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Research Papers

Alternative Designs of Acoustic Lenses Based on

Nonlinear Solitary Waves

Kaiyuan Li, Piervincenzo Rizzo and Xianglei Ni

[+] Author and Article Information

J. Appl. Mech 81(7), 071011 (Apr 25, 2014) (9 pages)

Paper No: JAM-14-1018; doi: 10.1115/1.4027327

History: Received January 07, 2014; Revised March 31, 2014; Accepted April 02, 2014

In the last decade, there has been an increasing attention on the use of highly- and weakly-nonlinear

solitary waves in engineering and physics. These waves can form and travel in nonlinear systems such as

one-dimensional chains of particles. When compared to linear elastic waves, solitary waves are much

slower, nondispersive, and their speed is amplitude-dependent. Moreover, they can be tuned by modifying

the particles' material or size, or the chain's precompression. One interesting engineering application of

solitary waves is the fabrication of acoustic lenses, which are employed in a variety of fields ranging from

biomedical imaging and surgery to defense systems and damage detection in materials. In this paper, we

propose the design of acoustic lenses composed by one-dimensional chains of spherical particles

arranged to form a line or a circle array. We show, by means of numerical simulations and an experimental

validation, that both the line and circle arrays allow the focusing of waves transmitted into a solid or liquid

(the host media) and the generation of compact sound bullets of large amplitude. The advantages and

limitations of these nonlinear lenses to attain accurate high-energy acoustic pulses with high signal-

to-noise ratio are discussed.

Journal of Applied Mechanics | Volume 81 | Issue 7 | research-article

1

ARTICLE REFERENCES FIGURES TABLES CITING ARTICLES

Abstract | Introduction | Array Design | Comparative Study | Experimental Study: Setup and Results |

Conclusions | Acknowledgements | References

Abstract

>

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(1)

(2)

(3)

Nonlinear solitary waves are compact nondispersive stress waves that can form and travel in highly

nonlinear systems, where they travel without changing form and amplitude due to balance of disperse and

nonlinear effects. The most common and simplest way to generate one or more solitary pulses is by

assembling a one dimensional chain of uniform spherical particles (beads). The nonlinearity arises from

the Hertzian type interaction in which the force F and deformation between two adjacent spheres isgoverned by the Hertz's law [1] F = A . Here the stiffness constant A is equal to

where R is the radius of the particles, and and E are the particles' Poissonratio and Young modulus, respectively. In a chain, solitary pulses can be excited by impacting the top

particle with a striker [2-14], by using a piezo-actuator [15], or by irradiating the top particle with laser

pulses [16]. When the chain is "weakly" compressed by means of its self-weight or by the action of some

form of static precompression, highly nonlinear solitary waves (HNSWs) form and propagate [17]. The term

weakly implies that the precompression is very small compared to the dynamic force associated with the

propagation of the waves.

When the chain of beads is under a static precompression force F , the initial strain of the system is

referred to as . In the continuum approximation (long-wavelength limit), the speed of the solitary wave Vhas a nonlinear dependence on the normalized maximum strain = / , or on the normalized force f =F /F in the discrete case. Here, and F indicate the total strain or total force made by the contributionof the static precompression plus the dynamic effect, which is associated with the propagation of a solitary

wave. When f (or ) is very large, the solitary wave velocity is [8]:

where a is the diameter of the beads, and , , and E are the density, Poisson's ratio, and Young'smodulus of the material, respectively. Moreover, the shape of the solitary wave can be closely

approximated by [3]:

where

and x is the coordinate along the wave propagation direction.

In the past ten years, there has been an increasing attention on the use of HNSWs for engineering

applications such as acoustic lenses [2], vibration absorbers [18], impurity detectors [19,20], acoustic

diodes [21], and nondestructive testing [22-25]. Spadoni and Daraio [2] introduced an acoustic lens made

of granular crystals. The lens consisted of a line array composed of chains of spherical particles with the

base of this array in contact with a linear medium (the host medium). The advantage of this lens is that the



Introduction

3/2

(E ) / [3 (1 )]2R 2

0

0 s

r m 0 r

m 0 m m

r r

= 0. 6802VS ( )2Ea (1 )3/2 21/3

F 1/6m

= ( ) cos ( x)5V 2s4c2 4 105a

c = 2E(1 ) 2

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(4)

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(6)

signals. Simultaneously, Daraio [26] illustrated the conceptual configurations of different line arrays to

control and/or redirect acoustic waves or pulses for focusing applications.

