International audienceThe equality of the energy and group velocities of bulk acoustic waves in a lossless piezoelectric medium is demonstrated, with the energy velocity defined from the generalized energy density and the generalized Poynting vector

Khelif, Abdelkrim

Laude, Vincent

Reinhardt, Alexandre

HAL - Université de Franche-Comté

Equality of the energy and group velocities of bulkacoustic waves in piezoelectric mediaVincent Laude, Alexandre Reinhardt, Abdelkrim KhelifTo cite this version:Vincent Laude, Alexandre Reinhardt, Abdelkrim Khelif. Equality of the energy and groupvelocities of bulk acoustic waves in piezoelectric media. IEEE Transactions on Ultrasonics,Ferroelectrics and Frequency Control, Institute of Electrical and Electronics Engineers, 2005,52, pp.1869. <hal-00074907>HAL Id: hal-00074907https://hal.archives-ouvertes.fr/hal-00074907Submitted on 24 May 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.LAUDE et al.: EQUALITY OF THE ENERGY AND GROUP VELOCITIES OF BULK ACOUSTIC WAVES IN PIEZOELECTRIC MEDIA 1Equality of the energy and group velocities of bulkacoustic waves in piezoelectric mediaVincent Laude, Alexandre Reinhardt, and Abdelkrim KhelifAbstract—The equality of the energy and group velocities ofbulk acoustic waves in a lossless piezoelectric medium is demon-strated, with the energy velocity defined from the generalizedenergy density and the generalized Poynting vector.Index Terms—Bulk acoustic waves; Piezoelectric media; En-ergy velocity; Group velocity.I. INTRODUCTIONIt is well known that the energy and the group velocitiesof bulk acoustic waves (BAW) in elastic media are equal [1],[2]. The energy velocity, ve, is defined as the ratio of thePoynting vector to the energy density, with the help of thePoynting theorem, while the group velocity, vg , is definedas the derivative of the phase velocity with respect to thedirection of propagation. The group velocity is thus definedon a purely geometrical ground and is by construction normalto the slowness surface. Further properties are the equality ofthe BAW kinetic and the potential (or strain) energies, and theorthogonality of the slowness vector to the wavefront surface.In the case of piezoelectric media, we are not aware of ageneral demonstration of equivalent properties, though gener-alized expressions have long been known for the energy den-sity and the Poynting vector [1]. Nevertheless, the collinearityof the generalized Poynting vector and the group velocity isroutinely used to predict the beam-steering angle, or directionof energy transport. However, Zaitsev and Kuznetsova [3]recently identified a possible discrepancy that can occur ifthe generalized Poynting vector is used to predict the di-rection of energy transport in strong piezoelectrics such aslithium niobate. Their argument relies on a dissymmetry of themechanoelectrical and electromechanical contributions to thegeneralized Poynting vector. Their work has motivated us toestablish the equality of the energy and group velocities of bulkacoustic waves in an arbitrary lossless piezoelectric medium,with the energy velocity obtained from the generalized formsof the Poynting vector and the energy density. This propertysettles the apparent discrepancy.II. BASIC RELATIONSWe summarize in this section some well known energeticrelations for bulk acoustic waves propagating in linear piezo-electric media. Let us first consider a perturbance of anarbitrary linear piezoelectric medium, characterized by theThe authors are with the Institut FEMTO-ST, de´partement LPMO, CNRSUMR 6174, 32 avenue de l’Observatoire, F-25044 Besanc¸n, France (e-mail:vincent.laude@lpmo.edu).strain Sij , the stress Tij , the electric field strength Ek andthe electric