An eigen theory of waves in piezoelectric solids

Based on the standard spaces of the physical presentation, both the quasi-static mechanical approximation and the quasi-static electromagnetic approximation of piezoelectric solids are studied here. The complete set of uncoupled elastic wave and electromagnetic wave equations are deduced. The results show that the number and propagation speed of elastic waves and electromagnetic waves in anisotropic piezoelectric solids are determined by both the subspaces of electromagnetically anisotropic media and ones of mechanically anisotropic media. Based on these laws, we discuss the propagation behaviour of elastic waves and electromagnetic waves in the piezoelectric material of class 6 mm.

2 coupled to the mechanical equations of motion. Such a fully dynamic theory is called piezoelectromagnetism by some researchers (Lee, 1991). Piezoelectromagnetic SH waves were studied by Li (Li, 1996) using scalar and vector potentials, which results in a relatively complicated mathematical model of four equations. Two of these equations are coupled, and the other two are one-way coupled. In addition, a gauge condition needs to be imposed. A different formulation was given by Yang and Guo (Yang et al., 2006), which leads to two uncoupled equations. Piezoelectromagnetic SH waves over the surface of a circular cylinder of polarized ceramics were analyzed. Although many works have been done for the piezoelectromagnetic waves in piezoelectric solids, the explicit uncoupled equations of piezoelectromagnetic waves in the anisotropic media could not be obtained because of the limitations of classical theory. In this chapter, the idea of eigen theory presented by author (Guo, 1999;2000;2001;2002;2007; is used to deal with both the Maxwell's electromagnetic equation and the Newton's motion equation. By this method, the classical Maxwell's equation and Newton's equation under the geometric presentation can be transformed into the eigen Maxwell's equation and Newton's equation under the physical presentation. The former is in the form of vector and the latter is in the form of scalar. As a result, a set of uncoupled modal equations of electromagnetic waves and elastic waves are obtained, each of which shows the existence of electromagnetic and elastic sub-waves, meanwhile the propagation velocity, propagation direction, polarization direction and space pattern of these sub-waves can be completely determined by the modal equations. In section 2, the elastic waves in anisotropic solids were studied under six dimensional eigen spaces. It was found that the equations of elastic waves can be uncoupled into the modal equations, which represent the various types of elastic sub-waves respectively. In section 3, the Maxwell's equations are studied based on the eigen spaces of the physical presentation, and the modal electromagnetic wave equations in anisotropic media are deduced. In section 4, the quasi-static theory of waves in piezoelectric solids (mechanical equations of motion, coupled to the equations of static electric field, or Maxwell's equations, coupled to the mechanical equations of equilibrium) are studied based on the eigen spaces of the physical presentation. The complete sets of uncoupled elastic or electromagnetic dynamic equations for piezoelectric solids are deduced. In section 5, the Maxwell's equations, coupled to the mechanical equations of motion, are studied based on the eigen spaces of the physical presentation. The complete sets of uncoupled fully dynamic equations for piezoelectromagnetic waves in anisotropic media are deduced, in which the equations of electromagnetic waves and elastic ones are both of order 4. The discussions are given in section 6.

Concepts of eigen spaces
The eigen value problem of elastic mechanics can be written as where C is a standarded matrix of elastic coefficients, i λ is eigen elasticity, and is invariables of coordinates, i ϕ is the corresponding eigen vector, and satisfies the orthogonality condition of basic vectors. www.intechopen.com From Eqs. (7) and (8) It is seen that Δ is a symmetrical differential operator matrix, and It is proved by author that the elastic dynamics equation (10) The calculation shows that the stress operators are the same as Laplace's operator (either two dimensions or three dimensions) for isotropic solids, and for most of anisotropic solids.
In the modal equations of elastic waves, the speeds of propagation of elastic waves are the following ,1 , 2 , , Eqs. (14) and (15) show that the number of elastic waves in anisotropic solids is equal to that of eigen spaces of anisotropic solids, and the speeds of propagation of elastic waves are related to the eigen elasticity of anisotropic solids.

Elastic waves in isotropic solids
There are two independent eigen spaces in isotropic solids (1) where i ξ is a vector of order 6, in which i th element is 1 and others are 0.
The eigen elasticity and eigen operator of isotropic solids are the following It is seen that there exist three elastic waves in cubic solids, one of which is the quasidilation wave, and two others are the quasi-shear waves.

Hexagonal solids
There are four independent eigen spaces in a hexagonal (transversely isotropic) solids ( Thus there exist four independent elastic waves in hexagonal solids, which can be described by following equations It is seen that there exist four elastic waves in hexagonal solids, two of which are the quasidilation wave, and two others are the quasi-shear waves.

