## Abstract

In this paper, the analytical and numerical studies of two-dimensional wave propagation in porous piezoelectric materials (PPMs) are carried out. The decoupling of waves, in such materials for various crystal classes, is studied analytically for different coordinate planes. It is found that, for wave propagation in a plane, the system is decoupled in some crystal classes, whereas it remains coupled in other crystal classes. It is established that the decoupled pure-shear wave, propagating in a PPM, can be stiffened or unstiffened with piezoelectric effects even if the PPM belongs to the same symmetry group but has a different crystal class. The skewing angles and mutual angles between the polarization directions of different waves are also computed numerically.

## 1. Introduction

Piezoelectric materials, with their ability to couple between the electrical and mechanical domains, have been widely recognized for their transduction capabilities and are being extensively used as transducers, actuators, sensors and filters in many areas of science and technology such as electronics, mechanical engineering, navigation, piezoelectric power supplies, medical ultrasonic imaging applications and other modern industrial fields. These materials act as very important functional components in sonar projectors, fluid monitors, pulse generators and surface acoustic wave devices. Reviews on the theory and the application of piezoelectric devices can be found in [1–3]. Owing to wide applications, the theory of electroacoustic waves in piezoelectric solids poses numerous challenging problems that attract the wide attention of researchers. The theory of acoustic wave propagation in piezoelectric materials commonly rests on the quasi-electrostatic description of electric fields accompanying elastic deformations [4]. Further details about wave propagation in piezoelectric materials can be found in [5,6]. Avetissian [7] studied the possibility of formulation of the plane and anti-plane problems in crystal bodies, taking into account the piezoelectric effect. Every [8] obtained the governing equations for electroacoustic waves in unbounded piezoelectric crystals for the weak and strong electromechanical coupling cases.

The characteristics of electroacoustic waves depend on the crystallographic properties of piezoelectric materials, the direction of propagation and the orientation of the cut plane. An essential feature of the wave propagation phenomenon is the anisotropy of the medium. Some special directions of propagation, which support purely longitudinal or purely transverse modes, can be found for certain types of material symmetry. Romeo [9] and Duda [10] investigated the specific directions along which such pure modes can exist. Chen & Lai [11] studied the correspondence between plane piezoelectricity and generalized plane strain in elasticity. Zou *et al.* [12] proposed a procedure, based on the solved axis-direction sets, to determine a linear piezoelectricity tensor and to trace the rotation transformation.

Ceramic continua, with small pores or voids, are directly related to the process used to manufacture ceramics and, in fact, a residual porosity is unavoidable in commercial piezoelectric ceramics. The porosity can be used to enhance acoustic impedance, hydrostatic figure of merits and design flexibility properties that are desirable for some technological applications. Porous piezoelectric materials (PPMs) have been of great interest and technological importance in ultrasonic applications such as hydrophones, actuators, underwater transducers and medical diagnostic devices [13–16]. PPMs have some merit over dense materials in many applications. A large amount of research has been carried out involving various manufacturing techniques to produce porous piezoelectric ceramics [17,18–23]. Different authors [24–31] developed experimental models to study the effects of pore size, porosity and sintering behaviour on the piezoelectric properties of PPMs. Gomez & Montero [32,33] introduced a new theoretical frame to study the characteristics of porous piezoelectric ceramics and 0–3/3–3 connectivity piezoelectric composites and governing equations were also derived. Experimental studies related to the elastic wave propagation and attenuation in PPMs [34,35] have also been carried out. Gomez *et al.* [36] presented a finite-element method to study the wave propagation in 0–3/3–3 connectivity composites with complex microstructure. Vashishth & Gupta [37] derived the constitutive equations and equations of motion for PPMs using the electric enthalpy density function and variational principle. The theory of wave propagation in PPMs with a crystal symmetry of 6mm was discussed by Vashishth & Gupta [38]. The effects of porosity, frequency, direction of propagation and piezoelectricity on the phase velocities and attenuation of waves propagating in porous piezoelectric medium were studied therein. Sharma [39] studied the effects of piezoelectricity and phase directions on the phase velocities and group velocities of waves propagating in anisotropic PPMs.

In this paper, two-dimensional wave propagation in PPMs is studied. Analytical proofs for the decoupling of waves in PPMs, possessing different crystal symmetry, in different planes of propagation are given. The skewing angles and mutual angles between the polarizations of different waves are also computed for the considered models.