displacement Dk. These fields are related by theconstitutive relations [1], [4]Tij = cijklSkl − ekijEk, (1)Dj = ejklSkl + ²jkEk, (2)where cijkl, ekij and ²jk are the elastic, piezoelectric anddielectric tensors, respectively. The repeated summation indexconvention is used throughout this letter. The time-averagedgeneralized energy density is given by [1]W =12Re(TijS∗ij +DkE∗k), (3)and the time-averaged generalized Poynting vector isPj =12Re(−Tijv∗i + (E×H∗)j), (4)where vi is the particle velocity, Hi is the magnetic fieldstrength and ∗ denotes complex conjugation.We then consider specifically a time-harmonic plane wavewith angular frequency ω and slowness vector sj . The dis-placements ui are then of the formui(x, t) = ui exp(ω(t− sjxj)), (5)and similar expressions hold for all fields. We specifically de-mand that the slowness vector be real-valued, i.e. the mediumis lossless and evanescent or inhomogeneous bulk acousticwaves are not considered.Considering further the quasi-static approximation [1],Maxwell’s equations giveEi = ωsiΦ, (6)(s×H)j = −Dj , (7)sjDj = 0, (8)where Φ is the electric potential. The constitutive relationsbecomeTij = −ω(cijklskul + ekijskΦ), (9)Dj = −ω(ejklskul − ²jkskΦ). (10)The generalized energy density simplifies toW =12Re(ωTijsju∗i ). (11)It can be seen that the electromagnetic part of the energydensity vanishes. The generalized Poynting vector becomesPj =12Re(ω(Tiju∗i +DjΦ∗)), (12)where vi = ωui has been used in Eq. (4).2 LAUDE et al.: EQUALITY OF THE ENERGY AND GROUP VELOCITIES OF BULK ACOUSTIC WAVES IN PIEZOELECTRIC MEDIAIII. ENERGY VELOCITYWe now introduce the generalized notationc¯ijkl =cijkl i, l = 1, 2, 3ekij l = 4, i = 1, 2, 3ejkl i = 4, l = 1, 2, 3−²jk i, l = 4. (13)This notation should not be confused with the piezoelectricallystiffened elastic constants [4]. With the generalized displace-ments, u¯i, and stresses, T¯ij , defined asu¯i = ui, i = 1, 2, 3, (14)u¯4 = Φ, (15)T¯ij = Tij , i = 1, 2, 3, (16)T¯4j = Dj , (17)the constitutive relations can be written in a compact way asT¯ij = −ωc¯ijklsku¯l, (18)and the Christoffel equation [1] governing the wave dynamicsassumes the formc¯ijklsjsku¯l = ρilu¯l, (19)withρil = ρ1 0 0 00 1 0 00 0 1 00 0 0 0 . (20)Contracting the Christoffel Eq. (19) with u¯∗i yields therelationc¯ijklsjsku¯lu¯∗i = ρilu¯lu¯∗i = ρu∗i ui. (21)Inserting Eq. (9) into Eq. (11) we obtainW =ω22Re(cijklsjskulu∗i + ekijsjskΦu∗i ), (22)or by inspectionW =ω22Re(c¯ijklsjsku¯lu¯∗i ). (23)Using Eq. (21), we observe that this quantity is real and thusW =ω22c¯ijklsjsku¯lu¯∗i =ω22ρu∗i ui. (24)This relation proves the equality of potential and kineticenergies for bulk acoustic waves as in the case of elastic media[2].The expression for the Poynting vector can similarly betransformed by inserting Eqs. (9-10) into Eq. (12), yieldingPj =ω22Re(cijklskulu∗i+ekijskΦu∗i+ejklskulΦ∗−²jkskΦΦ∗),(25)or simplyPj =ω22Re(c¯ijklsku¯lu¯∗i ). (26)Defining the energy velocity as for bulk acoustic waves inelastic media by the ratio of the Poynting vector to the energydensity, we obtain at once(Ve)j =PjW=Re(c¯ijklsku¯lu¯∗i )ρu∗i ui. (27)Furthermore, contracting this expression with sj , the followinguseful relation is obtainedsj(Ve)j = 1, (28)as in the case of elastic media [1], [2].IV. GROUP VELOCITYThe components of the slowness vector can be written si =sni where the ni are the components of a unit vector, i.e.nini = 1. With this notation, the Christoffel equation (19)becomes a generalized eigenvalue equation for the square ofthe phase velocity, V = s−1,c¯ijklnjnku¯l = V2ρilu¯l. (29)The group velocity is defined as [2](Vg)j =∂V∂nj, (30)which implies that the group velocity vector is normal to theslowness surface by construction. The group velocity vectorcan