In the study presented in this paper, we compare the design of nonlinear acoustic lenses made of elastic

spheres arranged into line and circle arrays. The novelty of this study is two-fold: first, we compare two line

arrays whose design is alternative to Ref. [2] by suggesting different materials and spacing between the

chains of granular particles forming the lens; second, we propose a circle array that solves some of the

shortcomings associated with the line arrays. To carry out a comprehensive study that can ascertain the

advantages and limitations of the four arrays, a numerical study was conducted. A discrete particle model

was used to derive the shape and amplitude of the force function at the chains-host medium interface.

Then, a commercial finite element software was used to model the propagation of the acoustic fields in the

host medium to predict the location and amplitude of the focused sound. A total of four arrays and two host

media, namely polycarbonate and water, were simulated. Moreover, the numerical results relative to the

circle array are validated experimentally.

The paper is organized as follows. Section 2 describes the principles and the design of the arrays as well

as a brief overview of the discrete particle model, utilized to predict the acoustic source at the chains-linear

medium interface, is given. Section 3 describes the results associated with a finite element simulation

adopted to compare the acoustic focusing attained by one line array and the circle array. Section 4

illustrates the results of an experiment, whereas Sec. 5 ends the paper with some concluding remarks.

The overall concept of the line array is schematized in Fig. 1. n chains are assembled to form the nonlinear

lens. n solitary pulses arrive at the interface between the lens and the host medium (the linear medium)

where they irradiate at the medium's speed of sound. Here, by constructive interference, the bulk waves

coalesce into a focal point of coordinates (0, y ). As the speed of the bulk wave is solely dependent on the

linear medium, the solitary pulse must reach the nonlinear/linear media interface with predetermined delay

that is dependent upon the relative position of the chain in the array; the waves propagating through the

peripheral chains being faster than the pulses along the middle chains.

Discrete Particle Model.

In this study, the delay was calculated by estimating the travel time of the nonlinear pulse across each

chain. The time was found by using a discrete particle model which enables the prediction of the

characteristics of the nonlinear solitary pulses traveling along a chain of spherical particles [12,17,27]. The

model considered a 1-D chain composed of N spherical particles. The first particle of the chain represents

the striker, while the last bead was in contact with a wall which is considered as a half-infinite medium.

Notice that the dissipative effects associated with the contact of the chain with the inner tube's surface are

not considered in the model. The equation of motion of the i-th particle is expressed as [12-13,27]

where

Here, the subscripts c and w refer to the point of contact between two neighboring particles and the point

of contact between the last particle and the half-infinite wall, respectively. The values of R, m, and u are,

respectively, the radius, mass, and axial displacement from the equilibrium position of the particle. A is

instead the contact stiffness between adjacent beads (A ) or between the last bead and the wall (A ). F is

the static precompression which may be generated by the self-weight of the particles or by the action of

some external static force applied ad hoc to the chain. The dot represents the time-derivative while the

operator [] returns the value of the variable if the variable is positive, otherwise it returns 0. Finally, E and

are the Young's Modulus and the Poisson's ratio, respectively, of the beads (E , ) and of the wall (E , ).



Array Design

0

= + ,miu i Ai13/2i1 Ai3/2i Fi i = 1, 2, ,N

=Ai

0,= ,Ac E 2R3(1 ) 2

= ,Aw 4 R3 ( + )1 2E 1 2wEw1

i = 0i = 1, 2, ,N 1

i = N

=i

[ ] ,u1 +[ ] ,ui ui+1 +

[ ] ,uN +

i = 0i = 1, 2, ,N 1

i = N

c w i

+

c c w

w

Sci-Hub URL , DOI,



velocity equal to 4 m/s, which was chosen in order to carry out a close comparison with the study in

Ref. [2] The other initial conditions were u = 0 and for i = 1, 2,, N. The fourth order RungeKutta

method was used to solve the differential equation (4) in order to calculate u and for each particle.

