# On Discovery of the Twelfth and Thirteenth New Nondispersive SH-SAWs in 6 mm Magnetoelectroelastics

## Abstract

**:**

^{2}. Here, the mechanically free interface between the solid and a vacuum was considered. This report discovers the twelfth (thirteenth) new SH-SAW for the magnetically closed (electrically open) case and continuity of both the normal component of the electrical (magnetic) displacement and the electrical (magnetic) potential when the coupling mechanism eα – hε (eµ – hα) works. The propagation velocities were obtained in explicit forms that take into account the contribution of the vacuum material parameters. The discovered waves were then graphically studied for the purpose of disclosing the dissipation phenomenon (the propagation velocity becomes imaginary) caused by the coupling with the vacuum properties. The obtained results can be useful for further investigations of interfacial and plate SH-waves, constitution of technical devices, nondestructive testing and evaluation, and application of some gravitational phenomena.

## 1. Introduction

_{2}O

_{3}can possess the ME effect. One year later, Astrov [3] experimentally confirmed the existence of this effect in Cr

_{2}O

_{3}. The ME effect can be found in a class of ME solids called piezoelectromagnetics (PEMs), also known as magnetoelectroelastics (MEEs). These continuous media can possess both the piezoelectric and piezomagnetic effects resulting in the existence of the ME effect. The ME solids relate to smart materials because any change in the magnetic subsystem can cause some change in the electrical subsystem via the mechanical subsystem, or vice versa. Up to today, various ME materials (composites) were found (created) and they were then investigated both experimentally and theoretically. However, only a few are suitable for commercial use. Some recent reviews on the ME materials and their applications are listed in References [4,5,6,7,8]. The reader can also find about one hundred reviews on the subject in the literature because the smart ME materials can be used for different applications including spintronics, i.e., the future electronics without free charge carriers.

_{3}Co

_{2}Fe

_{24}O

_{41}was discovered in 2010. This Z-type hexagonal ferrite actually possesses the realizable ME effect appropriate for practical and commercial employments.

^{2}. Besides the mass density ρ there are the following material constants for a piezoelectromagnetics: the elastic stiffness constant C, piezoelectric constant e, piezomagnetic coefficient h, dielectric permittivity coefficient ε, magnetic permeability coefficient μ, and aforementioned electromagnetic constant α. It is obvious that two first aforementioned CMEMC coupling mechanisms represent some exchange mechanisms and do not pertain entirely to one of three effects mentioned above. It is possible to assume that only the third mechanism is relevant to the ME effect. Exploiting one of the coupling mechanisms, suitable solutions can be found. This was discussed in reference [31]. This theoretical report investigates the SH-wave propagation in the transversely isotropic (6 mm) PEM solids and adds the twelfth and thirteenth new SH-SAWs to the family of already known eleven SH-SAWs briefly discussed above. To obtain the explicit forms for the new SH-wave velocities is the main purpose of the following three sections.

## 2. Theory Leading to an Extra Two New Results

_{1}, x

_{2}, x

_{3}} also known as the Cartesian coordinate system invented by Descartes, the famous mathematician. First of all, it is necessary to choose proper propagation direction in order to deal with the pure SH-wave coupled with both the electrical (φ) and magnetic (ψ) potentials. The fitting propagation direction for the 6 mm solid is described in reference [32]. For this purpose, the propagation direction is managed along the x

_{1}-axis and perpendicular to the 6-fold symmetry axis. The x

_{2}-axis is directed along the 6-fold symmetry axis and the x

_{3}-axis is managed along the normal to the free surface of the ME material. The coordinate beginning is situated at the interface between the solid and a vacuum and the x

_{3}-axis negative values are managed towards the depth of the piezoelectromagnetics. It is very important to state that the problem of acoustic wave propagation coupled with both the electrical and magnetic potentials is treated. Therefore, the quasi-static approximation must be used here because the speed of any acoustic wave is five orders slower than the speed of the electromagnetic wave in a vacuum or solid.

