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Tunable acoustic couplers for two fluids with large impedance mismatch

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2014 Appl. Phys. Express 7 067302

(http://iopscience.iop.org/1882-0786/7/6/067302)

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Tunable acoustic couplers for two fluids with large impedance mismatch

Cong Liu1, Xiao-dong Xu1,2*, Xiao-jun Liu1*, and Christ Glorieux2

1Key Laboratory of Modern Acoustics, Nanjing University, Nanjing 210093, China2Laboratorium voor Akoestiek en Thermische Fysica, Departement Natuurkunde, K. U. Leuven,Celestijnenlaan 200 D, B-3001 Leuven, BelgiumE-mail: xdxu@nju.edu.cn; liuxiaojun@nju.edu.cn

Received April 11, 2014; accepted May 13, 2014; published online May 28, 2014

The tunneling effects of acoustics have been found when the proposed solid–fluid superlattices (SLs) are used to match two fluids with a largeimpedance mismatch. One of the examples shows that, by embedding a rubber/water SL between air and crude oil, the transmission of acousticenergy reaches 0.97 at 5° at 8.73 kHz. According to the results of theoretical analyses and full-wave simulations, this energy tunneling is attributedto the acoustically resonant states induced by the SL at the interface. The tunneling frequencies can be tuned in real time by varying theinterspacing between the solid layers of the SL, which may have wide potential applications in matching two fluids, such as acoustic couplingbetween air and crude oil during petroleum exploration. © 2014 The Japan Society of Applied Physics

The study of acoustic and elastic waves in periodicstructures, especially superlattices (SLs), has receivedconsiderable attention over the past decades.1–13) The

most significant property of these structures is that they cansupport forbidden bands induced by the periodic modulationof the acoustic/elastic properties of constituent materials.In addition to this important property, extraordinary phenom-ena associated with the introduction of inhomogeneitieswithin SLs such as free surfaces, SL/substrate interfaces, andplanar defects have attracted much interest owing to potentialapplications.3–5,7,14,15) In previous studies, GaAs/AlAs SLswere placed between an AlAs substrate and a fluid,enhancing the transmission of L-mode phonons from thesubstrate into the fluid through resonant surface vibrations.6,8)

Further demonstrations were presented for obliquely incidentwaves, in which peaks of resonant transmission througha finite solid–solid SL appear when the frequency andparallel wave vector of the incident wave satisfy thedispersion relation of surface excitation.16) Compared withthe solid–solid SLs, the solid–fluid SLs have receivedless attention. The existence of surface modes for a solid–fluid SL (SMSSLs) was firstly investigated using a Green’sfunction approach in 2006.15) Further investigation in finite-size solid–fluid SLs is carried out to show peculiar propertiesas compared with solid–solid SLs.17) Unlike the solid–solidSLs, one obvious advantage of the solid–fluid SLs is itsreal-time adjustability: the structural parameters of a solid–fluid SL can be conveniently modified by adjusting thespacing between the solid layers within the SL. On thebasis of previous studies, by combining the adjustabilityof the solid–fluid SLs and the resonant-enhancing effectin the solid–solid SLs, tunable acoustic couplers can beconstructed to match two different fluids with largeimpedance mismatch.

In this paper, three types of solid–fluid SL have beenproposed to act as tunable couplers in order to achieve energytunneling through two fluids with large impedance mismatch.The mechanism of this acoustic tunneling, which is intro-duced by the designed solid–fluid SL when waves propagatethrough it from one fluid to another with a large impedancemismatch, has been theoretically analyzed, which is furtherconfirmed by full-wave simulations using the finite elementmethod (FEM). The real-time adjustability of the spacingratio of the solid–fluid SL can effectively compensate thebandwidth limitation of the tunneling.

A schematic configuration of the system is illustrated inFig. 1. A solid–fluid SL is inserted between two semi-infinitefluids. A longitudinal wave is assumed to strike the finitesolid–fluid SL from fluid A at the incident angle ª0 in the x–zplane (sagittal plane), defined by the normal to interfaces(labeled 0 and 1 in Fig. 1) and the wave vector k, and thenproceeds through the SL into fluid B. In this system, fluid Ais assumed to have a significantly smaller acoustic impedancethan fluid B or the constituent materials of the finite SL.

