 Open Journal of Acoustics, 2015, 5, 95-111 Published Online September 2015 in SciRes. http://www.scirp.org/journal/oja http://dx.doi.org/10.4236/oja.2015.53009 How to cite this paper: Zakharenko , A.A. (2015) A Study of New Nondispersive SH-SAWs in Magnetoelectroelastic Medium of Symmetry Class 6 mm. Open Journal of Acoustics, 5, 95-111. http://dx.doi.org/10.4236/oja.2015.53009 A Study of New Nondispersive SH-SAWs in Magnetoelectroelastic Medium of Symmetry Class 6 mm Aleksey Anatolievich Zakharenko International Institute of Zakharenko Waves (IIZWs), Krasnoyarsk, Russia Email: aazaaz@inbox.ru Received 20 July 2015; accepted 14 September 2015; published 17 September 2015 Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract Two additional solutions of new shear-horizontal surface acoustic waves (SH-SAWs) are found in this theoretical report. The SH-SAW propagation is managed by the free surface of a solid when it has a direct contact with a vacuum. The studied smart solid represents the transversely isotropic piezoelectromagnetic (magnetoelectroelastic or MEE) medium that pertains to crystal symmetry class 6 mm. In the developed theoretical treatment, the solid surface must be mechanically free. Also, the magnetic and electrical boundary conditions at the common interface between a vacuum and the solid surface read: the magnetic and electrical displacements must continue and the same for the magnetic and electrical potentials. To obtain these two new SH-SAW solutions, the natural coupling mechanisms such as and present in the coefficient of the magnetoe- lectromechanical coupling (CMEMC) can be exploited. Based on the obtained theoretical results, it is possible that a set of technical devices (filters, sensors, delay lines, lab-on-a-chip, etc.) based on smart MEE media can be developed. It is also blatant that the obtained theoretical results can be helpful for the further theoretical and experimental studies on the propagation of the plate SH- waves and the interfacial SH-waves in the MEE (composite) media. The most important issue can be the influence of the magnetoelectric effect on the SH-wave propagation. One must also be fa- miliar with the fact that the surface, interfacial, and plate SH-waves can frequently represent a common tool for nondestructive testing and evaluation of surfaces, interfaces, and plates, respec- tively. Keywords Piezoelectromagnetic Medium, Magnetoelectric Effect, New Shear-Horizontally Polarized SAWs  A. A. Zakharenko 1. Introductory Part There is single review work [1] concerning the piezoelectromagnetic shear-horizontal surface acoustic waves (SH-SAWs). Surface waves can propagate in a solid and are localized at the solid surface. The solid surface is free when there are no any external perturbations, but the contact with a vacuum above the surface. This report has an interest in an extra study of a smart solid (composite) material such as the transversely isotropic (hex- agonal class 6 mm) piezoelectromagnetic (PEM) continuum concerning any possibility of the surface wave pro- pagation managed by the free surface. The PEM continua are also called the magnetoelectroelastic media. They can simultaneously exhibit evidence of the followin g effects [2] [3]: magnetoelectric (ME), piezoelectric (PE), piezomagnetic (PM). The kno wledg e of PEM smart material features can serve for the creation of intelligent structures and a set of innovative tech- nical devices to record selected environmental and internal changes. Due to the extrinsic ME effect and the coupling interactio n bet ween the electrical and magnetic fields via the elastic field of the ME continuum [3] [4], the electrical polarization can occur upon the application of a magnetic field (direct ME effect) or the magneti- zation can occur upon the application of an electric field (converse ME effect). There is much review work cited in [3]-[45] on the ME effect, ME materials, and their applications because two-phase piezoelectric-piezomagnetic multiferroic composite materials with strong coupling between ferroe- lectric and ferromagnetic phases are frequently found [5] [6] and e xhau sti ve l y st udi ed [14]. The PEM SH-SAWs can be quite suitable for analyzing high-frequency technical devices and readily generated with the following nonco nta c t t ec hn iq ue [46]-[4 8] : electromagnetic acoustic transducers (EMATs). The use of this noncontact tech- nique can be preferable compared with the other traditional technique that uses the PE transduction [47] [48]. Two-phase PEM composite media possessing the PE and PM phases can be exploited in different technical devices because the ME coupling in such co mposites represents a product proper ty resulting from the mechani- cal interaction between the mentioned phases. Experimental studies of the ME effect in the two-phase compo- sites were begun in the 1970s with pioneer works [49]-[52] to synthesize the BaTiO3-CoFe2O4 composites by a unidirectional solidification process. As a result, it was found that the obtained composites possessing the PE phase BaTiO3 and PM phase CoFe2O4 can have two orders larger value of the ME coefficient than that of the pioneer single-phase ME Cr2O3 crystal. References [53] [54] provide the material characteristics of different BaTiO 3-CoFe2O4 hexago nal (6 mm) co mposite s perta ining to the (0-3) connecti vity whe n the PE p hase ser ving as the 3-D matrix contains the PM phase as 