_{1}and U

_{3}along the x

_{1}-axis and x

_{3}-axis, respectively. Besides, the second set is for the mechanical SH-wave possessing the mechanical displacement U

_{2}coupled with both the electrical and magnetic potentials. This theoretical report has an interest in the study of the second case. So, the differential form of the coupled equations of motion representing the partial differential equations of the second order can be composed as follows:

_{2}-axis is U = U

_{2}, and t is time. It is natural to write down the solutions of Equation (1) in the following plane wave form:

_{4}and the magnetic potential is ψ = U

_{5}. The unknown coefficients U

^{0}, φ

^{0}, and ψ

^{0}must be determined. The imaginary unity is defined by j = (– 1)

^{1/2}and ω is the angular frequency. The wavevector components are written as follows: {k

_{1},k

_{2},k

_{3}} = k{n

_{1},n

_{2},n

_{3}}, where k is the wavenumber in the propagation direction because the directional cosines are n

_{1}= 1, n

_{2}= 0, and n

_{3}≡ n

_{3}. A substitution of solutions (2) into Equation (1) leads to the tensor form of the coupled equations of motion known as the Green-Christoffel equation. The tensor form then reads:

_{ph}= ω/k stands for the phase velocity, V

_{t}

_{4}= (C/ρ)

^{1/2}denotes the purely mechanical bulk acoustic wave (BAW) with the shear-horizontal (SH) polarization, and $m=1+{n}_{3}^{2}$. All the suitable eigenvalues n

_{3}must be determined and the corresponding eigenvector components U

^{0}, φ

^{0}, and ψ

^{0}must be also obtained by resolving the set of three homogeneous Equations (3). For this purpose, an expansion of the determinant of the coefficient matrix in Equation (3) leads to a sixth order polynomial in one indeterminate n

_{3}. Therefore, six suitable eigenvalues n

_{3}must be found. They read as follows:

_{tem}represents the velocity of the SH-BAW coupled with both the electrical and magnetic potentials via the following coefficient of the magnetoelectromechanical coupling (CMEMC):

_{3}must have a negative sign. Let us use the first, third, and fifth eigenvalues as the suitable ones. Therefore, it is necessary to find now the eigenvector components by substituting the suitable eigenvalues into Equation (3). The problem of the suitability of the eigenvectors was also recently discussed in reference [38]. The reader can do the same and find the following forms of the first set of the eigenvector components:

^{(1)}, F

^{(3)}, and F

^{(5)}. Let us further use F

_{1}, F

_{2}, and F

_{3}instead of F

^{(1)}, F

^{(3)}, and F

^{(5)}, respectively. Also, it is natural to further exploit F = F

_{1}+ F

_{2}because there are two identical eigenvalues (4) that give two identical eigenvectors (11) or (18).

_{32}= 0. The electrical boundary conditions can be the continuity of the normal component of the electrical displacement D

_{3}, continuity of the electrical potential φ, D

_{3}= 0, or φ = 0. Besides, the possible magnetic boundary conditions are the continuity of the normal component of the magnetic displacement B

_{3}, continuity of the magnetic potential ψ, B

_{3}= 0, or ψ = 0. It is reasonable that all the boundary conditions will not be given in this report in their explicit final forms because the reader can find them in book [21] and open access publication [18,19].

_{0}= 10

^{−7}/(4πC

_{L}

^{2}) = 8.854187817 × 10

^{−12}(F/m), where C

_{L}= 2.99782458 × 10

^{8}(m/s) is the speed of light in a vacuum. The magnetic permeability constant is μ

_{0}= 4π × 10

^{−7}(H/m) = 12.5663706144 × 10

^{−7}(H/m). The constant ε

_{0}is the proportionality coefficient between the electric induction D

^{f}and the electric field E

^{f}: D

^{f}= ε

_{0}E

^{f}, where the superscript “f” is used for the free space (vacuum) and the electric field components can be defined as follows: E

^{f}

_{i}= − ∂φ

^{f}/x

_{i}. Similarly, the constant μ

_{0}is the proportionality coefficient between the vacuum magnetic induction B

^{f}and the vacuum magnetic field H

^{f}: B

^{f}= μ

_{0}H

^{f}, where the magnetic field components can be defined as follows: H

^{f}

_{i}= − ∂ψ

^{f}/x

_{i}. So, Laplace’s equations such as Δφ

^{f}= 0 and Δψ

^{f}= 0 can be used for the potentials in a vacuum. It is also required that both the potentials must exponentially vanish in a vacuum far from the free surface of the piezoelectromagnetics. Utilizing the vacuum parameters, let’s treat some additional new cases that were not recorded in book [21] and reference [18,19,22]. This is the main purpose of the following two sections.