For a finite SL with increasing number of periods, theoscillatory modes tend to form continuous bands. Thus, inorder to describe the dispersion characteristics of the bulkbands for the SL, a Green’s function approach the sameas in Ref. 15 is adopted. For an infinite SL composed ofalternating fluid layers (thickness df , longitudinal velocity vf ,and mass density µf ) and isotropic solid layers (thicknessds, longitudinal velocity v1, transverse velocity vt, and massdensity µs), the dispersion relation k(½) of the bulk bands canbe expressed as

cosðkzDÞ ¼ A2 � B2 þ a2 � b2 þ 2Aa

2Bb; ð1Þ

where

a ¼ � F coshð�fdf Þsinhð�fdf Þ ; b ¼ F

sinhð�fdf Þ ;

A ¼ � � coshð�ldsÞsinhð�ldsÞ � � coshð�tdsÞ

sinhð�tdsÞ ;

B ¼ �

sinhð�ldsÞ þ�

sinhð�tdsÞ ; ð2Þ

with �2f ¼ k2x � !2=v2f , F ¼ ��f!

2=�f , �2t ¼ k2x � !2=v2t ,

�2l ¼ k2x � !2=v2l , � ¼ ��sv

4t ðk2x þ �2

t Þ2=!2�l, and � ¼4�sv

4t �tk

2x=!

2. kz is the z-direction component of the wavevector k ¼ ðkx; kzÞ and D = ds + df is the period of the SL.

Fig. 1. Configuration of solid–fluid SL inserted between incidence fluidand transmission fluid.

Applied Physics Express 7, 067302 (2014)

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As described in Ref. 15, the dispersion relation for theSMSSLs of a semi-infinite solid–fluid SL that terminates withthe solid layer can be expressed by

aðB2 � A2Þ � Aða2 � b2Þ ¼ 0; ð3Þwith the condition «aB/Ab« > 1. The same expression asEq. (3) with the opposite condition «aB/Ab« < 1 gives theSMSSLs for the semi-infinite SL that terminates with thefluid layer. The two conditions are the inverse of each other,which means that if a surface mode exists on one SL, it doesnot exist on the surface of the complementary SL. Notethat the simple boundary condition of the SMSSL problem(i.e., vacuum-SL) is sufficient to model our system becausethe impedance of fluid A is much lower than those ofthe materials in the SL or fluid B. Therefore, the interfacevibrations near the entrance surface (labeled 0 in Fig. 1) ofthe finite SL can be approximately predicted with the samedispersion relation for the SMSSLs of the correspondingsemi-infinite SL.

As an example, Fig. 2(a) shows the bulk bands for aninfinite Al/Hg SL, together with the dispersion curves ofthe SMSSLs for the corresponding semi-infinite one, i.e.,the frequency as a function of the reduced wave vector kxD.The layer dimension of the SL is ds = df = 0.05m. In thisfigure, the grey areas denote the bulk bands where acousticwaves are allowed to propagate through the SL, whilewhite areas are the forbidden gaps. The curves formed bythe dots in these gaps represent the SMSSLs when the semi-infinite SL terminates with the Al layer (i.e., vacuum–

Al/Hg/Al/*). Different angles of incidence ª0 correspondto solid lines with the slope given by kxD=f ¼ 2�D sin �0=c

if ,

where cif represents the longitudinal velocity of the incidencefluid. As shown in Fig. 2(a), the two solid lines correspond toª0 = 5 and 10°, respectively.

The transmittance of acoustic waves through the layeredsystem in Fig. 1 can be obtained by a transfer matrix method.The transfer matrix M of the finite SL can be derived fromthe continuous boundary conditions for the normal stressTz, shear stress Tx, and normal velocity vz at each solid–fluidinterface. Examples of transmission coefficient, defined

as the ratio of the energy fluxes of the transmitted andincident waves, are illustrated for ª0 = 0, 5, and 10° inFigs. 2(b)–2(d), respectively. These figures all correspond toa finite Al/Hg SL of 10 periods plus one Al layer (i.e.,Al/Hg/*/Al) with the same dimension as the SL inFig. 2(a) inserted between water (fluid A) and Hg (fluid B).The oscillations of the transmission reflect the Fabry–Perotcavity, like standing wave behavior.

It can be clearly observed that the two frequencies of 12.21and 23.89 kHz for the transmission peaks in Fig. 2(b)(ª0 = 0°) coincide with the positions of the SMSSLs atkx = 0 (normal incidence) in Fig. 2(a). The transmissioncoefficients of the peaks at f = 12.21 and 23.89 kHz reach0.64 and 0.98, respectively, while the transmission coefficientof ordinary transmission in the bulk bands is only about 0.28(about the same value as in the case of direct acoustictransmission from water into Hg). In other words, acousticenergy can tunnel from water into Hg with an amplitude that ismuch higher than that in normal cases. This tunneling effectcan also be observed for oblique incidence. In Fig. 2(a), thesolid line representing ª0 = 5° crosses the dispersion curvesof the SMSSLs (dotted lines), producing an extraordinaryamplification at f = 12.26 and 23.83 kHz in Fig. 2(c).However, when ª0 increases to 10°, the corresponding solidline does not cross any SMSSL. Accordingly, no transmissionpeak is found, and only transmission gaps are observed inFig. 2(d). The positions of the energy tunneling correspondaccurately to the intersections of the line of incidence andthe dispersion curves of SMSSLs in Fig. 2(a). Because ofthe linearity of the system, the same tunneling effect can alsobe observed if the incidence comes in the reverse direction.