0-D inclusions. The reverse co nnectivity is also possible. Also, PE M composites can have the (2-2) co nnecti vit y. In thi s case, they represent a multi-layered (sandwich-like) struc tur e composed of linear homogeneous PE and PM layers with a perfect bonding at each interface. The study of such PEM laminated (composite) structures is up to date [55]. The PEM systems can demonstrate significant interactions between the elastic, magnetic, and electric fields. This allows direct applications in sensing and actuating devices. The material parameters of the freque ntl y used BaTiO 3-CoFe2O4 and PZT-5H-Terfenol-D laminated composites can be found in [56]-[59]. It is well-k no wn tha t the ME effect in the single-phase PEMs (for instance, Cr2O3, LiCo P O4, and TbPO4 [3]) is usua lly ve ry s mall . I n add it io n, no ne o f t he si ng le -phase ME materials can have c ombined large and robust magnetic and electric pola- rizations at room temperature. However, it is a pleasure to state that the Sr3Co2Fe24O41 Z-type hexaferrite [5] was discovered in 2010. It actually possesses the realizable ME effect apt for practical uses. It is thought that the first theoretical work on the propagation of the PEM SH-SAWs managed by the free surface was written by Melkumyan [60]. He has discovered several new SH-waves corresponding to different boundary conditions. Theoretical works [60] [61] are relevant to the studies of the SH-SAWs directed by the free surface of the hexagonal PEMs of symmetry class 6 mm and book [62] studies the PEM SH-SAW existence in cubic crystals. The theoretical work presented in this paper belongs to the existence of extra new SH-SAWs along the free surface of the aforementioned hexagonal medium. This paper has the purpose to discover some extra solutions (new SH-SAWs) for a set of the magnetic and electrical conditions applied at the vacuum-solid interface when the solid surface is the mechanically free. Continuity of both the electrical displacement and magnetic flux and continuity of both the ma gnetic and electrical potentials are treated here at the interface. Var- ious boundary conditions for the case when a medium simultaneously possesses both the PE and PM properties are perfectly described in paper [63]. The reader can find the comprehensive theory of SH-wave propagation in the transversely isotropic media in books [61] [64] [65], of which books [64] [65] study the interfacial and p late anti-plane polarized waves, respec-  A. A. Zakharenko tively. It is necessary to briefly review the theory developed in book [61] for the S H-SAW propagation along the free surface of the hexagonal (6 mm) PEMs. For a PEM medium, acoustic wave propagation coupled with both the electrical and magnetic potentials requires suitable thermodynamic functions and thermodynamic variables. The mechanical stress, electrical induction (D), a nd mag netic flux ( B) can be chosen as the appro priate thermo- dyna mic funct ions [61] in the c ase of linear ela sticity. As a result, the thermodynamic variables are the mechan- ical strain, electrical field (E), and magnetic field (H). In such thermodynamic treatment, all the material con- stants can be thermodynamically determined. The electrical field components (Ei) and the magnetic field com- pone nts ( Hi) can be defined by the electrical potential φ and magnetic potential ψ, respectively: Ei = – ∂φ/∂xi and Hi = – ∂ψ/∂xi, where xi represent the real space components and the index i runs from 1 to 3. Exploitation o f the equilibri um equatio ns and the cor responding Max well equatio ns written in the for m of the quasi -static approximation [66] [67] can constitute the coupled equations of motion representing partial second derivatives. Using the plane wave solution, the coupled equations of motion can be then written in the tensor form representing the Green-Christoffel equation [61]. The Green-Christof fel equation repr esenting a polyno mi- al is the main equation to study acoustic wave propagation co upled with both the electrical and magnetic po ten- tials. To resolve this equation means to determine the eigenvalues and corresponding eigenvectors. The re a re high s y m met r y pr o paga tio n d ir ec ti ons [67] [68] in which “pur e” wa ves wit h the in-lane pola rizatio n and “p ure” waves wit h the ant i-pale polarization (shear-horizontal polarization) can exist. When the pure waves with the a nt i-plane polarization are c oupled with both the ele c trical and magnetic potentials, the pure waves with the in-lane polarization represent purely mechanical waves, and vice versa. The appropriate cuts and propaga- tion directio ns for material s with an assor tment o f symmetr y classes are listed in works [67] [68] . It is ce ntral to state that each symmetry class has its own set of the material constants [69] [70]. For materials of s ymmetr y class 6 mm, the suitable propagation directions are mentioned in review paper [71] and review paper [1] exhibits the coordinate system that is fitting for PEMs, pure PEs, and pure PMs. It is ne- cessary to mention that in the pure PEs (or PMs) the surface Bleustein-Gulyaev waves [72] [73] can propagate. Using the rectangular coordinate system (x1, x2, x3), it is necessary to clarify that the SH-SAW propagation di- rection, sixfold symmetry axis of the PEM material, and the surface normal must be managed along the x1-, x2-, and x3-axes, respectively. Consequently, such