The delay was calculated by estimating the travel time of the nonlinear pulse across each chain. Owing to

the peculiar characteristics of HNSWs, the delay can be tuned by changing one or more of the following

parameters: static precompression force F applied to the particles; dynamic contact force F at the

contact points between two neighboring particles; size (term a) and material (terms E, , and ) of thespheres.

Designs of Line Arrays.

The arrays consisted of n = 21 chains made of twenty-one 9.525 mm (3/8 in.) diameter stainless steel 302

particles arranged to create a sound bullet 100 mm below the chains-linear medium interface. The host

medium was a semi-infinite polycarbonate and it was chosen in order to carry out a close comparison

between our results and those in Ref. [2].

In the first design, the chains were equally spaced with a center-to-center distance (the parameter d in Fig.

1) equal to 9.65 mm. Each chain was assumed to be impacted by a striker having same geometry and

mass of the particles forming the array. The impact velocity of 4 m/s was set for all 21 chains, i.e., the

impact energy was evenly distributed across the array. To achieve focusing, each chain was

precompressed differentially such that the solitary pulses reach the interface with appropriate delays.

These delays were calculated based on the lengths of the travel paths in the linear media and the bulk

wave speed. Using the discrete particle model, the delay was expressed as a nonanalytic function of the

precompression. The results are presented in Table 1 where the delay is expressed with respect to the

fastest pulse. The middle chain (chain 0 in Fig. 1) is subjected to gravity precompression only while chains

10, located at the edges of the array, are subjected to the largest precompression (60 N) to enable thepropagation of the fastest wave.

In the second and third designs which are to the best of the authors' knowledge novel, the focusing was

achieved by changing the beads' material, i.e., tuning the parameters E, , and . The materials listed inTable 2 were chosen based on the commercial availability of 9.525 mm spheres, thus on the feasibility to

construct the array.

In the second arrangement, the particles were subjected to gravity-induced precompression only and the

dynamic contact force was imposed to be the same across the entire array. The latter condition implies

that the speed of the strikers cannot be identical, otherwise the dynamic contact force cannot be the same

because the strikers made of different materials have different masses. At a given velocity of strikers, the

dynamic contact force and the speed of the solitary wave at each chain were obtained by solving the

equation of motion of particles (Eq. (4)). By considering a range of velocity of the strikers (from 0 to 10m/s), the relationship among the velocity, the dynamic contact forces, and the phase delays were

established. Then at a given dynamic contact force the parameter d was determined. Figure 2(a) shows

the center-to-center distance of each chain with respect to the central chain, made of Tungsten particles.

The distance is plotted as a function of the dynamic contact force. Only 11 curves are shown because of

the array's symmetry. The figure shows that the distance is not uniform among neighboring particles and

that overall the heavier particles are located toward the center whereas the lighter materials constitute the

peripheral chains.

Finally, we modified the second design by assuming that the velocity was equal for all the strikers. This

implies that all the strikers have the same velocity. However, since the strikers are made of different

materials, the resulting dynamic contact force at the impact point across the line array is not the same.

Different dynamic contact force yields to different speed of the solitary waves along the chains. Therefore

focusing can be attained by spacing the chains appropriately. Figure 2(b) shows the distance between a

chain made of a certain material and the middle chain, as a function of the strikers' velocity. It should be

noted that the arrangement of the materials in the array between the second and the third design is

different. Here the central portion of the lens is made of copper particles. It must be remarked that for

certain materials, the center-to-center distance between adjacent chain is smaller than the diameter (9.65

mm) of the particles forming the chains. This can be resolved by slightly misaligning the overlapping

chains.