## 2. Basic equations

The constitutive equations for piezoelectric materials are modified for PPMs [37] and can be written as
*σ*_{ij}(*σ**) and *Φ*(*Φ**) are electric displacement components and electric potential for the solid (fluid) phase of porous bulk material, respectively. *c*_{ijkl} are the elastic stiffness coefficients. The elastic constant *R* measures the pressure to be exerted on fluid to push its unit volume into the porous matrix. *m*_{ij};

The equations of motion and Gauss equations for an anisotropic PPM saturated with viscous fluid, in the absence of body forces, are
*f*), density of the porous aggregate (*ρ*), pore fluid density (*ρ*_{f}) and the inertial coupling parameters. The dissipation coefficients (*b*_{ij}) steer the effect of wave frequency (*υ*), fluid viscosity (*μ*), permeability coefficients (*χ*_{ij}) and the porosity.

For propagation of plane harmonic waves, we consider
*ι*^{2}=−1, *ω* is the circular frequency of harmonic waves, *v* is the velocity of wave propagation and *t* is the time, *n*_{i} are the components of the direction of propagation, *B*_{i}, *F*_{i}, *G*, *H* are the associated amplitudes.

Making use of equations (2.1)–(2.4) and (2.9), equations (2.5)–(2.8) become
*m*_{ij}, *R*, *ζ*_{kij}, *A*_{il}, *b*_{ik}, *F*_{k}, *H* as zero.

The condition of existence of the non-trivial solution of the system (2.10)–(2.13) leads to a biquadratic equation in *v*^{2}, whose solution gives the wave velocity (*v*_{j}) of four waves propagating in a medium. Let *j*th wave propagating in a PPM. The angles between the mutual polarization vectors of waves are defined as
*i*=1, 2, 3, 4 corresponds to *qP*_{1}, *qP*_{2}, *qS*_{1} and *qS*_{2} waves, respectively.

## 3. Wave propagation in the *x*_{1}–*x*_{3} plane

For waves propagating along a direction **n**(*n*_{i}) in the *x*_{1}–*x*_{3} plane, we have
*θ* is the angle made by the propagation direction with the *x*_{3}-axis.

For the direction of propagation under consideration, the algebraic system of equations (2.10)–(2.13) is deduced analytically for each of the 19 crystal classes mentioned in table 1.

It is found that, of these 19 crystal classes, the algebraic system (2.10)–(2.13) remains coupled in some crystal classes, whereas it is decoupled into subsystems in other crystal classes. The decoupling of systems in different crystal classes, for the direction of propagation under consideration in the *x*_{1}–*x*_{3} plane, is summarized in table 2.

### (a) Coupled waves in porous piezoelectric material of group I

For a monoclinic PPM with crystal symmetry of m-type, the elastic, piezoelectric and dielectric constants have the following structure:

In order to support these theoretical results, numerical computations were carried out, and the results are presented for the particular models considered. The data, taken for the numerical computation, are mainly based on Auld [5]. The variation of mutual angles (*θ*_{ij}) with the angle *θ* for the crystal class m is shown in figure 1. It is clear from figure 1 that, for no values of *θ*, *B*^{(i)} and *B*^{(j)} are perpendicular, which implies that waves in the case of PPMs of this class are not mutually orthogonal. Similar results were also obtained for other crystal classes of group I, but the plots are omitted to save repetition.

### (b) Decoupling of waves in porous piezoelectric material of group II

Here, the algebraic system (2.10)–(2.13) is deduced for the 2mm crystal class and the waves are studied for their decoupling. For a PPM with crystal symmetry 2mm, the elastic, piezoelectric and dielectric constants can be represented as
*y*_{i}(*i*=1,2,…,11) are given in the electronic supplementary material, appendix A.

The condition of existence of the non-trivial solution of system (3.4) leads to
*T*_{1},*T*_{2},*T*_{3} and *T*_{4} are listed in the electronic supplementary material, appendix A.

The elimination of *B*_{2} and *F*_{2} from system (3.5) gives

The three roots of equation (3.6) give the complex velocities of three coupled waves (*qP*_{1}, *qP*_{2} and *qS*_{1}) and are stiffened with piezoelectric effects. Equation (3.7) gives the velocity of an unstiffened *qS*_{2} wave that is decoupled from three coupled waves and is polarized along a direction orthogonal to the sagittal plane containing these three coupled waves. Calculations were performed for other crystal classes of group II and similar conclusions followed.