propagation directions can support the coupling of the elastic SH-waves with both the electrical and magnetic potentials. For the case of the SH-wave propagation, the Green-Christoffel equation is simplified and all the apt eigenvalues and the corresponding eigenvectors in the form of can be analytically determined. The following section starts with the Green-Christoffel equation for the case of the SH-wave propagation in the suitable direction when there is the coupling of the SH-waves with both the magnetic and electrical poten- tials. It also develops the theory leading to extra two solutions of new SH-SAWs (see final formulae (60) and (66 ) c or r espo nd ing t o t he c o up l ing mec ha ni s m such as [74]) for the certain set of the boundary condi- tions mentioned above. It is also worth noting that the eighth and ninth new SH-SAWs corresponding to the other coupling mechanism such as [74] were discovered in theoretical work [75]. Consequently, one has to look through the further analysis to be familiar with the explicit forms of the discovered tenth and ele- venth new SH -SAWs. These extra two new solutions are possible because only one of three suitable eigenvalues depends on the phase velocity and all the apt corresponding eigenvector components do not depend on the ve- locity that will be demonstrate d in the following section. 2. Theory Leading to Two New Results In the appropriate propagation directions mentioned in the previous section, the SH-wave propagation can be coupled with both the magnetic and electrical potentials. In the case of surface wave, the anti-plane polarized SH-waves can be localized at the free surface of the transversely isotropic PEM continuum of symmetry class 6 mm. T he treated SH-waves propagates along the x1-axis of the rectangular coordinate system (x1, x2, x3). In t his case there is only the single mechanical displacement component U = U2 directed along the x2-axis. These propagation directions allow the existence of the following independent nonzero material constants: the elastic stiffness constant C, piezomagnetic coefficient h, piezoelectric constant e, dielectric permittivity coefficient ε, magnetic permeability coefficient μ, and electromagnetic constant α. They are defined by the following equali- ties: , , , , , and [61]. The propagation direction can be defined by the directional cosines (n1,n2,n3) respectively directed along the  A. A. Zakharenko (x1, x2, x3) axes, where n1 = 1, n2 = 0, and n3 ≡ n3. They are coupled with the compone nts ( k1,k2,k3) o f t he wa v e- vector K as follows: () () 123 123 ,, ,, kk kknn n = , where k is the wavenumber in the pr opagation direc tion. All the suitable eigenvalues n3 and the corresponding eigenvectors () () 000000 245 ,, ,,U UUU ϕψ = must be disclosed. When only the SH -wave propagation i s coupled with both the electrical p otential φ and the magnetic potential ψ, the corresponding coupled equations of motion representing the partial differential equations of the second order are written and they have the plane wave fo rm solutio ns [61]. Substituting the solutions in to the differential form for the coupled equations of motion, the tensor form of the coupled equations of motion can be expressed by three homogeneous equations. The differential and tensor for ms of the coupled equations of motion are identical and therefore, it is possible to start the analysis with the tensor form known as the modified Green-Christoffel equation. Using the corresponding nonzero Green-Christoffel tensor components [61], the following three ho- mogeneous equations natur ally written in the matri x form can be compo s ed: ( ) 2 0 4 0 0 0 0 0 ph t C mVVemhmU emm m hmm m ε αϕ α µψ − −− = −− (1) where ; ρ and Vph are the PEM mass density and the phase velocity, respectively. The velocity Vph is defined by Vph = ω/k, where ω is the angular frequency. Also, Vt4 stands for the speed of the shear-horizontal bulk ac o us tic wave ( SH-BAW). This velocit y is uncoupled with bo th the electrical a nd magnetic po tentials. T his speed of the purely mechanical SH-wave is defined by (2) All the appropriate eigenvalues n3 can be determined when the determinant of the coefficient matrix in Equa- tions (1) va nishes. E xpandi ng the ma trix det ermina nt, the follo wing se cular equation consisting of three factors can be readily obtained: ( ) ( ) 2 24 10 emph t mmKmV V ×× +−= (3 ) In expression (3), stands for the coefficient of the magnetoelectromechanical coupling (CMEMC) de- fine d by the following formula: ( ) () () ( ) 22 2 22 2 em eeh heh e heh KCC µα αε µε α εµ αεµ α −− − +− = = −− (4) It is obvious that CMEMC (4) can be represented as the material parameter depending on the following three different coupling mechanisms [74] that naturally contain the electromagnetic constant α: (5) (6) (7) It is apparent that equality (3) is satisfied as soon as one of three factors on the left-hand side is equal to zero. Thus, the first and second factors can become equal to zero for the following identical eigenvalues [61] [64] [75]: (8) The third facto r in equation (3) provides the following suitable eigenvalue [61] [64] [ 75] : (9) where the velocity Vtem represents the speed of the SH-BAW co upled with both the electrical and magnetic po- tentials and is defined as follows:  A. A. Zakharenko (10) Found sui ta bl e e igenval ue s ( 8 ) a nd (9 ) p o sses s nega ti ve i ma gina r y p a rt s to have