6. Explicit theory for hexagonal crystals

In this section based on [11], we present the explicit analytical theory of the effect under consideration for a hexagonal medium with transversely isotropic elastic properties, which makes it possible to specify the above general relations and express the geometric conditions for mode conversion in terms of the moduli of elasticity of crystal. Analytical considerations will be supplemented with numerical calculations for some of hexagonal crystals.

To describe a hexagonal crystal, we use a standard crystallographic system of coordinates with the z axis oriented parallel to principal symmetry axis 6 and the x and y axes orthogonal to the z axis and lying in the basal plane of transverse isotropy [14, 15]. We choose the crystal boundary P<sup>0</sup> to be parallel to the axis 6 so that the normal n<sup>0</sup> to this boundary is directed along the y axis. In this geometry, an EBW with the polarization A02||z can propagate along the crystal surface in the direction m0||х with a speed:

$$
\hat{w}\_0 = \sqrt{c\_{44}/\rho}. \tag{29}
$$

For transverse isotropy, we may change the initial crystal surface orientation P<sup>0</sup> ! P, rotating its normal vector n around m<sup>0</sup> (i.e., choosing in Figure 2a the angle χ = π/2) by a small angle ψ

$$\mathbf{n}\_0 \to \mathbf{n} = (\mathbf{0}, \cos \psi, \sin \psi). \tag{30}$$

In addition, as before, we introduce a perturbed propagation direction m rotated relative to the vector m<sup>0</sup> by a small angle ϕ in the new surface plane P:

$$\begin{aligned} \mathbf{m}\_0 \to \mathbf{m} &= \mathbf{m}\_0 \cos \varphi + [\mathbf{n} \times \mathbf{m}\_0] \sin \varphi, \\ [\mathbf{n} \times \mathbf{m}\_0] &= (\mathbf{0}, \sin \varphi, -\cos \varphi). \end{aligned} \tag{31}$$

Based on the standard equations of crystal acoustics [14, 15], one can determine the wave parameters entering superposition (1)

$$\begin{aligned} \mathbf{A}\_{1,4} &= (\pm p, \mathbf{1}, \mathbf{0}) \sqrt{c\_{66}/c\_{44}}, \\ \mathbf{L}\_{1,4} &= \{2c\_{66} - c\_{44}, -2c\_{66}p, (\pm p\wp - \varrho)c\_{44}\} \sqrt{c\_{66}/c\_{44}}, \end{aligned} \tag{32}$$

$$\begin{aligned} \mathbf{k}\_{1,4} &= k(\mathbf{1}, \pm p, -\rho); \\ \mathbf{A}\_2 &= (\rho d/\Delta\_{14}, 0, 1), \\ \mathbf{L}\_2 &= \{\boldsymbol{\wp}, \beta \boldsymbol{\wp}, \delta \boldsymbol{p} + \tilde{\beta} \boldsymbol{\wp} \boldsymbol{\wp}\} \boldsymbol{c}\_{44}, \\ \mathbf{k}\_2 &= k(\mathbf{1} - \boldsymbol{\wp}^2/2, \delta \boldsymbol{p} + \varrho \boldsymbol{\wp}, -\varrho); \end{aligned} \tag{33}$$

Figure 4 shows the optimization of the parameters δα<sup>m</sup> and η<sup>m</sup> for the lithium niobate crystal that corresponds to the value K<sup>2</sup> = 5 at the variation of the orientations of the normal n to the surface, i.e., the angles ψ and χ. For each n direction, the surface K2(φ, δα) similar to that shown in Figure 3 was plotted from which the φ<sup>m</sup> and δα<sup>m</sup> values corresponding to the extremal point at the "crest" with the amplitude K<sup>2</sup> = 5 were determined. Consequently, each point on the δαmð Þ ψ; χ and ηmð Þ ψ; χ surfaces in Figure 4 corresponds to a certain angle φm. As is seen in the figure, the variations of the angles ψ and χ can significantly increase δα<sup>m</sup> and ηm.