The variation of angles (*θ*_{ij}) with *θ* is exhibited in figure 2 for the crystal class 2mm. This shows that, in all the directions of propagation, the *qS*_{2} wave is polarized along a direction mutually orthogonal to the polarization directions of the *qP*_{1}, *qP*_{2} and *qS*_{1} waves. Numerical computation was also carried out for other crystal classes of this group and the same behaviour was found for those classes.

Figure 3 shows the variation of skewing angles *θ* for the 2mm crystal class. The solid curves (dotted curves) correspond to the results for PPM (pure piezoelectric material). The results for the crystal classes 4mm, 6mm and *qP*_{1} wave. It is observed that the *qS*_{2} wave is polarized along a direction perpendicular to the *x*_{1}–*x*_{3} plane which identifies the existence of the pure transverse mode. Along two directions (*θ*=0° and *θ*=90°), the *qP*_{1} and *qS*_{1} waves can be identified as pure longitudinal and pure transverse modes, respectively. The *qP*_{2} wave is pure longitudinal mode along one direction (*θ*=90°) for the crystal classes 2mm, 4mm, 6mm, whereas in the case of crystal class *qP*_{2} wave is pure longitudinal mode along two directions (*θ*=0° and *θ*=90°). It is also observed that, in the case of crystal class *qS*_{1} wave is nearly transverse mode along all the directions of propagation. The possible directions of existence of pure modes remain unaffected owing to the porosity, but the magnitude of skewing angles increases when porosity effects are taken into account.

### (c) Decoupling of waves in porous piezoelectric material of group III

Here, the algebraic system (2.10)–(2.13) is deduced for the 222 crystal class. For a PPM with crystal symmetry 222, the elastic, piezoelectric and dielectric constants have the following structure:
_{2}]_{7×7} is the submatrix of CM_{2} consisting of the first seven rows and seven columns. [CM_{2}]_{6×6} is the submatrix of CM_{2} consisting of the last six rows and six columns.

Using equations (3.1) and (3.8), the system (2.10)–(2.13) decouples into the following two subsystems:

The condition of existence of the non-trivial solution of system (3.9) leads to
*T*_{1}, *T*_{2}, *T*_{3} and *T*_{4} are given in the electronic supplementary material, appendix B.

Similarly, the condition of existence of the non-trivial solution of system (3.10) leads to

Thus, the three roots (*v*_{1},*v*_{2},*v*_{3}) of equation (3.11) give the complex velocities of three unstiffened coupled waves termed the *qP*_{1}, *qS*_{1} and *qP*_{2} waves. Equation (3.12) gives the complex velocity of the stiffened *qS*_{2} wave, which is decoupled from the other three coupled waves and is polarized along a direction orthogonal to the sagittal plane containing these three coupled waves. It is interesting to note that the decoupled *qS*_{2} wave is stiffened in the case of the 222 class, unlike the 2mm class. The behaviour of decoupling in other crystal classes of this group is also found to be the same as in the case of the crystal class 222.

The mutual (*θ*_{ij}) for the crystal class 222 are shown in figure 4. It is clear from figure 4 that the *qS*_{2} wave is mutually orthogonal to the coupled *qP*_{1}, *qS*_{1} and *qP*_{2} waves. Numerical computation was also carried out for other crystal classes of group III, and a similar behaviour was observed for those crystal classes.

The variation of skewing angles *γ*_{1}, *γ*_{2}, *γ*_{3} and *γ*_{4} with angle *θ* for the crystal class 222 is shown in figure 5. The results for the crystal classes 422, *qS*_{2} wave is pure transverse mode for all crystal classes. For the crystal classes 222 and *θ*=0°, 45°, 90° and 135°) are possible along which the *qP*_{1} wave can be identified as pure longitudinal waves, whereas there are two such directions (*θ*=0° and 90°) in the case of crystal classes 422, *qP*_{2} wave can be identified as pure longitudinal modes along the two directions (*θ*=0° and 90°) in the case of the 222, 422, *θ*=0°, 45°, 90° and 135°) in the case of the *θ*=0° and 90°), the *qS*_{1} wave can be identified as pure transverse mode in all the crystal classes of this group. In the case of the crystal classes 622 and *qS*_{1} wave is nearly pure transverse mode along all the directions of propagation. Unlike the 2mm class, the porosity decreases the skewing angle of the *qP*_{1} wave in magnitude.