wave damping towards depth of the PEM material. Using the m in equations (1), it is possible to reveal the corr esponding eigenvecto rs [61]. It is natural to use the first equation in matrix form (1) to demonstrate the dependence of the eigenvector compo- nent U0 on both the components φ0 and ψ0. Usi ng this U0 for the second and third equations, it is possible to get two homogeneous equations. Therefore, one can write (11) 200 0 me meh AA εϕαψ +++ = (12) 2 00 0 meh mh AA αϕ µψ +++ = (13) (14) It is natural to utilize Equations (11) and (13) to obtain the eigenvector components such as U0, φ0, and ψ0. Using for eigenvalues (8), it is possible to have the following eigenvector components [61] discussed in [76]: ( )( )( ) ( ) ( )( )( ) ( ) ( ) 01 0101030303 ,,,,0, ,UU ϕ ψϕψµα == − (15) For eigenva lue (9) wit h m ≠ 0, the corresponding eigenvector components are as follows: ( )( )() ( ) ( ) () () ( ) 2 05 0505 22 2 22 22 2 ,,, , 1,, . em emem em mem em e hheh UCK CKCK ehCKK KK K α µα ϕψµ α µα µα − =−+ − =−−− − (16) whe r e ( )( )()( ) 0101 0505 e heheh ϕ ψϕψµα +=+=− . (17) In expression (16), the nondimensional parameters and are defined by (18) (19) where the last is called the coefficient of the magnetomechanical coupling (CMMC). The obtained eigenvalues and the corresponding eigenvectors are employed to compose the complete me- chanical displacement UΣ, complete electrical potential φΣ, and complete magnetic potential ψΣ. Using the weight factors F(1), F(3), and F(5), these parameters are also written in the plane wave forms as follows: ( )()( ) ( ) 0113 3 1,3,5 exp j pp pph p UFUk nxnxVt Σ = = +− ∑ (20) ( )( )( ) ( ) 0113 3 1,3,5 exp j pp pph p Fk nxnxVt ϕϕ Σ = = +− ∑ (21) ( )( )( ) ( ) 0113 3 1,3,5 exp j pp pph p Fk nxnxVt ψψ Σ = = +− ∑ (22)  A. A. Zakharenko where t is time and j is the imaginary unity, j = (–1)1/2. These weight factors can be found when the boundary conditions are applied. The mechanical, electrical, and magnetic boundary conditions are perfectly described in theoretical work [63]. For the further analysis, let’s use F1, F2, and F3 instead of F(1), F(3), and F(5), respectively. It is worth mentioning that in this case there are two identical eigenvalues (8). This fact allows the utilization of F = F1 + F2 that will be naturally used below and plays a crucial role in the discovery of new SH-SAWs. Therefore, it is possible to state that this three-partial wav e with F1, F2, and F3 looks like a hidden two-partial wave with F and F3. At the interface between the PEM medium and a vacuum (x3 = 0) the mechanically free surface requires that the nor mal compo nent of the stress te nsor must vani sh, namel y σ32 = 0 [61]. The electrical boundary conditions at the interface include the continuity of both the electrical induction (D3 = Df) and the ele ctrica l pote ntial (φ = φ f) where the superscript f relates to a vacuum. Besides, the magnetic boundary conditions at the interface include the continuity of both the magnetic flux (B3 = Bf) and the magnetic po te ntial ( ψ = ψf). For the PEM medium, the parameters corresponding to the mechanical, electrical, and magnetic boundary conditions can be expressed as follows: ( )()( )( )( )( ) ( )()( )( )( )( ) ( )()( )( )( )() 1011 01101 32 1333 3033 03303 233 3 5055 05505 333 3 . FCk Uekhk FCk Uekhk F Ck Uekhk σ ϕψ ϕψ ϕψ = ++ + ++ + ++ (23) ( )()()()()() ( )()()( )( )( ) ( )()( )( )( )( ) 01 1101101 3 1333 03 33 03303 23 3 3 05 55 05505 33 3 3 . DFek Ukk Fek Ukk FekUkk εϕαψ εϕαψ εϕαψ = −− + −− + −− (24) ( )()() 0103 05 12 FF F ϕϕϕ ϕ =++ (25) ( )()( )( )( )( ) ( )()() ()()() ( )()( )( )( )( ) 01 11 01101 3133 3 03 33 03303 2333 05 55 05505 333 3 . BFhkUkk Fhk Ukk Fhk Ukk αφ µψ αφ µψ αφ µψ = −− + −− + −− (26) ( )()() 01 03 05 12 3 FF F ψψ ψ ψ =++ (27) where F1 = F(1), F2 = F(3), and F3 = F(5). The corresponding vacuum parameters read: (28) (29) ( 30) (31) where FE and FM are the electrical and magnetic weight factors, respectively. In order to elucidate the vacu um material par ameters, it is esse ntial to state that the elastic constant C0 of the free space (vacuum) is as high as C0 = 0.001 Pa [77 ]. It is clearly seen that it must be multiplied b y a factor of 1013 in order to be comparable with the corresponding material parameter of a solid. Thus, the neglec t of this vacuum parameter is understandable in the calculations. However, the other two material parameters must be account ed. They are the magne tic permeabil ity constant, [ ] –7 62 0 4π10H m1.2566370614410NA µ − =×= × , and the dielectric permittivity c onstant, ( ) [ ] –7 210 0 10 4π0.0885418781710F m L C ε − = =× where is the speed of light in the free space. For the magnetic and electrical potentials in  A. A. Zakharenko a vacuum, it is natural to write the corresponding Laplace equations of types ∆ψf = 0 and ∆φf = 0, where ∆ de- notes the differential operator called the Laplacian that forms a vector field from a scalar one. Utilizing the equations corresponding to the mechanical, magnetic, and electrical boundary conditions written above, the following ma t rix for m of three homogeneo us equa tions [61] can be inscribed: ()()()()( )( )()( )()( )( )( ) ( )( )( ) ( ) ( )()()()( )( ) ( ) ( )()()( )( )( ) ( ) ( )()( ) ( )( )()()( ) ( ) ( )()( )( )( )( ) ( ) ( ) 10101013030303505 0505 33 3 01 0305 110110133033 0355055 05 33033303 3303 01 03 11 0110133 03303 3330 3330 jjj jj n CUehnCUehnCUeh en UnnenUnnenUnn hn UnnhnUnnhn ϕψϕψ ϕψ εεϕαψεεϕαψεεϕαψ α ϕµµψαϕµµψ ++++ ++ −− −−−−−−− − −−−−− ( )( )()( )( ) ( ) ( ) 05 55 05505 3 330 1 2 3 j 0 0. 