<sup>p</sup> versus δα for the series of acoustic crystals (1—BaTiO3,

Angle of incidence δα<sup>m</sup> and efficiency η<sup>m</sup> versus the angles ψ and χ at K2 = 5 for lithium niobate (LiNbO3)

crystal at m0||x2, n0||y, x<sup>2</sup> = (cosθ2, 0, sinθ2), and θ<sup>2</sup> = 0.46.

Figure 4.

Acoustics of Materials

Figure 5.

168

Numerical plot of the product Kmax

2

ffiffiffiffiffi δα

2—paratellurite, 3—graphite, 4—CdCe, 5—ZnS, 6—LiF, 7—LiNbO3, 8—Si).

$$\begin{aligned} \mathbf{A}\_3 &= \{ \mathbf{1}, iq, (-\varrho + iq\wp)d/\Delta\_{14} \} \sqrt{c\_{11}/c\_{44}}, \\ \mathbf{L}\_3 &= \{ 2\mathbf{i}c\_{66}q, c\_{44} - 2c\_{66}, -c\_{44}(\wp + i\kappa q\wp) \} \sqrt{c\_{11}/c\_{44}}, \\ \mathbf{k}\_3 &= k(\mathbf{1}, iq, -\varrho). \end{aligned} \tag{34}$$

The following designations are introduced in formulas (32)–(34):

$$p = \sqrt{\Delta\_{46}/c\_{66}} \quad q = \sqrt{\Delta\_{14}/c\_{11}} \tag{35}$$

<sup>λ</sup>″φ<sup>2</sup> � <sup>a</sup>λ<sup>0</sup>

the rotation angles of the boundary (ψ) and sagittal plane (ϕ):

� � � � �

Resonance Compression of Acoustic Beams in Crystals DOI: http://dx.doi.org/10.5772/intechopen.82364

second equation in (43) yields:

δp � con <sup>¼</sup> <sup>φ</sup><sup>2</sup> con 1 <sup>λ</sup><sup>0</sup> <sup>∓</sup> γ β

maximum value δα �

as:

ψ 2

171

real:

<sup>ψ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>,

The first equation yields two versions of the mode conversion relationship between

ffiffiffiffiffiffi λ″ aλ<sup>0</sup>

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>c</sup>66ð Þ <sup>c</sup><sup>44</sup> � <sup>c</sup><sup>66</sup> <sup>p</sup>

> con <sup>¼</sup> <sup>φ</sup><sup>4</sup> con 2p

> > con 2p

γ<sup>2</sup> � �s2=s<sup>4</sup>

1 <sup>λ</sup><sup>0</sup> <sup>∓</sup> γ β \_ � �<sup>2</sup>

, 0 and the lower sign at β

1 <sup>λ</sup><sup>0</sup> <sup>þ</sup> γ β \_ � � � � � �

� �<sup>2</sup>

<sup>c</sup>44=c<sup>66</sup> <sup>p</sup> : (48)

4 1 <sup>þ</sup> <sup>a</sup><sup>2</sup>γ<sup>4</sup> ð Þ : (49)

<sup>¼</sup> <sup>0</sup>: (43)

: (44)

� � � � �

: (45)

=2p (39). The

: (46)

. 0):

: (47)

,

\_

<sup>φ</sup><sup>2</sup> � <sup>λ</sup><sup>0</sup> <sup>δ</sup><sup>p</sup> <sup>þ</sup> φψ β � �\_

ψcon ¼ � γφcon, γ ¼

Substituting the parameter values from Eq. (42) into the radicand of Eq. (44), we can easily see that for с<sup>44</sup> > с<sup>66</sup> in the case under consideration, the γ value is always

<sup>γ</sup> <sup>¼</sup> ð Þ <sup>c</sup><sup>44</sup> � 2c<sup>66</sup> ½ � <sup>c</sup>44ð Þþ <sup>c</sup><sup>11</sup> � 2c<sup>66</sup> <sup>c</sup>13ð Þ <sup>c</sup><sup>44</sup> � 2c<sup>66</sup>