In summary, in the linear medium the speed of the acoustic wave departing from the interface is fixed and

cannot be tuned. If each chain is fixed in space, the path between the bottom of each chain and the focal

point is determined and therefore focusing can be attained by imposing an appropriate delay among the

solitary pulses reaching the interface. Conversely, if the time delay is imposed by the tunable

u 0i

= 0u iu

0 m

u 0

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simulations, this distance was derived and plotted on the vertical axis of Figs. 2(a) and 2(b).

Novel Design of Circle Arrays.

The three line arrays require tuning the granular crystals. It will be discussed in the conclusions that each

design has advantages and shortcomings. To circumvent some of the shortcomings, we propose a lens

where the n chains are assembled in a circle (Fig. 3) and focusing is attained along the axis (hereafter

indicated as the focal axis) passing through the center of the circle and orthogonal to the interface. Here

the delay of the waves radiating from the interface is null, i.e., the nonlinear characteristics of the

transducers are uniform. The diameter D of the array is strictly dependent upon the center-to-center

distance d between two adjacent particles and the number n of the chains in the array, i.e.,

Owing to the geometric attenuation of the bulk waves in the linear medium, the amplitude of the sound at

the focal axis is a function of the Euclidean distance between the radiating point and the position along the

focal axis and of the constructive interference (superposition) of the linear waves propagating at the speed

of sound of the host medium.

Solid Host Medium.

To portray, quantify, and compare the focusing effect between the line and the circle arrays, we coupled

the discrete particle model used to predict the time of arrival of the solitary wave pulses, to a finite element

analysis. Two host media were considered: polycarbonate and water. The finite element models were built

using the commercial software ANSYS v13.0. For polycarbonate, the 3D 8-node SOLID185 element with

three degrees of freedom per node was used. The material properties were density = 1230 kg/m ; Young's

modulus = 3.45 GPa; Poisson's ratio = 0.35; sound speed = 1675 m/s. For the water, the 3D 8-node

FLUID30 element with pressure as the only degree of freedom was used. The material's properties were:

density = 1000 kg/m ; sound speed = 1460 m/s. To guarantee the accuracy of the numerical result, the

element size was 2 mm which is of the impact wavelength, and the implicit integral time step was 2

s which is of the impact time duration. A convergence test was conducted by doubling the spatial

and the time resolution of the model. Same numerical result was obtained to prove the convergence of the

finite element model.

First, we calibrated our analysis by considering a line array made of chains subjected to the different

precompressions as listed in Table 1 and in contact with a polycarbonate block. Essentially, this simulation

reproduced the work in Ref. [2] and it served to calibrate our finite element model. Figure 4 shows the

force profiles at the interface between the array's base and the polycarbonate medium. The nonzero

dynamic force visible prior to the rise of the pulse is due to the static force associated with the

precompression. The asymmetric shape of the pulses is due to the interference of the tail of the incoming

wave with the front of the wave reflected at the interface. The rightmost pulse, i.e., the one labeled as

pulse from chain 0, represents also the input force at the interface between the circle array and the host

medium. For the polycarbonate simulation, the dynamic contact force profile between the last sphere and

the linear media was calculated using the discrete particle model. The dynamic contact force was applied

to the polycarbonate as point force load on the node where the sphere was located. For the simulation with

water, the dynamic contact pressure profile was calculated using the discrete particle model and was

applied as point pressure on the finite element node where the sphere is located.

For the circle array the diameter D was equal to 64.5 mm and the center-to-center distance d was kept to

9.65 mm, i.e., equal to the distance d of the line array. The two arrays were in contact with a semi-infinite

solid made of polycarbonate. For both arrays the striker's velocity was assumed equal to 4 m/s. To portray

the formation of the acoustic focus, the contour maps of the Von Mises stress at different instances are

presented in Fig. 5. The panels at the left of Fig. 5 refer to the line array (design 1) at 290 s, 310 s, and

342 s, whereas the panels at the right are relative to the circle array at 290 s, 310 s, and 346 s. Based

upon the sound speed in the polycarbonate and the geometry of the lens, the arrival of the focused signal

at the designed point (x = 0 mm; y = 100 mm) is expected at 342 s. Figure 5(e) shows that the highest

energy is located 15 mm above the designed focal point. This is due to wave attenuation associated with

the beam spreading.