## 4. Wave propagation in the *x*_{2}–*x*_{3} plane

For waves propagating in the *x*_{2}–*x*_{3} plane, along a direction **n**(*n*_{i}) making an angle *θ* with the *x*_{3}-axis, we have

For the direction of propagation under consideration in the *x*_{2}–*x*_{3} plane, the algebraic system (2.10)–(2.13) is deduced analytically for each of the 19 crystal classes. The behaviour of coupling or decoupling of the system (2.10)–(2.13) in this case remains the same as it was in the case of the *x*_{1}–*x*_{3} plane for all the crystal classes except crystal classes 32 and 3m. The coupling and decoupling of the algebraic system (2.10)–(2.13) in different crystal classes is summarized in table 3.

### (a) Decoupling of waves in porous piezoelectric material of group V

Here, the system (2.10)–(2.13) is deduced for crystal class 3m. For a PPM with crystal symmetry 3m, the elastic, piezoelectric and dielectric constants can be represented as
*q*_{i}(*i*=1,2,…,11) are detailed in the electronic supplementary material, appendix C.

The conditions of existence of the non-trivial solution of the system (4.3) and (4.4) lead to
*T*_{1}, *T*_{2}, *T*_{3} and *T*_{4} are given in the electronic supplementary material, appendix C.

As discussed earlier, the three roots of equation (4.5) give the complex velocities of three stiffened coupled waves, namely the *qP*_{1}, *qP*_{2} and *qS*_{1} waves. Equation (4.6) gives the velocity of the decoupled unstiffened *qS*_{2} wave. Similar calculations were carried out separately for each crystal class of this group, and the same decoupling behaviour was obtained.

The mutual angles (*θ*_{ij}) in a PPM of the 3m crystal class are shown in figure 6. It is clear from figure 6 that the *qS*_{2} wave is orthogonal to three coupled waves. Figure 7 shows the polar diagram of skewing angles *γ*_{1}, *γ*_{2}, *γ*_{3} and *γ*_{4} for the crystal class 3m. It is observed that the *qS*_{2} wave is pure transverse modes for all the directions in the considered plane. Two directions (*θ*=0°, 90.2°) for possible propagation of pure longitudinal modes can be identified with the *qP*_{1} and *qP*_{2} waves. The directions of existence of pure modes remain unchanged in both the cases, but the skewing angle of the *qP*_{1} wave decreases in the presence of porosity.

### (b) Decoupling of waves in porous piezoelectric material of group VI

Here, the system (2.10)–(2.13) is deduced for the crystal class 32. For a PPM with crystal symmetry 32, the elastic, piezoelectric and dielectric constants can be represented as
_{4}]_{7×7} is the submatrix of CM_{4} consisting of the first seven rows and seven columns. [CM_{4}]_{6×6} is the submatrix of CM_{4} consisting of the last six rows and six columns.

Using equations (4.1) and (4.7), the system (2.10)–(2.13) is decoupled into the following two subsystems:

The conditions of existence of the non-trivial solution of systems (4.8) and (4.9) lead to
*T*_{1}, *T*_{2}, *T*_{3}, *T*_{4} are listed in the electronic supplementary material, appendix D.

The complex velocities of three coupled stiffened waves, the *qP*_{1}, *qP*_{2} and *qS*_{1} waves, are obtained from equation (4.10). Equation (4.11) gives the velocity of the unstiffened *qS*_{2} wave, which is decoupled from the other three coupled waves and polarized in a direction orthogonal to the plane of propagation containing the three coupled waves. The algebraic system (2.10)–(2.13) was also deduced for other crystal classes of this group and the same conclusion followed.

The mutual angles (*θ*_{ij}) in a PPM of the 32 crystal class are shown in figure 8. It is clear from figure 8 that the *qS*_{2} wave is mutually orthogonal to the other three coupled waves. Figure 9 shows the polar diagram of skewing angles *γ*_{1}, *γ*_{2}, *γ*_{3} and *γ*_{4} for crystal class 32. It is again observed that the *qS*_{2} wave is a pure transverse wave in all propagation directions in the *x*_{2}–*x*_{3} plane. However, the *qP*_{1} and *qP*_{2} waves become pure longitudinal mode only along specific directions, three (*θ*=0°, 59.4°, 113.7°) in the case of the *qP*_{1} and two (*θ*=0°, 90.8°) in the case of the *qP*_{2} wave. It is observed that inclusion of porosity in the pure piezoelectric materials decreases the magnitude of the skewing angle corresponding to the *qP*_{1} mode, whereas directions of the pure *qP*_{1} mode remain unaffected.

