be employed to collimate and redirect waves
to specific regions where sensors arelocated.

ACKNOWLEDGEMENTS

JS-D and FC acknowledge financial support by the RDI grant PID2020-112759GB-2I100 funded
by MCIN/AEI/10.13039/501100011033.

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