Analytical Investigation of Surface Wave Characteristics of Piezoelectromagnetics of Class 6 mm

This short work copes with theoretical investigations of some surface wave characteristics for transversely isotropic piezoelectromagnetic composites of class 6mm. In the composite materials, the surface Bleustein-Gulyaev-Melkumyan wave and some new shear-horizontal surface acoustic waves SH-SAWs recently discovered by the author can propagate. The phase velocities Vph of the SH-SAWs can have complicated dependencies on the coefficient of the magnetoelectromechanical coupling K2 em CMEMC which depends on the electromagnetic constant α of the composites. Therefore, the analytical finding of the first and second partial derivatives of the Vph α represents the main purpose of this study. It is thought that the results of this short letter can help for theoreticians and experimentalists working in the research arena of opto-acoustoelectronics to completely understand some problems of surface wave propagation in piezoelectromagnetics.


Introduction
Two-phase composite materials, which possess both the piezoelectric and piezomagnetic phases, are very promising composites with the magnetoelectric effect.They are very interesting for various applications in space and aircraft technologies.Several books concerning composite materials are cited in 1-3 .The geometry of a two-phase composite material can be denoted by the following connectivities: 0-0, 0-1, 0-2, 0-3, 1-1, 1-2, 1-3, 2-2, 2-3, and 3-3, where 0, 1, 2, and 3 are the dimensions of piezoelectric-piezomagnetic phases.Some latterly published papers concerning the magnetoelectric effect in different composite materials can be found in 4-9 .For example, the composite structures 10 called 2-2 composites represent laminated composite materials in which alternate layers of two different materials are bonded together to form a stratified continuum.Also, 10-13 cope with some laminate composites in which the popular Terfenol-D material is used as the piezomagnetic phase.For ISRN Applied Mathematics the study of the magnetoelectric effect, much work was described in the review paper 14 by Fiebig.
It is thought that some of the main characteristics of piezoelectric, piezomagnetic, and composite materials are the speeds of the shear-horizontal surface acoustic waves SH-SAWs .In 1998, Gulyaev 15 has written a review of SH-SAWs in solids.However, in the beginning of this millennium, Melkumyan 16 has discovered twelve new SH-SAWs in piezoelectromagnetic composite materials.In 2010, the author of this paper has additionally discovered seven new SH-SAWs in the piezoelectromagnetic composites of class 6 mm, see the book 17 .One of the new surface Melkumyan waves 16 written in the following section was called the surface Bleustein-Gulyaev-Melkumyan wave 17 .Note that the classical SH-SAWs in purely piezoelectric materials and purely piezomagnetic materials are called the surface Bleustein-Gulyaev waves simultaneously discovered by Bleustein 18 and Gulyaev 19 to the end of the last millennium.The new SH-SAWs discovered in 17 by the author of this theoretical work depend on the speed of light in a vacuum and can represent an interest for acoustooptics and photonics researchers.Also, some peculiarities of the new SH-SAW propagation will be briefly discussed in the following section.This peculiarity allows one to assume a restriction for the electromagnetic constant α of the complex piezoelectromagnetic composite materials.
It is also noted that SH-SAWs can easily be produced by electromagnetic acoustic transducers EMATs , a nontrivial task for common piezoelectric transducers 20 .The EMATs can offer advantages over traditional piezoelectric transducers.Comprehensive monographs 21, 22 on the EMATs collect the research activities on this topic.Therefore, it is thought that this short theoretical work can be also useful as a small step towards new applications of the EMATs technologies.Indeed, it is believed that some characteristics of the SH-SAWs in piezoelectromagnetic composite materials can be revealed by the utilization of the electromagnetic acoustic transducers.Therefore, the following section describes the analytical finding of the first and second partial derivatives of the phase velocity V ph with respect to the electromagnetic constant α.

Theoretical Investigations
According to the recent work by Melkumyan 16 concerning wave propagation in piezoelectromagnetic materials of class 6 mm, the velocity V BGM for the shear-horizontal surface Bleustein-Gulyaev-Melkumyan wave can be written in the explicit form 17 as follows:

2.1
In 2.1 , the velocity V tem of the piezo-magnetoelectromechanical shear-horizontal bulk acoustic wave SH-BAW and the coefficient of the magnetoelectromechanical coupling K 2 em CMEMC are defined as follows: In 2.2 , the velocity V t4 of the purely mechanical SH-BAW is determined as follows: where ρ is the mass density.In 2.3 and 2.4 , there are the following material constants: the elastic stiffness constant C, piezoelectric constant e, piezomagnetic coefficient h, dielectric permittivity coefficient ε, magnetic permeability coefficient μ, and electromagnetic constant α.The material constants are described in the well-known handbook 23 on electromagnetic materials.Formula 2.1 for the surface Bleustein-Gulyaev-Melkumyan BGM wave corresponds to the first case of the electrical and magnetic boundary conditions at the interface between the composite surface and a vacuum.This case is for the electrically closed surface electrical potential ϕ 0 and the magnetically open surface magnetic potential ψ 0 using the mechanical boundary condition of the mechanically free interface.The realization of different boundary conditions is described in an excellent theoretical work 24 .In addition to the first case, it is also possible to treat the second case of electrical and magnetic boundary conditions for the mechanical boundary condition.This second case represents the continuity of both the normal components of D 3 and B 3 at the interface, where D 3 and B 3 are the components of the electrical displacement and the magnetic flux, respectively.This leads to the following velocities for the SH-SAWs discovered by the author in the recent theoretical work 17 :

2.6
In expressions 2.5 and 2.6 there is already dependence on the vacuum characteristics such as the dielectric permittivity constant ε 0 10 −7 / 4πC 2 L 8.854187817 × 10 −12 F/m and the magnetic permeability constant μ 0 4π × 10 −7 H/m 12.5663706144 × 10 −7 H/m , where C L 2.99782458 × 10 8 m/s is the speed of light in a vacuum: Also, expression 2.5 depends on the well-known coefficient of the electromechanical coupling K 2 e CEMC for a purely piezoelectric material see below , and expression 2.6 ISRN Applied Mathematics depends on the well-known coefficient of the magnetomechanical coupling K 2 m CMMC for a purely piezomagnetic material:

2.8
Therefore, it is possible to obtain the first and second derivatives of the velocities V BGM , V new1 , and V new2 with respect to the electromagnetic constant α as the results of the theoretical investigations for this short report.Note that these investigations were not carried out in the recent book 17 due to some mathematical complexities.Therefore, this report continues the theoretical investigations of the book 17 .These investigations are useful because it is possible that the functions V BGM α > 0 , V new1 α > 0 , and V new2 α > 0 can have some peculiarities, namely, the SH-SAWs cannot exist for some large values of α 2 → εμ when K 2 em → ∞; see formula 2.3 .Note that papers 25, 26 studied some composites with the electromagnetic constant α < 0, for which these peculiarities do not exist.Therefore, it allows one to suppose that the right sign for the electromagnetic constant α is negative.
The first partial derivatives of the velocities V BGM , V new1 , and V new2 with respect to the constant α can be written in the following forms: where In 2.9 -2.12 , the first partial derivative of the CMEMC K 2 em with respect to the electromagnetic constant α is defined by Using 2.5 and 2.6 , the functions b 1 and b 2 in 2.10 and 2.11 are determined as follows: Therefore, the first partial derivatives of the b 1 and b 2 with respect to the constant α can be expressed in the following forms: The second partial derivatives of the velocities V BGM , V new1 , and V new2 with respect to the electromagnetic constant α read where In 2.16 and 2.19 , the second partial derivative of the K 2 em with respect to the constant α is defined as follows: In 2.17 and 2.18 , the second partial derivatives of the b 1 and b 2 with respect to the α are where

2.22
It is obvious that the first partial derivatives of the velocities V BGM , V new1 , and V new2 with respect to the electromagnetic constant α have dimension of m/s 2 and can represent some squares in the corresponding two-dimensional 2D spaces of velocities.Analogically, the second partial derivatives of the velocities with respect to the constant α can represent some volumes with dimensions of m/s 3 in the corresponding 3D spaces of velocities.Indeed, it is also possible to graphically investigate the complicated first and second partial derivatives of the velocities obtained in formulae 2.9 -2.11 and from 2.16 to 2.18 .However, this does not represent the purpose of this short report.

Conclusion
This short theoretical report further developed the study of the recently published book 17 .In this work, on some wave properties of composite materials, the propagation peculiarities of new shear-horizontal surface acoustic waves SH-SAWs recently discovered in book 17 were theoretically studied and briefly discussed.Therefore, the analytical finding of the first and second partial derivatives of the phase velocity with respect to the electromagnetic constant α represented the main purpose of this study.This theoretical work can be useful for theoreticians and experimentalists working in the arena of acoustooptics, photonics, and opto-acoustoelectronics.Also, the theoretical study of this short paper can be useful for investigations of cubic piezoelectromagnetics like the researches carried out for cubic piezoelectrics 27 and cubic piezomagnetics 28 .