## 3. The New Results for the Case of B_{3} = 0 and Continuity of Both D_{3} and φ

_{3}. Therefore, the vacuum electric constant ε

_{0}introduced in the previous section must be included in the further calculations. Using the first set of the eigenvector components given in Equations (11) and (12), the coupled three equations for this case (Equations (188), (189), and (190) in book [21]) are then rewritten as follows:

_{0}is taken into account.

_{new}

_{5}of the fifth new SH-SAW discovered in book [21]:

_{n}

_{5}already includes the vacuum electric constant ε

_{0}that is missing in the book. Therefore, this parameter can be expressed as follows:

_{0}= 0, this parameter can be simplified and written similar to the result obtained in book [21] and studied in reference [23]:

_{EM}= (εμ)

^{−1/2}and V

_{α}= 1/α stand for the electromagnetic wave speed and the exchange speed, respectively.

_{n}

_{5}= 0 in Equations (27) and (28) and V

_{new}

_{5}= V

_{tem}occurs in Equation (26). This means that the magnetoelectric effect is vital for the existence of the fifth new SH-SAW. Indeed, as soon as α <> 0, the fifth new SH-SAW can propagate. This is similar to the eighth and tenth new SH-SAWs discovered in reference [18,19], respectively, and studied in reference [20]. However, the existence conditions for the latter two new SH-SAWs are more complicated. These SH-SAWs cannot propagate in some PEM (composite) solids of class 6 mm [20] and require large enough value of the electromagnetic constant α. Generally, the value of the α is very small for real piezoelectromagnetics that are available today for commercial aims.

_{0}in calculations setting ε

_{0}= 0, Equation (30) reduces to the following form:

_{n}

_{12}(α = 0) reduces to the following form:

## 4. The New Results for the Case of D_{3} = 0 and Continuity of Both B_{3} and ψ

_{3}allows one to include the vacuum magnetic constant µ

_{0}. Using the second set of the eigenvector components given in Equations (18) and (19), the three coupled equations (namely, Equations (185), (186), and (187) from book [21]) can be rewritten in the following forms:

_{0}, the velocity V

_{new}

_{5}of the fifth new SH-SAW defined by Equation (26) depends on the parameter b

_{n}

_{5}. This parameter b

_{n}

_{5}already includes the vacuum magnetic constant µ

_{0}instead of ε

_{0}. So, the parameter b

_{n}

_{5}reads:

_{new}

_{13}of the thirteenth new SH-SAW that can be written as follows:

_{0}in (36), the parameter b

_{n}

_{13}reduces to the following form:

_{n}

_{13}reduces to the following form:

_{new}

_{12}≠ V

_{tem}due to b

_{n}

_{12}(α = 0) ≠ 0 in Equation (32) and V

_{new}

_{13}≠ V

_{tem}due to b

_{n}

_{13}(α = 0) ≠ 0 in Equation (38). It is obvious that some graphical investigations of the new results obtained in both this section and the previous sections are required. Also, it is possible to demonstrate the influence of the inclusion of the corresponding vacuum parameter (ε

_{0}or µ

_{0}) in the calculations. This is the main purpose of the following section.

## 5. Numerical Results and Discussion

_{3}–CoFe

_{2}O

_{4}and PZT-5H–Terfenol-D) given in references [39,40]. The PZT-5H–Terfenol-D material parameters are: C = 1.45 × 10

^{10}(N/m

^{2}), e = 8.5 [C/m

^{2}], h = 83.8 (T), ε = 75.0 × 10

^{−10}(F/m), μ = 2.61 × 10

^{−6}(N/A

^{2}), ρ = 8500 (kg/m

^{3}). This composite is known as one of the strongest piezoelectromagnetics. The composite material BaTiO

_{3}–CoFe

_{2}O

_{4}possesses significantly weaker piezoelectromagnetic properties. Therefore, the reader has a contrast for comparison and better understanding of the problem of the SH-wave propagation in the piezoelectromagnetic materials. The BaTiO

_{3}–CoFe

_{2}O

_{4}material parameters are: C = 4.4 × 10

^{10}(N/m

^{2}), e = 5.8 (C/m

^{2}), h = 275.0 (T), ε = 56.4 × 10

^{−10}(F/m), μ = 81.0 × 10

^{−6}(N/A

^{2}), ρ = 5730 (kg/m

^{3}).