Eigen spaces of electromagnetic solids
In anisotropic electromagnetic solids, the dielectric permittivity and magnetic permeability are tensors instead of scalars. The constitutive relations are expressed as follows where the dielectric permittivity matrix e and the magnetic permeability matrix μ are usually symmetric ones, and the elements of the matrixes have a close relationship with the selection of reference coordinate. Suppose that if the reference coordinates is selected along principal axis of electrically or magnetically anisotropic solids, the elements at non-diagonal of these matrixes turn to be zero. Therefore, Eqs. Projecting the electromagnetic physical qualities of geometric presentation, such as the electric field intensity vector E, magnetic field intensity vector H, magnetic flux density vector B and electric displacement vector D into eigen spaces of physical presentation, we get (43) and (44) in the form of scalar, we have n ≤ are number of electromagnetic independent subspaces. These are the electromagnetic physical qualities under the physical presentation. Substituting Eqs. (43) and (44) into Eqs. (37) and (38) respectively, and using Eqs. (45) and (46) The above equations are just the modal constitutive equations of electromagnetic media in the form of scalar.

Matrix expression of Maxwell's equation
The classical Maxwell's equations in passive region can be written as , Now we rewrite above equations in the form of matrix as follows   (37) and (38) into Eqs. (52) and (54) Substituting Eq. (55) into (56) or Eq. (56) into (55) yield

Electrically anisotropic solids
In anisotropic dielectrics, the dielectric permittivity is a tensor, while the magnetic permeability is a scalar. So Eqs. (57) and (58) can be written as follows Substituting Eqs. (43), (44) and (47) into Eqs. (60) and (61), we have is defined as the eigen matrix of electromagnetic operators under the eigen spaces, is a diagonal matrix. Thus Eqs. (62) and (63)  Eqs. (64) and (65) are the modal equations of electromagnetic waves in anisotropic dielectrics.

Magnetically anisotropic solids
In anisotropic magnetics, the magnetic permeability is a tensor, while the dielectric permittivity is a scalar. So Eqs. (57) and (58) can be written as follows Substituting Eqs. (43), (44) and (48) into Eqs. (66) and (67), we have is defined as the eigen matrix of electromagnetic operators under the eigen spaces, is a diagonal matrix. Thus Eqs. (68) and (69) Eqs. (70) and (71) are the modal equations of electromagnetic waves in anisotropic magnetics.

Electromagnetic waves in anisotropic solids
In this section, we discuss the propagation behaviour of electromagnetic waves only in anisotropic dielectrics.

Isotropic crystal
The matrix of dielectric permittivity of isotropic dielectrics is following We can see from the above equations that there is only one eigen-space in isotropic crystal, which is a triple-degenerate one, and the space structure is following The basic vector of one dimension in a triple-degenerate subspace is The eigen-qualities and eigen-operators of isotropic crystal are respectively shown as below ( ) Eq. (79) is just the Helmholtz's equation of electromagnetic wave.

Uniaxial crystal
The matrix of dielectric permittivity of uniaxial dielectrics is following www.intechopen.com We can see from the above equations that there are two eigen-spaces in uniaxial crystal, one of which is a double-degenerate space, and the space structure is following The basic vectors in two subspaces are following The eigen electric strength qualities of uniaxial crystal are respectively shown as below It is seen that there are two kinds of electromagnetic waves in uniaxial crystal.

Biaxial crystal
The matrix of dielectric permittivity of biaxial dielectrics is following We can see from the above equations that there are three eigen-spaces in biaxial crystal, and the space structure is following The eigen-qualities and eigen-operators of biaxial crystal are respectively shown as below Therefore, the equations of electromagnetic waves in biaxial crystal can be written as below ( ) The velocities of electromagnetic waves are respectively as follows It is seen that there are three kinds of electromagnetic waves in biaxial crystal.

Monoclinic crystal
The matrix of dielectric permittivity of monoclinic dielectrics is following We can see from the above equations that there are also three eigen-spaces in monoclinic crystal, and the space structure is following www.intechopen.com It is seen that there are also three kinds of electromagnetic waves in monoclinic crystal. In comparison with the waves in biaxial crystal, the electromagnetic waves in monoclinic crystal have been distorted.

Modal constitutive equation of piezoelectric solids
For a piezoelectric but nonmagnetizable dielectric body, the constitutive equations is the following where h is the piezoelectric matrix. Substituting Eqs. (5), (43) and (44) into Eqs. (132)-(134), respectively, and multiplying them with the transpose of modal matrix in the left, we have www.intechopen.com

Modal equation of elastic waves in piezoelectric solids
When only acoustic waves are considered, we can use the of quasi-static electromagnetic approximation. In this case, the mechanical equations are dynamic, the electromagnetic equations are static and the electric field and the magnetic field are not coupled. The static electric field equations and dynamic equations are given as follows

Eigen properties of polarized ceramics
In this section, we discuss the propagation laws of piezoelectromagnetic waves in an polarrized ceramics poled in the 3 x -direction. The material tensors in Eqs.(132)-(134) are represented by the following matrices under the compact notation  There are four independent mechanical eigenspaces as follows

Elastic waves in polarized ceramics
There are four equations of elastic waves in polarized ceramics, in which the propagation speed of elastic waves is the folloing, respectively It is seen that two of elastic waves are the quasi-dilation wave, and two others are the quasishear waves, and only two waves were affected by the piezoelectric coefficients, which speeds up the propagation of second and fourth waves.