## 5. Wave propagation in the *x*_{1}–*x*_{2} plane

For waves propagating in the *x*_{1}–*x*_{2} plane, along a direction- **n**(*n*_{i}) making an angle *ϕ* with the *x*_{1}-axis, we have

For the direction of propagation under consideration in the *x*_{1}–*x*_{2} plane, the algebraic system (2.10)–(2.13) is derived for each of the 19 crystal classes. The behaviour of the algebraic system (2.10)–(2.13) in different crystal classes, for propagation in the *x*_{1}–*x*_{2} plane, is summarized in table 4.

### (a) Decoupling of waves in porous piezoelectric material of group VIII

Here, the system of equations for the crystal class m is derived and then the decoupling of waves is studied. The equations corresponding to the other crystal classes of this group can be deduced from the equations for class m.

Using equations (3.2) and (5.1), the system (2.10)–(2.13) reduces to the following two subsystems:
*s*_{i}(*i*=1,2,…,11) are given in the electronic supplementary material, appendix E.

The condition of existence of the non-trivial solution of system (5.2) leads to
*T*_{1}, *T*_{2}, *T*_{3} and*T*_{4} are listed in the electronic supplementary material, appendix E.

The elimination of *B*_{3} and *F*_{3} from system (5.3) gives

The three roots of equation (5.4) give the complex velocities of coupled *qP*_{1}, *qP*_{2} and *qS*_{1} waves. These waves are stiffened waves. Equation (5.5) gives the phase velocity of the decoupled unstiffened wave, which is identified as the *qS*_{2} wave. Repeating the same steps of calculations for other crystal classes of this group, a similar conclusion followed.

Figure 10 exhibits the variation of mutual angles between the different pairs of the polarization directions of *qP*_{1}, *qP*_{2}, *qS*_{1} and *qS*_{2} waves with angle *ϕ* for crystal classes m. It is observed that the *qS*_{2} wave is polarized along a direction orthogonal to the sagittal plane containing three coupled waves, namely *qP*_{1}, *qP*_{2} and *qS*_{1}.

The variation of skewing angles *γ*_{1}, *γ*_{2}, *γ*_{3} and *γ*_{4} with angle *ϕ* in the *x*_{1}–*x*_{2} plane for crystal class m is depicted in figure 11. The results for crystal classes *qP*_{1} and *qP*_{2} waves exist as pure longitudinal waves are two (*ϕ*=72.7°, 173.3° for the *qP*_{1} wave and *ϕ*=67.52°, 172.01° for the *qP*_{2} wave) in the m class and six (*ϕ*=13.86°, 43.3°, 72.8°, 102.9°, 133.3°, 163.9°) in the *qP*_{1} mode is affected owing to porosity effects, whereas that of the *qS*_{1} and *qS*_{2} modes remains unaffected.

### (b) Decoupling of waves in porous piezoelectric material of group IX

Here, the system (2.10)–(2.13) is derived for crystal class 2. For a PPM with crystal symmetry 2, the elastic, piezoelectric and dielectric constants are
_{1}]_{7×7} is the submatrix of CM_{1} consisting of the first seven rows and seven columns. [CM_{1}]_{6×6} is the submatrix of CM_{1} consisting of the last six rows and six columns. Using equations (5.1) and (5.4), the system (2.10)–(2.13) reduces to the following two subsystems:

The condition of existence of the non-trivial solution of system (5.7) leads to
*T*_{1}, *T*_{2}, *T*_{3} and*T*_{4} are given in the electronic supplementary material, appendix F.

Similarly, the condition of existence of the non-trivial solution of system (5.8) leads to

The three roots of equation (5.9) give the complex velocities of coupled unstiffened *qP*_{1}, *qP*_{2} and *qS*_{1} waves. Equation (5.10) implies that one wave is decoupled from the other waves and is identified as the *qS*_{2} wave. Similar calculations were carried out for other crystal classes of this group IX, and the same kind of decoupling behaviour was observed. The algebraic system (2.10)–(2.13) corresponding to the other crystal classes of this group can be reduced from the system corresponding to class 2.