The panels at the right of Fig. 5 refer to the circle array, for which the focusing was achieved at 346 s (Fig.

5(f)) instead of 342 s, due to the slight difference in the travel path length. By comparing Figures 5(f) with

5(e), we see that the area with the highest energy is smaller, i.e., better focusing is achieved, and the array

is more compact (65 mm wide instead of 193 mm). To enhance the readability of the contour plot, the color



Comparative Study

3

3

1/801/50

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However, it can be clearly seen that as the acoustic waves generated at the solid-granular array interface

propagate in the polycarbonate, their amplitude decrease with the increase of the depth because of the

geometric spreading.

To quantify the acoustic energy localized at the central axis of the polycarbonate, i.e., x = 0, Fig. 6 is

presented. It shows the values of the maximum Von Mises stress as a function of the depth for both the

line and the circle array. Depth 0 identifies the coordinate of the array-polycarbonate interface. By

comparing the values computed 20 mm and 40 mm below the interface, it can be seen that the stress

generated by the novel array nearly doubles with respect to the 20 mm depth and then decreases.

Moreover, deeper than 32 mm, the circular arrangement provides larger stress than the line arrangement.

At (x = 0 mm; y = 100 mm) the stress is 464 kPa versus 780 kPa. This is due to the following. First, the

geometric attenuation is such that the points closer to the interface are subjected to larger stress. Second,

the pulse transmitted into the linear medium is directional, i.e., the wave amplitude varies with the direction

of radiation.

Fluid Host Medium.

The same analysis was conducted by replacing the polycarbonate with water. In the finite element model,

the profile of the solitary wave input source at the fluid structure interface (FSI) represented the pressure.

Thus, the wave propagating in water was omnidirectional. Figure 7 shows the pressure field in the linear

medium overlapped with instantaneous pressure amplitude along the symmetry axis at three different

instants for both arrays.

The area with the highest pressure was above the designed focal point, due to beam's spreading in the

liquid. Figures 7(e) and 7(f) show the pressure distribution at 368 s and 372 s for the line array and the

circle array, respectively. At these two instants, the area associated with the largest pressure reached its

minimum, i.e., focusing was achieved.

To compare the maximum pressure attained at each point of the axis of symmetry by both arrays, Fig. 8 is

presented. Figure 8(a) shows the first peak associated with the propagating wave at the focusing point,

located 100 mm below the interface. Figure 8(b) shows instead the pressure as a function of the depth

associated with both arrays. A monotonic decrease is observed for both, but overall the circle array

provides higher amplitude than the line array. Interestingly, the profile of the maximum pressure associated

with the water medium is different than the solid medium (Fig. 6). The difference is related to the

directionality of wave propagation. In water the wave is almost omnidirectional whereas in the

polycarbonate the presence of shear, longitudinal, head, and surface waves (the latter at the interface) is

such that the out-of-plane stress is dominant along the propagation direction perpendicular to the interface.

When the measured point is near the interface, the contribution to stress from the peripheral chains is

lower than the corresponding contribution from the chains closer to the center of the array.

In the line arrays alternative to the design introduced in Ref. [2], the focusing was achieved by changing

the beads' material, i.e., tuning the parameters E, , and of the particles forming the chains, or byarranging the center-to-center distance between neighboring chains. In the circle array the chains were

arranged to form a circle and the particles were identical and differential precompression was not

necessary. Overall, the array proposed in Ref. [2] requires precise precompression and it assumes that the

dynamic force is assumed to be identical among all the chains when calculating the wave speed.