The energy tunneling effect of the solid–fluid SL may havepotential applications. As an example shown in Fig. 3, thiseffect can be applied in sensing devices to realize acousticpenetration from air into crude oil during petroleum explora-tion, overcoming the obstacle of a large impedancemismatch. Figure 3(a) gives the bulk bands (grey areas) ofan infinite water/rubber SL and the corresponding SMSSLs(dotted lines) induced by the free surface of the waterlayer. The thicknesses of the layers are ds = df = 0.05m.Figures 3(b)–3(e) show the transmission for a water/rubberSL of 5 periods (i.e., water/rubber/*/rubber) embedded asa matching coupler between air and crude oil. Here, as apreliminary study, the viscosity and sound absorption of thecrude oil (mass density µ = 960 kg/m3 and longitudinalvelocity v = 1200m/s) were ignored.18) Figures 3(b)–3(e)correspond to ª0 = 0, 1, 2, and 5°, respectively, which arealso denoted by the solid lines in Fig. 3(a). Although thetransmission efficiency for waves going directly from air intooil is extremely low (acoustic impedance ratio Zoil/Zair µ2800), with the assistance of the matching SL, acousticpenetration can be achieved. In Figs. 3(b)–3(e), the ampli-tude of the first peak increases from 0.31 at ª0 = 0° to 0.97 atª0 = 5°. The weak tunneling of the second peak, as denotedby the insets, can be observed in Figs. 3(c) (near 9.58 kHz)and 3(d) (near 9.8 kHz). The comparison of Figs. 3(a) and3(b)–3(e) reveals that the positions of the transmission peaksagree excellently with respective intersection points of thesolid lines representing different incident angles and thedotted curves representing the SMSSLs. Clearly, the energytunneling peaks of transmission are associated with the

Fig. 2. (a) Bulk bands of infinite Al/Hg SL (grey areas) and SMSSLs(dotted lines) of corresponding semi-infinite SL that terminates with Al layer.Frequency dependence of transmission coefficient through finite Al/Hg SLof 10 periods plus one Al layer for (b) ª0 = 0°, (c) ª0 = 5°, and (d) ª0 = 10°.

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corresponding SMSSLs, that is, the interface vibrationsexcited by the incident waves near the interface of fluid Aand the matching SL. It should be noted that the performanceof the transmission enhancement of tunneling closelydepends on the number of SL layers, which is why SLs ofdifferent lengths are chosen in the above cases in order toachieve high transmission. Figure 3(f ) gives the dependenceof the transmission peak value on the number of periods,N, for the first peak (at 9.7 kHz) in Fig. 3(e). The resultsshow that the optimal transmission efficiency for this peak isobtained at N = 5. Whenever N decreases or increases fromthe optimal number, the transmittance becomes weaker.

To further understand the tunneling effect induced by theinterface vibrations of the solid–fluid SL, the displacementfield distribution within a glass/Hg SL of 2 periods plus oneglass layer (glass/Hg/*/glass) inserted between waterand Hg was investigated with ds = 0.1m and df = 0.035m.Figure 4(a) depicts the corresponding allowed bands and alsoSMSSLs of the semi-infinite SL terminating with the glasslayer. On the other hand, Fig. 4(b) shows the transmissioncoefficient for the incident angle of ª0 = 15° from water.As shown in Fig. 4(a), since the solid line of incidencehas two intersections on SMSSLs (10.24 and 18.535 kHz,respectively), two respective peaks of transmissions areexpected. However, the weak tunneling effect of the secondpeak is hardly visible. Four displacement fields are depictedin Figs. 5(a)–5(d), corresponding to four different types ofsituation labeled 1, 2, 3, and 4 in Fig. 4(b), i.e., extraordinarytunneling associated with interface vibrations at 10.24 kHz,ordinary transmission in the allowed band at 13 kHz, zero

transmission in the forbidden gap at 16 kHz, and zero trans-mission coinciding with the SMSSL incapable of supportingefficient tunneling at 18.535 kHz, respectively. In thesefigures, the black lines denote the solid–fluid interfaceswithin the structure.