The mutual angles between the polarization directions of the *qP*_{1}, *qP*_{2}, *qS*_{1} and *qS*_{2} waves with angle *ϕ* for crystal class 2 is exhibited in figure 12. It is clear from figure 12 that, for crystal class 2, the polarization direction of the *qS*_{2} wave is mutually orthogonal to the polarization directions of the other three coupled waves. Similar results were also obtained for other crystal classes of this group and the decoupling behaviour was verified numerically for these crystal classes.

The polar diagrams of skewing angles *γ*_{1}, *γ*_{2}, *γ*_{3} and *γ*_{4} in the *x*_{1}–*x*_{2} plane for crystal class 2 are not significantly different from crystal class m and thus plots are not shown here. The results were also obtained for the remaining classes of this group, but plots are not shown here for those classes. The polarization direction of the *qS*_{2} wave is always found to be orthogonal to the direction of propagation in the considered *x*_{1}–*x*_{2} plane. It is observed that the *qP*_{1} and *qP*_{2} modes can be identified as pure modes along two directions (*ϕ*=0° and 90°) in the case of crystal classes 222, 2mm and *qP*_{1} and *qP*_{2} modes can be identified as pure modes for the remaining classes of this group are summarized in table 5.

Next, the algebraic subsystems for the crystal class 422 are reduced from the corresponding subsystems (5.7) and (5.8) after imposing the following conditions on the elastic, piezoelectric and dielectric constants:

Thus, the piezoelectric effects are completely decoupled from the mechanical one for crystal class 422 in the *x*_{1}–*x*_{2} plane. All four waves propagating in this case are unstiffened. Similar results are also obtained when subsystem (5.8) is deduced for the 622 crystal class.

To support this theoretical result, the effects of variation of *e*_{14} on the phase velocity of the stiffened *qS*_{2} wave for crystal classes 2 and 422 are shown in figure 13. It is clear from figure 13 that the phase velocity of the stiffened wave is affected owing to piezoelectricity in the case of crystal class 2, whereas it remains unaffected in the case of crystal class 422.

The algebraic system for pure piezoelectric materials can be obtained from the algebraic system (2.10)–(2.13), derived for PPMs, by substitution of *m*_{ij}, *R*, *ζ*_{kij}, *A*_{il}, *b*_{ik}, *F*_{k}, *H* as zero. Following the same procedure as described for porous piezoelectric materials, the algebraic system corresponding to pure piezoelectric materials was studied for different crystal classes and different planes of propagation. The behaviour of coupling or decoupling of the algebraic system remains the same in pure piezoelectric and PPM cases.

The mutual angles *θ*_{13}, *θ*_{14} and *θ*_{34}, corresponding to the pure piezoelectric case, have been computed for all the above-mentioned crystal classes and planes of propagation. Unlike the PPMs all three waves are found to be mutually orthogonal in all the crystal classes and planes of propagation (figure 14).

## 6. Conclusion

The decoupling of waves in 19 crystal classes is studied analytically and numerically for propagation in the *x*_{1}–*x*_{2}, *x*_{2}–*x*_{3} and *x*_{1}–*x*_{3} planes. The skewing angles and mutual angles between the polarization vectors of different waves are also obtained.

For the wave propagation along a direction in the *x*_{1}–*x*_{3} plane, the algebraic system remains coupled in crystal classes 2, m, 4,

For a given propagation direction in the *x*_{2}–*x*_{3} plane, the algebraic system remains coupled in crystal classes 2, m, 4,

For propagation in the *x*_{1}–*x*_{2} plane, the algebraic system remains coupled in the 3, 32 and 3m crystal classes. In the case of crystal classes 2, 222, 2mm, 4, *x*_{1}–*x*_{2} plane.

The system remains coupled in the trigonal (3) crystal class in all the planes of propagation. The decoupled shear wave is pure transverse waves in all the planes of propagation. The crystal symmetry affects the number of possible directions of propagation along which quasi-longitudinal modes can be identified as pure modes.

The behaviour of coupling or decoupling of algebraic system remains the same in pure piezoelectric and PPM cases. The directions of existence of pure modes remain unchanged owing to the inclusion of porosity in the model, but the magnitude of skewing angles changes owing to porosity.

- Received March 24, 2014.
- Accepted June 9, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.