Nonetheless, the use of static precompression allows the nonlinear acoustic lenses to vary their focal

length and focal position dynamically in a 3D space, relying on simple mechanical means. The second

design overcomes the problem related with the precompression but at the expense of the size, which is at

least fourfold wider. Although differential precompression is not required, the occurrence of identical

dynamic contact force is hindered by the presence of different materials. Moreover, the control of 11

differential strikers' speeds is impractical. This is overcome in the third array where the velocities of the

strikers can be easily arranged to be identical. However, for both the second and third array the distance

between any two adjacent chains made of stainless steel 316, 302, 440C and iron is less than the

particles' diameter, which can be resolved by slightly misaligning the overlapping chains. All of the above

arrays require tuning of the granular crystals. Although each design is feasible, its practical implementation

could be challenging. To circumvent this problem we proposed a lens where the n transducers were

assembled in a circle. Besides the fact that the circular arrangement does not require any particles' tuning,

we found by a quantitative comparison between the first line array and the circle array, that the latter array

enables larger pressure and comparable focusing area. The circle array differs in dimensionality with

respect to the line array and this could be a disadvantage in terms of bulkiness. Moreover, the 1D array is

expected to produce a boomerang shaped focal field along the plane normal to the lens, while the circle

array, being two-dimensional and axisymmetric, will generate a symmetric focus.

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To investigate the feasibility of the circle array an experiment was conducted and its results are presented

here. The acoustic lens consisted of an ultra-high-molecular-weight polyethylene block containing 20

through-thickness holes, 9.6 mm in diameter. The holes were filled with 21 general purpose stainless steel

beads, 9.525 mm (3/8 in.) in diameter. The chains were distributed along a circle of 101.6 mm (4 in.) in

diameter. This was a good trade-off between the objective of validating experimentally the numerical setup

and the need of machining through-thickness holes without shrinking the adjacent holes. To prevent the

free fall of the particles, a 2.54 mm thick aluminum sheet was bonded at the bottom of the acoustic lens.

An aluminum plate 6.35 mm thick was placed above the chains. Two photos of the lens are presented in

Fig. 9.

The solitary waves on the chains were generated by means of the impact of an iron rod remotely controlled

by means of an electromagnet. The rod was set such to impact the centroid of the thick plate. The host

linear medium was water. The acoustic pressure field generated by the solitary waves reaching the FSI

was measured by means of a commercial hydrophone (RESON TC4013-1). The overall experimental

setup is shown in Fig. 10. The pressure field was measured along the symmetric axis of the lens (scanning

line V in Fig. 10(a) and moving the hydrophone along a line path parallel to the interface 50 mm below

(scanning line H in Fig. 10(a)). For each scanning point, ten measurements were taken to improve the

statistical significance of the experiment.

Figure 11(a) shows the time waveform recorded by the hydrophone when located 50 mm below the

liquidsolid interface at the symmetric axis of the system. The signal is oscillatory after 900 s, which is

likely the result of the vibration of the aluminum baffle. To alleviate this oscillation Donahue et al. [28], in a

paper just released proposed a method of adjusting baffle properties to absorb the vibration energy

generated at the interface between the linear and nonlinear medium. The Gabor Wavelet scalogram of Fig.

11(a) is presented in Fig. 11(b). The wavelet transform decomposes the original time-domain signal by

computing its correlation with a short-duration wave called the mother wavelet that is flexible in time and in

frequency. We used the Gabor mother wavelet as it provides the best balance between time and

frequency resolution, i.e., the smallest Heisenberg uncertainty box [29-31]. The bulk wave associated with

the pressure generated at the interface by the propagating solitary waves is centered at about 8 kHz.

To estimate the focusing effect and evaluate how the pressure field is distributed along the two directions

measured here, the amplitude of the negative time waveform peak was considered. Figure 12(a) shows

the water pressure recorded by the hydrophone along the scanning line H schematized in Fig. 10(a). The

values are normalized with respect to the peak pressure value of 1.38 kPa which was recorded at the

central axis. The vertical bars represent the standard deviation associated with the ten measurements per

scanning point. The experimental data are overlapped to the results of a finite element analysis. For

simplicity, the analysis did not included the presence of the thin and the thick plates at the bottom and top,

respectively, of the lens, and the input force profile was considered identical to the one shown in Fig. 4

(case of pulse from chain 0). The agreement between the numerical and the experimental results is

remarkable. It can be seen from Fig. 12(a) that as the observation point is moved from the periphery (x =

60 mm to the symmetric axis, the value of the pressure field doubles. Owing to the different amplitude of

the solitary waves generated experimentally and numerically, a quantitative comparison between the two

investigations cannot be carried.