The most important common feature of the displacementfields in Figs. 5(a) and 5(d) is the strong localization at theentrance interface of the SL, followed by the rapid decayaway from this interface. In fact, the vibration of the SMSSLsis characterized by the decay of the displacement field aswaves penetrate from one solid (fluid) layer into the adjacentsolid (fluid) layer. These two fields differ though in that theincident wave in Fig. 5(a) excites a resonant state, which isefficient enough to ensure that the decaying waves reach theopposite interface with sufficient amplitude, while the wavein Fig. 5(d) does not. In Fig. 5(d), the localization in thefirst layer clearly is not capable of conveying energy tothe adjacent layer as the vibrational motion near the rightinterface is almost parallel to the interface. This behavior isattributed to the rapid attenuation of the normal vibration

Fig. 4. (a) Bulk band of infinite glass/Hg SL (grey areas) and SMSSLs(dotted lines) of corresponding semi-infinite one that terminates with glasslayer. (b) Frequency dependence of transmission coefficient through finiteglass/Hg SL of 2 periods plus one glass layer for ª0 = 15°. Four differenttypes of transmission are labeled 1, 2, 3, and 4 in panel (b).

Fig. 5. Total displacement field excited by incident wave (with unitamplitude) of ª0 = 15° at (a) 10.24, (b) 13, (c) 16, and (d) 18.535 kHz. Boththe amplitude field (color variations) and vector field (arrows) are shown. Theblack solid lines indicate the solid–fluid interfaces.

Fig. 3. (a) Bulk band of infinite water/rubber SL (grey areas) andSMSSLs (dotted lines) of corresponding semi-infinite one that terminateswith water layer. Frequency dependence of transmission coefficient throughfinite water/rubber SL of 5 periods for (b) ª0 = 0°, (c) ª0 = 1°, (d) ª0 = 2°,and (e) ª0 = 5°. The insets in (c) and (d) show the zoomed area of theweak tunneling. (f ) Dependence of transmission peak value on number ofperiods, N.

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within the first layer (because of a comparatively highfrequency and a large angle of incidence). This behavior isalso a distinctive feature compared with the solid–solid casebecause the solid–solid interface is capable of conveyingshear vibrations.

As for the zero transmission in the forbidden gap (16 kHz)in Fig. 5(c), the incoming wave hardly penetrates the SLowing to the forbidden gap. In Fig. 5(b), the case of ordinarytransmission in the allowed band shows almost no decay aswaves propagate along the perpendicular direction. Theseresults of comparisons further confirm the resonant assistanceof the interface vibrations in Fig. 5(a).

By adjusting the thickness of the fluid layers of the SL(i.e., by changing the spacing between the solid layers), thereal-time tunability of the tunneling frequency can be easilyachieved. Figure 6 depicts an example of the spacing ratio(df/ds) dependence of the transmission spectra between 5 and25 kHz for ª0 = 0° and ds = 0.05m in the Al/Hg systemanalyzed in Fig. 2. As one can see, the tunneling frequenciesdecrease monotonically with df/ds with rather constanttransmission magnitudes. These characteristics indicatepotential applications in the active control of impedancematching, which can effectively remedy the inherentbandwidth limitation of the resonant tunneling.

In conclusion, acoustic tunneling can be achieved bysimply inserting a finite solid–fluid SL between two fluids

with a large impedance mismatch. Tunneling occurs when thevector and frequency of the incidence wave coincide with thedispersion relations of the SMSSLs for corresponding semi-infinite SLs. The transmission coefficient of the tunnelingis substantially larger than the ordinary transmission in thebulk bands, and further calculated displacement fields confirmthe existence of the acoustic resonant state owing to theinteraction of the incident waves with the interface vibrationsnear the entrance surface of the finite SL. By adjusting theinterspacing between the solid layers of the solid–fluid SL,active control of the acoustic tunneling can be easily achieved.These merits suggest potential applications in matching twofluids with acoustic impedance contrast, and thus enhancingthe acoustic transparency at the interface zone.

Acknowledgments This work was supported by the National BasicResearch Program of China under Grant No. 2012CB921504, the NSFC underGrant Nos. 11074126, 11274171, 11274099, and 11204145, Jiangsu OverseasResearch and Training Program for University Prominent Young and Middle-agedTeachers and Presidents, and K. U. Leuven (Project OT/11/064).

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Fig. 6. Transmission coefficient variation as a function of frequency andspacing ratio df/ds of SL. Different colors denote different levels oftransmission.

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