_{new}

_{5}of the fifth new SH-SAW given by Equation (26) and discovered in book [21] several years ago. It depends on the parameter b

_{n}

_{5}defined by Equations (27), (28), and (34). Equations (27) and (34) contain the constants ε

_{0}and µ

_{0}, respectively. Figure 1 shows the dependence of the normalized velocity V

_{new}

_{5}/V

_{tem}on the electromagnetic constant α, namely on the normalized parameter α

^{2}/εµ. It is obvious that the value of V

_{new}

_{5}/V

_{tem}must be smaller than unity because one deals here with one of the surface acoustic waves. Also, the value of the α

^{2}/εµ must be confined between zero and unity to satisfy the limitation condition such as α

^{2}< εµ [4,5]. The insertion in Figure 1 demonstrates that at α = 0 (i.e., α

^{2}/εµ = 0) there is V

_{new}

_{5}= V

_{tem}and then for the larger values of the α

^{2}/εµ, the SH-SAW speed V

_{new}

_{5}is slightly slower than the SH-BAW speed V

_{tem}. However there is V

_{new}

_{5}= V

_{tem}anew at α

^{2}/εµ ~ 0.1565 for BaTiO

_{3}–CoFe

_{2}O

_{4}and at α

^{2}/εµ ~ 0.2793 for PZT-5H–Terfenol-D symbolizing the fact that there is no instability of the SH-BAW at such large enough values of α

^{2}/εµ. Indeed, V

_{new}

_{5}= V

_{tem}can occur when b

_{n}

_{5}changes its sign at some certain value of α

^{2}/εµ > 0 given above for both the studied solids. One can also read paper [23] on the study of the fifth new SH-SAW, in which the parameter Δ = V

_{tem}– V

_{new}

_{5}was used instead of V

_{new}

_{5}/V

_{tem}. For α

^{2}→ εµ there is V

_{tem}→ ∞ due to ${K}_{em}^{2}\to \infty $ and therefore, V

_{tem}>> V

_{new}

_{5}occurs resulting in V

_{new}

_{5}/V

_{tem}→ 0.

_{new}

_{12}/V

_{tem}on the parameter α

^{2}/εµ. This is shown in Figure 2. For the case of result (29) with (30) when the parameter ε

_{0}is included in the calculations, the velocity V

_{new}

_{12}touches the SH-BAW speed V

_{tem}at the same nonzero value of α

^{2}/εµ that is given in the context above for the case of V

_{new}

_{5}= V

_{tem}. After that right away, the speed V

_{new}

_{12}rapidly decreases down to zero and then even becomes imaginary, illuminating the dissipation phenomenon. The velocity V

_{new}

_{12}stays imaginary for the values of α

^{2}/εµ in the narrow α-range between α

^{2}/εµ ~ 0.1569 and ~0.1571 for BaTiO

_{3}–CoFe

_{2}O

_{4}and between ~ 0.27975 and ~ 0.28046 for PZT-5H–Terfenol-D. These are quite narrow α-ranges for the existence of the dissipation phenomenon due to taking into account the vacuum electric constant ε

_{0}. It is expected that a narrow α-range for the dissipation phenomenon can exist even for a very weak PEM material and this α-range can be naturally shifted towards smaller values of the normalized parameter α

^{2}/εµ.

_{new}

_{12}= V

_{tem}occurs when b

_{n}

_{12}= 0, i.e., b

_{n}

_{12}defined by Equation (30) changes its sign. Indeed, the value of b

_{n}

_{12}rapidly approaches an infinity and also quickly returns from an infinity already with the changed sign. It is clearly seen in Figure 2 that these peculiarities of V

_{new}

_{12}= V

_{tem}and the imaginary velocity V

_{new}

_{12}within the narrow α-range are absent for the case of Equation (29) with (31). Also, both cases (30) and (31) have the imaginary values of the velocity V

_{new}

_{12}at some large values of α

^{2}/εµ due to large values of (b

_{n}

_{12})

^{2}> 1: α

^{2}/εµ > ~ 0.96826 for BaTiO

_{3}–CoFe

_{2}O

_{4}and α

^{2}/εµ > ~ 0.85933 for PZT-5H–Terfenol-D. Note that for the case without ε

_{0}shown by the solid lines in Figure 2, this dissipation commences at slightly larger value of α

^{2}/εµ for PZT-5H–Terfenol-D. Such small discrepancy was not recorded for the weaker PEM composite BaTiO

_{3}–CoFe

_{2}O

_{4}.