Electromagnetic waves in polarized ceramics
There are two equations of electromagnetic waves in polarized ceramics, in which the propagation speed of electromagnetic waves is the folloing, respectively It is seen that two electromagnetic waves are all affected by the piezoelectric coefficients, which slow down the propagation of electromagnetic waves.

Fully dynamic waves in piezoelectric solids
When fully dynamic waves are considered, the complete set of Maxwell equation needs to be used, coupled to the mechanical equations of motion as follows From the above results, we can see that when considering the mutual effects between mechanical subspaces and electromagnetic subspaces, the electromagnetic wave equations become four-order partial differential ones, meanwhile elastic wave equation still keep in the form of four-order partial differential ones.

Conclusions
In the first place, we analyzed here the elastic waves and electromagnetic waves in anisotropic solids. The calculation shows that the propagation of elastic waves in anisotropic solids consist of the incomplete dilation type and the incomplete shear type except for the pure longitudinal or pure transverse waves in isotropic solids. Several novel results for elastic waves were obtained, for example, there are two elastic waves in isotropic solids, which are the P-wave and the S-wave. There are three elastic waves in cubic solids, one of which is a quasi-P-waves and two are quasi-S-waves. There are four elastic waves in hexagonal (transversely isotropic) solids, half of which are quasi-P-waves and half of which are quasi-S-waves. There are five elastic waves in tetragonal solids, two of which are quasi-P-waves and three of which are quasi-S-waves. There are no more than six elastic waves in orthotropic solids or in the more complicated anisotropic solids. For electromagnetic waves, the similar results were obtained: 1) the number of electromagnetic waves in anisotropic media is equal to that of eigen-spaces of anisotropic media; 2) the velocity of propagation of electromagnetic waves is dependent on the eigen-dielectric permittivity and eigen-magnetic permeability; 3) the direction of propagation of electromagnetic waves is related on the eigen electromagnetic operator in the corresponding eigen-space; 4) the direction of polarization of electromagnetic waves is relevant to the eigen-electromagnetic quantities in the corresponding eigen-space. In another word, there is only one kind of electromagnetic wave in isotropic crystal. There are two kinds of electromagnetic waves in uniaxial crystal.
There are three kinds of electromagnetic waves in biaxial crystal and three kinds of distorted electromagnetic waves in monoclinic crystal. Secondly, the elastic waves and electromagnetic waves in piezoelectric solids both for static theory and for fully dynamic theory are analyzed here based on the eigen spaces of physical presentation. The results show that the number and propagation speed of elastic or electromagnetic waves in anisotropic piezoelectric solids are determined by both the subspaces of electromagnetically anisotropic media and ones of mechanically anisotropic media. For the piezoelectric material of class 6mm, it is seen that there exist four elastic waves, respectively, but only two waves were affected by the piezoelectric coefficients. There exist two electromagnetic waves, respectively, but the two waves were all affected by the piezoelectric coefficients. The fully dynamic theory of Maxwell's equations, coupled to the mechanical equations of motion, are studied here. The complete set of uncoupled dynamic equations for piezoelectromagnetic waves in anisotropic media are deduced. For the piezoelectric material of class 6mm, it will be seen that there exist eight electromagnetic waves and also eight elastic waves, respectively. Furthermore, in fact of I j cv , except for the classical electromagnetic waves and elastic waves, we can obtain the new electromagnetic waves propagated in speed of elastic waves and new elastic waves propagated in speed of electromagnetic waves. SAW devices are widely used in multitude of device concepts mainly in MEMS and communication electronics. As such, SAW based micro sensors, actuators and communication electronic devices are well known applications of SAW technology. For example, SAW based passive micro sensors are capable of measuring physical properties such as temperature, pressure, variation in chemical properties, and SAW based communication devices perform a range of signal processing functions, such as delay lines, filters, resonators, pulse compressors, and convolvers. In recent decades, SAW based low-powered actuators and microfluidic devices have significantly added a new dimension to SAW technology. This book consists of 20 exciting chapters composed by researchers and engineers active in the field of SAW technology, biomedical and other related engineering disciplines. The topics range from basic SAW theory, materials and phenomena to advanced applications such as sensors actuators, and communication systems. As such, in addition to theoretical analysis and numerical modelling such as Finite Element Modelling (FEM) and Finite Difference Methods (FDM) of SAW devices, SAW based actuators and micro motors, and SAW based micro sensors are some of the exciting applications presented in this book. This collection of up-to-date information and research outcomes on SAW technology will be of great interest, not only to all those working in SAW based technology, but also to many more who stand to benefit from an insight into the rich opportunities that this technology has to offer, especially to develop advanced, low-powered biomedical implants and passive communication devices.