0 Unn F F F α ϕµµψ − −− ⋅= (32) where the corr espond ing eige nvalues n3 are used instead of k3 = kn3. The vacuum material parameters ε0 and μ0 are already accounted in equations (32) because the vacuum weight factors FE and FM can be naturally e xcluded; see the boundary conditions written above. It is natural to use expressions (15) and (16) to simplify equations (32). As a result, the following equalities can be used to significantly simplify equations (32): ()( )()()()( ) 01 0101030303 CUehCUehe h ϕ ψϕψµα ++ =++ =− (33) ( )( )( ) ( ) 2 05 0505 2 1 em em K CUehe hK ϕ ψµα + ++ =− (34) ( )( )( )( )( )( ) 01 03 01 0103032 eU eU εϕαψεϕαψεµ α − −=−−=−+ (35) ( )()( ) 22 05 05 052 22 2 0 em emem eehheh eU CK CKCK µα εα εϕ αψεµα − −−=+−−+ = (36) ( )( )()( )( )( ) 01 03 01 010303 0hU hU αϕµψαϕµψαµ µα −−=−−=−+ = (37) ( )( )( ) 22 05 0505 22 2 0 em emem eh hheh hU CK CKCK µα αµ αϕ µψαµαµ − −−=+−−+= (38) The exploitation of equalities from (33) to (38) and eigenvalues (8) and (9) for the matrix form (32) allows one to rewrite three homogeneous equations (32) in the following simplified forms: () 2 2 123 2 110 ph em tem em V K e hFFFV K µα + − ++−= (39) ( ) ( ) 22 2 1 20302 0 em m em KK FFF K εεµαεµ − ++ −+= (40) 22 123 2 0 em em KK FFFK α α − − ++= (41) It is apparent that three homogeneous equations written in matrix form (32) with their simplified forms ob- tained in equations from (39) to (41) are identical. Therefore, it is possible to compose the determinant of the coefficient matrix called the determinant of the boundary conditions that must vanish to obtain a certain phase velocity Vph satisfying the boundary conditions. Analyzing equations from (39) to (41), it is possible to reveal that the composed determinant will have two identical columns such as the first and second ones. This peculiar- ity allows the conclu sion such that the matrix deter minant will b e equal to zero at any value of the phase veloci- ty Vph. This means that there is uncertainty for the velocity Vph. However, the value of the suitable SH-SAW speed must not exceed the value of the SH -BAW speed Vtem. Indeed, all the apt SH-SAW speeds sati sfying t he  A. A. Zakharenko boundary conditions must be disclosed. To avoid this uncertainty for the phase velocity Vph, it is natural to use F = F1 + F2 and to rewrite equations from (39) to (41) as follows: ( ) 2 2 32 110 ph em tem em V K e hFFV K µα + −+− = (42) ( ) 22 2 030 2 0 em m em KK FF K εεµαεµ − + −+= (43) 22 32 0 em em KK FF K α α − −+= (44) It is visible t hat the set o f equations from (42) to (44) is the same to t he set of equations from (39) to (41) be- cause F = F1 + F2 was used. There is already no uncertainty of the phase velocity Vph for the set of equations fro m (42 ) to (44) . Thi s is tr ue b eca use t hey re pre sen t thre e h omo geneo us e qua tio ns in t wo unk nown weig ht fa c- tors suc h as F and F3. It is well-known t hat s uch ne w syste m of eq uatio ns can ha ve a c ertai n soluti on o f the ve- locity Vph when three equations from (42) to (44) are dependable from each other. This happens as soon as one equation represents a sum of two others. Therefore, three equations from (42) to (44) must be properly trans- formed into a set of suitable consis te nt equations. In order to obtain three consistent equations, it is natural to treat equation (43) as the main one and multiply Equations (42) and (44) by and α, respectively. So, one can get the following three ho- mogeneous equations that are already consistent and have the same dimension: ( ) 2 2 03 2 110 ph em tem em V K FF V K εεµ + ++− = (45) ( ) 22 2 030 2 0 em m em KK FF K εεµαεµ − + −+= (46) 22 2320 em em KK FF K α α − −+ = (47) So, three equations from (45) to (47) in two unknowns F and F3 are consistent because the left-hand side of main equation (46) can become equal to zero as soon as equations (45) and (47) are successively subtracted from equatio n (46 ). This subtr actio n leads to the certain velocity of new SH-SAW. Also, main equation (46) can be used to d isclose the explicit for ms of the weight facto rs F and F3. It is also conve nient to re write three equa- tions from (45) to (47) as the following set of two equations in two unknowns F and F3: ( )( ) 2 22 2 22 03 0 22 110 ph em em tem em em V KK K FF V KK α εεµααεεµ −+ +− +++−= (48) ( ) 22 2 030 2 0 em m em KK FF K εεµαεµ − + −+= (49) where Equation (48) represents a sum of equations (45) and (47). It is clearly seen in the first term of equatio ns (4 8) and ( 49) that the factor at F such as can be interpreted as coupling mechanism (7) of CMEMC (4) such as (εμ – α2) when the vacuum electrical parame- ter ε0 must be also included. To determine the SH-SAW velocity, it is necessary to subtract equation (48) from Equation (49), or vice versa, because they must be identical. So, the value of the velocity Vnew2 of the new SH-SAW recently discovered in book [61] (the new SH-SAW velocity is defined by expression (120) in the book, see also papers [1] [78]) can be calculated with the following explicit formula:  A. A. Zakharenko ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12 2 2222 22 0 2 2 0 12 2 2 00 222 0 1 11 12 em mLem new tem em tem KKC KK VV K ehe h VCCeheh α µ αµ ε ε µαεµ ααεε ε εµεµαµεα −+ − = − ++ −+− + = − +− ++− (50) In formula (50) there are t wo possib ilitie s when