Figure 12(b) presents the experimental normalized values of the pressure amplitude as a function of the

hydrophone position below the FSI along the symmetric axis (scanning line V in Fig. 10(a)). Here, we also

compare the experiment with the finite element analysis. Interestingly, Fig. 12 suggests that the numerical

model underestimate the amplitude decay of the acoustic pressure. The cause of such a discrepancy

between experimental and experimental results is being investigated, and therefore, a conclusive remark

cannot be made.

We can generalize the expected response associated with the normalized amplitude decay. Lenses having

the same number of chains but smaller diameter, i.e., closely packed chains, are expected to give raise to

higher decay than the larger lenses. In fact, values as those shown in Fig. 12(b) represent the amplitude of

the pressure normalized with respect to the corresponding maximum pressure computed 20 mm below the

FSI. By looking at the similarity of the triangles generated by the travel path of a single acoustic source

departing from the FSI of the lens, it can be demonstrated that the ratio of the two line of sights connecting

the acoustic source at the FSI to y = 120 mm and y = 20 mm, respectively, decreases as the diameter of the

lens becomes larger.



Conclusions

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In this paper, we presented three arrays made of granular crystals and arranged to achieve acoustic

focusing in a linear medium. The three arrays consisted of one-dimensional chains of spherical particles

assembled to form a line or a circle. The granular material forming each chain supports the propagation of

highly nonlinear solitary waves, which are nondispersive and compact stress waves. The focusing

properties of these acoustic lenses were compared to the focusing properties achieved by a line array

proposed [2] and based on one-dimensional chains of spherical particles subjected to differential

precompression. Although it adds a dimensionality to the overall shape of the lens, the circle array seems

to be the most practical as it does not require differential precompression, proper selection of particle

materials, or control of the striker impact. Besides the fact that the circular arrangement does not require

any particles' tuning, we found by a quantitative comparison between the first line array and the circle

array, that the latter array enables larger pressure and comparable focusing area.

One interesting characteristics of the proposed system is that focusing is not achieved by deflecting the

wave incident at the nonlinear-linear medium interface at a certain angle (refraction angle) or by delaying

the signal emitted at each element of a phased-array controlled by a function generator. Indeed, focusing

was achieved by the constructive interference of waveforms departing from the interface with a certain

delay (line array designs) or simultaneously (circle array design). Time delay distribution is necessary to

obtain a focal point in air, liquid, or solid when a line array is adopted. For both designs, the wave field at

the focal point is the result of acoustic energy coalescence and the resulting sound field is composed of a

symmetric pressure maximum and minimum. As such, the proposed lens is different than the method used

in most of the optic and acoustic lens where focusing is achieved by tuning the propagation direction with

inhomogeneous refraction index.

To improve the comparison between the experimental and the finite element results, future works shall

include the baffle and the upper plate in the numerical model. In fact, it is expected the resonant modes of

these elements may have an effect on the formation of the sound bullet. Furthermore, future studies shall

consider alternative ways to excite the nonlinear solitary waves to reduce secondary waves generated by

bending or global vibration of the lens. The novel lens may also be compared to the acoustic lens attained

by using the tensegrity units, instead of granular materials, as proposed in Ref. [32]. Finally, other studies

may find the optimal combination of particles' diameters, modulus, and number per chain able to achieve

the largest acoustic intensity (amplitude per unit area) and compare the results with a similar optimization

study applied to line arrays.

X.N. performed this work while postdoctoral scholar at the University of Pittsburgh supported by the

University of Pittsburgh's Mascaro Center for Sustainable Innovation. Part of this work was also supported

by the U.S. National Science Foundation, Dynamical Systems program (Grant CMMI 1200259).




Acknowledgements



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