^{2}/εµ that are smaller than some threshold value (α

^{2}/εµ)

_{th}for the 8th and 10th new SH-SAWs discovered in references [18,19], respectively. Some propagation peculiarities of the 8th and 10th new SH-SAWs were studied in reference [20]. If α

^{2}/εµ > (α

^{2}/εµ)

_{th}, the 8th and 10th new SH-SAWs can propagate. The value of the electromagnetic constant α is very small for real piezoelectromagnetic monocrystals and even composite materials. Therefore, the existence condition of α

^{2}/εµ > (α

^{2}/εµ)

_{th}for the 8th and 10th new SH-SAWs is extremely important. Also, the existence of the dissipation phenomenon with the imaginary propagation velocity at a narrow α-range, 0 < α

^{2}/εµ < (α

^{2}/εµ)

_{th}, is caused by the corresponding vacuum parameter, ε

_{0}or µ

_{0}. Theoretical work [20] also states that the 8th and 10th new SH-SAWs are apt for constitution of technical devices to study (to sense) the magnetoelectric effect.

_{new}

_{13}/V

_{tem}in dependence on the nondimensional parameter α

^{2}/εµ for composite materials BaTiO

_{3}–CoFe

_{2}O

_{4}and PZT-5H–Terfenol-D. The figure insertion illuminates the dependence of V

_{new}

_{13}(α) for small values of α

^{2}/εµ. It is clearly seen for one of the strongest PEM composites such as PZT-5H–Terfenol-D that the inclusion of the vacuum parameter µ

_{0}in the calculations with Equations (35) and (36) can result in disappearance of the dissipation phenomenon existing at the large values of α

^{2}/εµ ~ 1. For the significantly weaker PEM composite such as BaTiO

_{3}–CoFe

_{2}O

_{4}, this phenomenon cannot exist. Also, the insertion clearly demonstrates that at the very small values of α

^{2}/εµ, the speed V

_{new}

_{13}(α → 0) is slower for the case of Equations (35) and (37) without the vacuum parameter µ

_{0}in comparison with the case of Equations (35) and (36). This means that the penetration depth for the 13th new SH-SAW is larger for the latter case.

^{2}<< εµ [4] and [5]. This means that the aforementioned dissipation phenomenon cannot be reachable for all known PEM monocrystals and composites. Indeed, this research is primary and allows the further theoretical investigations of various SH-waves in PEM plates. In general, the plate waves have the following peculiarity: at large values of the plate thickness the speed of the plate SH-wave must approach the speed of the corresponding SH-SAW. Therefore, it is helpful to know the characteristics of SAWs before development of any theory of SH-wave propagation in plates. It is well-known that plates are used for further miniaturization of various technical devices with a higher level of integration. Also, various SH-waves such as SH-SAWs and plate SH-waves can be significantly more sensitive to various chemicals. This fact is called for implementation in sensor technologies. These SH-waves can be also used in nondestructive testing and evaluation of the certain surfaces of various PEM solids.

_{GC}= (γη)

^{−1/2}, where γ and η are the gravitic constant and cogravitic constant, respectively, instead of the electromagnetic wave speed V

_{EM}= (εμ)

^{−1/2}. For the purely electromagnetic case, it is also possible to state that speeds (29) and (35) depend on the speed of light in a vacuum, C

_{L}= (ε

_{0}μ

_{0})

^{−1/2}because ε

_{0}= 1/(μ

_{0}C

_{L}

^{2}) and μ

_{0}= 1/(ε

_{0}C

_{L}

^{2}). In the purely gravitational case, there are the vacuum gravitic constant γ

_{0}= 1/(η

_{0}C

_{L}

^{2}) and the vacuum cogravitic constant η

_{0}= 1/(γ

_{0}C

_{L}

^{2}) due to C

_{L}= (γ

_{0}η

_{0})

^{−1/2}. However, it is expected that sound experimental verifications of some gravitational phenomena can be released only in several decades. For instance, the theoretically predicted existence of gravitational waves by Albert Einstein in 1916 [45] was verified only in 2016 [46] with very expensive experiments carried out by an international group consisting of over 1000 researchers. This happens due to the weakness of the gravitational phenomena. However, these phenomena are extremely important. Therefore, it is expected that it is preferable for the first time to deal with the electromagnetic phenomena regarding the experimental proof. In the case of the acoustic wave propagation in solids, taking into account both the gravitational and the electromagnetic phenomena can result in the dependence of the acoustic wave velocity on both the speeds V

_{EM}and V

_{GC}as well as on the speeds of the new fast waves that can propagate in both solids and a vacuum. In both continuous media the propagation speeds of the new fast waves can be thirteen orders faster than the speed of light. The propagation of these new fast waves [47,48] in a vacuum represents a great interest due to a possibility to develop the instant interplanetary communication [47]. For this purpose, it is necessary to focus on solutions of many theoretical, mathematical, experimental, and engineering problems that can lead to the development of perfect infrastructures for the instant interplanetary communication.