be obtained by differentiating with respect to nj theidentity formed by contracting Eq. (29) with u¯∗i , orc¯ijklnjnku¯lu¯∗i = V2ρilu¯lu¯∗i , (31)which is merely a restatement of Eq. (21). In this equation, u¯lis a function of the unit vector n since it is the eigenvectorassociated with the eigenvalue V 2 of the Christoffel equation(29) for a fixed propagation direction. The differentiation ofEq. (31) with respect to nj results in2c¯ijklnku¯lu¯∗i + c¯iβklnβnk∂u¯l∂nju¯∗i + c¯iβklnβnku¯l∂u¯∗i∂nj= 2V (Vg)jρilu¯lu¯∗i + V2ρil∂u¯l∂nju¯∗i + V2ρilu¯l∂u¯∗i∂nj. (32)This equation can be simplified by considering two differentcontractions of the Christoffel equation (29). First, the con-traction by∂u¯∗i∂njyieldsc¯iβklnβnku¯l∂u¯∗i∂nj= V 2ρilu¯l∂u¯∗i∂nj. (33)Second, complex conjugation of Eq. (29) and subsequentcontraction by ∂u¯i∂njresults inc¯iβklnβnku¯∗l∂u¯i∂nj= V 2ρilu¯∗l∂u¯i∂nj. (34)Permutation of indices i and l, and β and k, respectively, andconsideration of the symmetries c¯iβkl = c¯lkβi and ρil = ρlilead toc¯iβklnβnku¯∗i∂u¯l∂nj= V 2ρilu¯∗i∂u¯l∂nj. (35)Eventually, Eq. (32) simplifies to(Vg)j =c¯ijklnku¯lu¯∗iV ρuiu∗i=c¯ijklsku¯lu¯∗iρuiu∗i. (36)This is identical with the expression (27) for the energyvelocity and incidentally shows that the Re(.) operators inexpressions (26) and (27) can be dropped. We have thus shownthe equality of energy and group velocities for bulk acousticwaves in a lossless piezoelectric medium.LAUDE et al.: EQUALITY OF THE ENERGY AND GROUP VELOCITIES OF BULK ACOUSTIC WAVES IN PIEZOELECTRIC MEDIA 3V. CONCLUSIONWe have demonstrated that the energy and group velocitiesfor bulk acoustic waves are equal in a lossless piezoelectricmedium. Our derivation closely parallels and generalizes forpiezoelectric media the derivation of Ref. [2], that was limitedto purely elastic media. The possible discrepancy pointed outin Ref. [3], that the mechanoelectrical and electromechanicalpower flows calculated from the generalized form of the Poynt-ing vector for piezoelectric media do not compensate eachother, has then no significance for the estimation of the powerflow direction. This observation is moreover supported by theexperimental results of Havlice et al. [5], who demonstratedthat the generalized Poynting vector indeed predicts the correctbeam-steering angle of longitudinal waves along the y axis oflithium niobate. We finally observe that attempting to separatethe generalized energy density or the generalized Poyntingvector into their purely electrical, purely mechanical, mecha-noelectrical and electromechanical parts can be confusing,since the wave displacements are obtained as the solution ofan eigenvalue problem involving the mixed elastic, electricaland piezoelectric properties of the medium.REFERENCES[1] B. A. Auld, Acoustic Fields and Waves in Solids. New-York: Wiley,1973, vol. 1.[2] D. Royer and E. Dieulesaint, Elastic waves in solids, vol. 1. New York:Wiley, 1999.[3] B. D. Zaitsev and I. E. Kuznetsova, “The energy density and power flowof acoustic waves propagating in piezoelectric materials,” IEEE Trans.Ultrason., Ferroelec., Freq. Control, vol. 50, no. 12, pp. 1762–1765, 2003.[4] “IEEE standard on piezoelectricity 176-1987,” IEEE Trans. Ultrason.,Ferroelec., Freq. Control, vol. 43, no. 5, p. 717, 1996.[5] J. F. Havlice, W. L. Bond, and L. B. Wigton, “Elastic Poynting vector ina piezoelectric medium,” IEEE Trans. Sonics and Ultrason., vol. SU-17,no. 4, pp. 246–249, 1970.

Equality of the energy and group velocities of bulk acoustic waves in piezoelectric media

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Equality of the energy and group velocities of bulk acoustic waves in piezoelectric media

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