Vnew2 = Vtem occurs: and . Also, the speed of light CL in a vacuum is defined by (51) It is worth mentioning that the SH-SAW defined by expression (50) represents one of the seven new SH- SAWs recently discovered in book [61]. This new SH-SAW can propagate along the free surface of the hex- agonal PEM medium of symmetry class 6 mm. Also, it is natural to demonstrate the case when the piezomag- netic and electromagnetic constants vanish, namely h = 0 and α = 0. In this case, the PEM SH-SAW d efine d b y expression (50) reduces to the well-known velocity VBGpe of the surface Bleustein-Gulyaev waves [72] [73] propagating in a purely piezoelectric solid. The velocity VBGpe is defined by ( ) ( ) 12 2 2 20 111 e BGpe te e K VV K εε = − ++ (52) In definition (52), Vte and respectively stand for the SH-BAW velocity coup led with the e lectric al po ten- tial and the coefficient of the electro mechanical coupling (CEMC). They represent very important material cha- racteristics of a pure piezoelectrics and read as follows: (53) (54) In formula (53) , the velocity Vt4 is defined by expression (2). Formula (50) for the new SH-SAW velocity discovered in book [61] is given in this paper for comparison with the other new results obtained below. The main purpose of this paper is to discover additional new SH- SAWs that can propagate in the PEM using the same set of the boundary conditions at the vacuum-solid inter- face: σ32 = 0 , φ = φf, D = Df, ψ = ψf, and B = Bf. Therefore, two new solutions for new SH-SAW propagation are obtained in subdivisions (i) and (ii) belo w. Also, it is necessary to state t hat this theoreti cal study can be useful for constitution of a set of technical devices based on smart PEM solids and the further researches on the propa- gation of the plate and interfacial SH-wa ves. (i) Similar to the theory developed above for the PEM SH-wave propagation (see also in book [61]) it is possible to start with the analysi s of three homogeneo us equation s fro m (42) to (44) in two unknowns F and F3. The second possibility for coupling mechanism (7) of CMEMC (4) such as (εμ – α2) can be also treated when there is also the coupling with the vacuum electrical constant ε0. Therefore, Equation (43) with the factor at F such as is the main equation that is not changed for this case. This main equation must couple Equations (42) and (44) together forming a system of three homogeneous equations in two unknown weight factors F and F3. It is flagrant that these three equations can become dependent on each other as soon as Equation (42) is multiplied by and Equation (44) is multiplied by –ε0μ/α. Accordingly, three equations from (42) to (44) can be rewritten as follo ws:  A. A. Zakharenko ( ) 2 2 232 110 ph em tem em V K FF V K εµ α + −+−= (55) ( ) 22 2 030 2 0 em m em KK FF K εεµαεµ − + −+= (56) 22 03 2 0 em em KK FF K α εµ − += (57) It is fl agra nt tha t it is co nve nient to cop e with t he set o f two homoge neou s equat ions i n two unkno wns F and F3 instead of three equations from (55) to (57) written above. A sum of Equations (55) and (57) leads to the fol- lowing convenient set of equations: ( ) ( ) 2 22 2 22 030 22 110 ph emem tem em em V KK K FF V KK α εεµα εµεµα −+ + −++−−= (58) ( ) 22 2 030 2 0 em m em KK FF K εεµαεµ − + −+= (59) Therefore, the velocity Vnew10 of the t ent h new anti -plane polarized SAW propagating along the free surface of the PEM medium is obtained by a subtraction of Equation (58) from Equation (59). Also, the veloc ity Vnew10 can be obtained by a successive subtraction of Equations (55) and (57) from main Equation (56) . Thus, this reads: ( ) 12 2 22 0 10 22 12 2 0222 11 12 m new tem em tem KK VV K ee h h VeCCeheh α εµ εµ α µα ε αεµα µεα − =− −+ − = −− − ++− (60) In expression (60), the following equality results in Vnew10 = Vtem. So, it is possible to state that the ne w SH-S AW d i sco vered in this paper can propagate with the velocity Vnew10 expressed by formula (60). This is the new solution that was not considered in book [61]. Concerning the SH-wave propagation in the PEM plates [65], solution (60) discovered in this paper is more preferable and con- venient than solution (50) discovered in book [61]. Also, solution (60) looks like simple one compared with so- lution (50). However, one can find a very interesting pec uliarity: results in the fact that the ne w SH-SAW defined by expression (60) cannot exist because the expression under the square root in formula (60 ) ca n have a negati ve s ig n re sult i n g i n an i ma gi nary va lu e o f t he SH -S AW ve lo ci t y. This p ec uli ar it y d o es no t exist for solution (50) allowing SH-SAW propagation for very small values of the electromagnetic constant α, even fo r α = 0. (ii) Consider the second case that also leads to discovery of an extra new SH-SAW. However , it is essentia l to clarify why this case must be considered. The author of this paper has understood that this case can be possible after the study of the interfacial SH-wave propagation directed by the perfectly bonded interface between two dissi milar P EMs o f symme tr y class 6 mm [64]. T his case is relevant to coupling mechanism (5) of CMEMC (4) such as (eμ – hα). Let’s start with the analysis of three equations from (42) to (44). It is indispensable to treat Equation (42) as the main equation that couples Equations (43) and (44) in a set of three homogeneous equations to make them consistent. For