## 6. Conclusions

_{0}or µ

_{0}, on the propagation velocity of the new SH-SAWs. For the fifth new SH-SAW discovered in book [21], it was found that the influence of the inclusion of the ε

_{0}or µ

_{0}is not significant and these vacuum parameters can be neglected in calculations. However, this is not true for the 12th and 13th new SH-SAWs discovered in this paper.

_{0}for the 12th new SH-SAW there can be found one extra narrow α-range for α

^{2}/εµ << 1, in which the propagation velocity becomes imaginary, i.e., there is some dissipation. This dissipation phenomenon can also exist at large values of the nondimensional parameter α

^{2}/εµ < ~ 1. For the 13th new SH-SAW there is the other picture: the presence of the vacuum magnetic constant µ

_{0}can result in the disappearance of the dissipation phenomenon for the values of the α

^{2}/εµ being just below unity. This was found only for PZT-5H–Terfenol-D representing one of the strongest PEM solids. The obtained theoretical results can be useful for the further investigations of the SH-wave propagation in PEM plates, nondestructive testing and evaluation, constitution of various technical devices based on PEM SH-SAWs. Also, some gravitational phenomena discussed in recently developed theory [43] and [44] can be applied as a supplementary to the electromagnetic properties or instead of them.

## Funding

## Conflicts of Interest

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**Figure 1.**The influence of the inclusion of the vacuum parameters ε

_{0}and µ

_{0}in the dependence of the normalized velocity V

_{new}

_{5}/V

_{tem}on the electromagnetic constant α, i.e., on the nondimensional parameter α

^{2}/εµ. With Equation (26) using (28), the gray and black lines are calculated for BaTiO

_{3}–CoFe

_{2}O

_{4}and PZT-5H–Terfenol-D, respectively. The dotted lines are calculated by using (27) with ε

_{0}and (34) with µ

_{0}. Note that all three cases (27), (28), and (34) coincide which is clearly seen in the inclusion.

**Figure 2.**The normalized velocity V

_{new}

_{12}/V

_{tem}versus the nondimensional parameter α

^{2}/εµ. The gray and black lines are for BaTiO

_{3}–CoFe

_{2}O

_{4}and PZT-5H–Terfenol-D, respectively. Using Equation (29), the dotted and solid lines are for the cases with the parameter ε

_{0}in Equation (30) and without ε

_{0}in Equation (31), respectively. The insertion shows the dissipation phenomenon by the dotted black line at α

^{2}/εµ ~ 0.28 for PZT-5H–Terfenol-D.

**Figure 3.**The normalized velocity V

_{new}

_{13}/V

_{tem}versus the nondimensional parameter α

^{2}/εµ. The gray and black lines are for BaTiO

_{3}–CoFe

_{2}O

_{4}and PZT-5H–Terfenol-D, respectively. The dotted lines correspond to the calculations with Equation (35) when the parameter µ

_{0}is included in Equation (36). The solid lines are for case (37) without the parameter µ

_{0}. The insertion shows the dependence for small values of α

^{2}/εµ.

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**MDPI and ACS Style**

Zakharenko, A.A.
On Discovery of the Twelfth and Thirteenth New Nondispersive SH-SAWs in 6 mm Magnetoelectroelastics. *Acoustics* **2019**, *1*, 749-762.
https://doi.org/10.3390/acoustics1040044

**AMA Style**

Zakharenko AA.
On Discovery of the Twelfth and Thirteenth New Nondispersive SH-SAWs in 6 mm Magnetoelectroelastics. *Acoustics*. 2019; 1(4):749-762.
https://doi.org/10.3390/acoustics1040044

**Chicago/Turabian Style**

Zakharenko, Aleksey Anatolievich.
2019. "On Discovery of the Twelfth and Thirteenth New Nondispersive SH-SAWs in 6 mm Magnetoelectroelastics" *Acoustics* 1, no. 4: 749-762.
https://doi.org/10.3390/acoustics1040044