this purpose, it is necessary to multiply the left-hand side of Equation (43) by and the o ne of Equation (44) by the piezomagnetic constant h because it already has the factor such as the constant α. As a r esult, three equations from (42) to (44) can be rewritten as follows: ( ) 2 2 32 110 ph em tem em V K e hFFV K µα + −+− = (61)  A. A. Zakharenko ( ) 22 0 322 0 0 em m em KK e FFK εµ µεεµα − += +− ( 62) 22 32 0 em em KK h FFK α α − −+ = (63) The set of three equations from (61) to (63) can be rewritten as a set of two corresponding equations for con- venience. Indeed, a sum of Equations (62) and (63) allows one to compose the following apt set of two homo- geneous eq uations: ( ) 2 2 32 110 ph em tem em V K e hFFV K µα + −+− = (64) ( ) ( ) 2222 0 32 22 0 0 emem m em em KK KK FehFhe KK α εµ µα αµ εεµα −− − +−+= +− (65) These t wo equations lead to the following explicit form for the velocit y Vne w11 of the eleventh new SH-SAW: ( ) ( ) ( ) ( ) ( )() 12 2 22 22 0 11 2 22 0 12 2 2 00 2 222 0 111 12 emem m new tem em em tem KK KK he VV eh eh KK eeh heh VCCe heh α εµ αµ µα µα εεµα εµµαεµεµααε εµε µαεµαµεα −− = −−+ −− + +−+ −−+−− = − +−− ++− (66) It is obvio us that the followi ng equality Vnew11 = Vtem occurs in expression (66) as soon as one treats the case of ( ) ( ) ( ) 2 00 0eeh heh εµµαεµεµ ααε −−+−−= . This equatio n ca n be a lso rewritten as . For the case of a very small value of the constant α (α → 0) formula (66) reduces to the following for m: ( ) () 12 2 22 11_ 0022 22 0 1111 me new tem em em KK VV KKKK εε = −−+ ++ ++ + (67) whe r e ( ) 12 22 04 1 temte m V VKK= ++ (68) (69) With h = 0, formula (67) reduces to formula (53) for the velocity VBGpe of the surface Bleustein-Gul yaev wa ve [72] [73] existing in a purely piezo electr ic solid . So, it is possib le to co ncl ude that the con sideration of CMEMC coupling mechanism (5) such as (eμ – hα) results in the ne w SH-SAW propagating with the velocity Vnew11 de- fined by explicit form (66). Solutions (50), (60), and (66) are true because they are based on the natural coupling mechanisms of the CMEMC. 3. Numerical Results and Discussion Comparative numerical calculations are listed in Table 1 for the different BaTiO3-CoFe2O4 composites. For comparison, the table provides results of the calculations of the propagation speeds for six known SH-SAWs that can exist for the same set of the electrical and magnetic boundary conditions: the velocities Vnew1, Vnew2, Vnew8, Vnew9, Vnew10, and Vnew11 of the first, second, eighth, ninth, tenth, eleventh new SH-SAWs. The reader can find that all the speeds of the studied new SH-waves are slower than the SH-BAW speed Vtem. The interesting issue is the exi ste nce of the ei ghth and te nth new SH-SAWs. The eighth can exist only for the 20% volume fraction  A. A. Zakharenko Table 1. The material and wave characteristics of the piezoelectromagnetic composites consisting of BaTiO3-CoF e2O4 of class 6 mm. Following the results given in papers [53] [54] , the material constants are given as percentage volume fraction (V F) of B aTiO3 in the composites consisting of BaTiO3-CoFe2O4. It is worth noting that the magnetic per meability of a va- cuum is () () –1 71 7 0 4π10H m~12.56637110VsA m µ −− − = ×⋅×⋅⋅⋅ ; 10–12 N⋅s/(V ⋅C) = ps/m; F = C/V; T = Tesla = N⋅(A⋅m)–1. The mass densi ty is assumed the same ρ = 5730 [kg/m3]. Comp os ite VF 0% 20% 40% 60% 80% 100% C, 1010 [N/m2] 4.53 4.50 4.50 4.50 5.00 4.30 e [C/m2] 0 0.1 0 .2 0 .3 0 .4 1 1.6 h [T] 5 60 340 220 1 80 80 0 ε, 10–10 [F/m] 0 .8 3.3 8 .0 9.0 1 0.0 112.0 μ, 10–6 [N/A2] –590 –390 –250 –150 –80 5.0 α, 10–12 [N⋅s/V⋅C] 0 2.8 4 .8 6.0 6 .8 0 - −0.00591346 104 −0.00319 106 4178 −0.00257767 111 0.0016 001 08799 - Vt4 [m/s ] 2 811.7 181868 6 2802 .3923 9604 2802 .3923 9604 2802 .3923 960 4 2953.9809 563 4 2739.4 092 4320 Vtem [m/ s] - 2794.0941 9095 2797.9 175162 9 2798 .7782 425 5 2956.3433 571 5 - Vnew1 [m/ s] - 2 794.0 941903 1 2797.9 175156 2 2798 .7782 4023 2956 .3433 5618 - Vnew2 [m/ s] - 2 794.0 941905 1 2797.9 175160 8 2798 .7782 4189 2956 .3433 5598 - Vnew8 [m/ s] - 1 354.5 114292 7 does not exist does not exis t does not exist - Vnew9 [m/ s] - 2 794.0 935086 6 2797.9 157093 6 2798 .7710 4004 2956 .3280 2670 - Vnew10 [m / s] - does not exist does not exist does not exist 1 640.1 049 3727 - Vnew11 [m / s] - 2794 .0331 806 4 2797.8916 0405 2 798.7 461259 8 2956.3 397166 2 - of B aTiO 3 in the composites c onsisting of BaTiO3-CoFe2O4. On the other ha nd, the tent h can exi st only for the 80% volume fraction. This fact can be explained by a strong dependence of the nondimensional parameter α2/εµ. This interesting found moment by the numerical study can be researched in the future more widely because this short report theoretically predicts and numerically demonstrates the existence of the new SH-SAWs. Let’s continue our debates on the following problem: how many surface SH-waves can exist in the trans- versely isotropic piezoelectromagnetics. This is an important question because there is an opinion that in this particular case, an infinite number of analytical solutions obtained in explicit forms can be found. For today, the author can certainly state that as many as fourteen different SH-SAWs can propagate treating different sets of the electrical and magnetic boundary conditions. Indeed, the author of this theoretical report can agree only with three SH-SAWs of twelve theoretically discovered by Melkumyan [60]. Seve n ne w SH-SAWs were theoretical- ly discovered in book [61], two new SH-SAWs were discovered in paper [75], and extra two new SH-SAWs were discovered in this report. The number of possible SH-waves is big but it does not approach an infinity. The reader can check the relatively simple mathematics used above in this report to analytically find possible extra solutions. Concerning the boundary conditions used in this report, they represent the most complicated case. Therefore, more possibilities can exist and as many as six new SH-SAWs were discovered: the first and second in book [61], the e ight h a nd ni nt h i n pa p er [7 5], and t he te nt h a nd e l eve nt h i n t hi s re p or t. T he e xis te nce o f s i x SH -SAWs for the same set o f the bounda ry conditio ns at the solid -vacuum boundary (σ32 = 0, φ = φf, D = Df, ψ = ψf, and B = Bf) can be naturally explain by the existence of the different possible coupling mechanisms of the CMEMC such as , , and . It is possible to assume that the first two mechanisms represent ex- change ones and only the third can relate to the magnetoelectric effect. Therefore, different solutions correspond to different CMEMC coupling mechanisms. The other important factor that influence of the number of possible SH-SAWs is the fact that there are two different sets of the eigenvector components for the same eigenvalue.  A. A. Zakharenko Utilization of two different sets of the eigenvector components can frequently lead to two different solutions. This fact certainly increases the number of possible SH-SAWs that can propagate in the transversely isotropic piezoelectromagnetics. It is also discussed that so many solutions for the transversely isotropic case are possible because only one of three suitable eigenvalues defined by formulae (8) and (9) depends on the phase velocity and all the apt corresponding eigenvector components do not depend on the velocity. The different picture there is for the problem of SH-SAW propagation in the cubic piezoelectromagnetics. It was numerically found in book [62] that two different sets of the eigenvector components lead to the same result for the propagation ve- locity. As a res ult, only seven po ssible new SH-SAW s were numer icall y found in [62]: two different sets of the eigenvector components actually lead here to the same result for the propagation velocity for each possible set of the electrical and magnetic boundary conditio ns. The interesting issue is that the solutio ns can be obtained only by an analytical method for the transversely isotropic case but the solutions (after expanding the determinant of the boundary conditions) can be ob tained only b y a numeric al method for the cubic case. The author will be glad if one can find analytical solutions for the cubic case. Also, the solution for the surface Bleustein-Gulyae v- Melkumyan wave can be also found analytically in the cubic piezoelectromagnetics [62]. Finally, some inco rrec t solutio ns obta ined in pap ers [79] [80] and me ntioned in r evi ew [1] for the propagation problem of SH-wave managed by the PEM free surface are also exist. It is possible to discuss them in a few words. These incorrect solutions pertain to the same set of the boundary conditions at the solid-vacuum boun- dary: σ32 = 0, φ = φf, D = Df, ψ = ψf, and B = Bf. T he authors of theoretical articles [79] [80] have use d t he o ther theoretical methods leading to the other forms [1] that are different from formulae (50), (60), and (66). Moreo- ver, their results discussed in review [1] even differ from each other. Review paper [1] has also exhibited that their results are incorrect because they can definitely mix two different eigenvectors for the same material. The eigen vect or mixi ng is p ossi ble whe n one eige nvec to r is use d for o ne material and the second for the second ma- terial for the pr oblem of the interfacia l SH-wave propagation along the common interface between two dissimi- lar piezoelectromagnetics that was analytically demonstrated in paper [81]. Besides, the authors of papers [79] [80] did not demonstrate that they have found suitable e igenvectors. It is worth noting that to find all the suitable eigenvalues and the corresponding eigenvectors is the main mathematical procedure to resolve the coupled equ- ations of motion. Therefore, the authors of papers [79] [80] did not find any true solutions for the coupled equa- tions o f motion and the aut hor o f this pap er cannot agr ee with the ir solut ions. T heir metho d can be use d for the study of this particular and particularistic case and cannot be used for the other cases, for instance, for the prob- lem of the SH-S AW existence in the cubic piezoelectromagnetics. 4. Conclusion This theoretical work has demonstrated that extra two new solutions of new SH-SAWs propagating along the free surface of the transversely isotropic PEM medium of symmetry class 6 mm can be found. The discovered t wo S H-SAWs relate to the c ase of σ32 = 0, φ = φf, D = Df, ψ = ψf, and B = Bf rep rese nting t he bo undar y cond i- tions at the interface between the PEM and a vacuum. The found solutions correspond to two natural coupling mechanisms such as and in the coefficient of the magnetoelectromechanical coupling (CMEMC). Comparative numerical calculations are listed in the table for the different BaT iO3-CoFe2O4 co mp o- sites. 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