In other words, at с<sup>44</sup> > с66, there are always two crystal cuts, i.e., two versions of coupled orientations of the boundary and sagittal planes, which ensure mode conversion near angle α^ of total internal reflection. Certainly, each set of chosen angles

Thus, the resonance width with respect to the incidence angle is indeed very small: δα ∝ φ4. As we have seen, it is preferable to choose the sign corresponding to the

\_

, δαcon <sup>¼</sup> <sup>φ</sup><sup>4</sup>

Given the found relations for reflectances (40) and (41), we can obtain the gain

<sup>s</sup><sup>1</sup> <sup>≈</sup> <sup>s</sup>4, s2=s<sup>4</sup> <sup>≈</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Here we took into account that for the studied reflection geometry with the sagittal plane close to transverse isotropy, the ray speeds s<sup>α</sup> are approximately equal to the phase speeds of the bulk waves involved in the reflection. Under the total conversion conditions (44)–(47), the coefficient K<sup>1</sup> is 0; therefore, the excitation efficiency of beam r2 is maximum: η = 1 = 100%. In this case, the gain K<sup>2</sup> can be written

, Gcon <sup>¼</sup> <sup>μ</sup><sup>2</sup> <sup>þ</sup> <sup>μ</sup>~<sup>2</sup>

The above analysis based on expansion of the equations in small parameters φ<sup>2</sup>

, δp, and δα is approximate. Therefore, the range of applicability of the results obtained may differ, depending on the degree of crystal anisotropy and other factors. In Figure 6, the analytical linear relation between the conversion angles φ

2c44ð Þ c<sup>11</sup> � c<sup>44</sup>

<sup>ψ</sup> <sup>=</sup> �γφ corresponds to its own definite angle of incidence: δα <sup>¼</sup> <sup>δ</sup>p<sup>2</sup>

con (the upper sign at β

ψcon ¼ �γφcon sgn β

(K2) and loss (K1) coefficients (8) where one can put

Kcon <sup>2</sup> <sup>¼</sup> <sup>G</sup>con φ2 con

\_ � �, δα �

\_

$$\beta = \frac{c\_{12}d - \Delta\_{13}\Delta\_{14}}{c\_{44}\Delta\_{14}}, \ \ \tilde{\beta} = \frac{c\_{13}d - \Delta\_{34}\Delta\_{14}}{c\_{44}\Delta\_{14}}, \ \ \kappa = 1 + \frac{d}{\Delta\_{14}};\tag{36}$$

$$
\Delta\_{\vec{\imath}\vec{\jmath}} = \mathcal{c}\_{\vec{\imath}\vec{\imath}} - \mathcal{c}\_{\vec{\jmath}\vec{\jmath}}, \quad \mathcal{d} = \mathcal{c}\_{44} + \mathcal{c}\_{13}.\tag{37}
$$

We assume the parameters q and p to be real, which holds true at c<sup>11</sup> . c<sup>44</sup> . c66. Note that the inequality c<sup>11</sup> . c<sup>66</sup> is always satisfied (this is the crystal stability condition [14]) and the inequality c<sup>11</sup> . c<sup>44</sup> is almost always satisfied (we do not know exclusions). Meanwhile, the condition c<sup>44</sup> . c66, which indicates that EBW (29) belongs to the middle sheet of the slowness surface (Figure 2b), is valid in far from all crystals (say, in a half of them).

As before, the angle of incidence is chosen near the angle α^ of total internal reflection (Figure 2b). This angle corresponds to the limiting speed, which now can be directly related to the EBW speed v ^<sup>0</sup> (29):

$$
\hat{\boldsymbol{\nu}}^2 = \hat{\boldsymbol{\nu}}\_0^2 \left( \mathbf{1} - \stackrel{\frown}{\boldsymbol{\beta}} \,\, \boldsymbol{\varrho}^2 \right), \qquad \hat{\boldsymbol{\beta}} = \frac{\,\, \hat{\boldsymbol{d}}^2 - \Delta\_{14} \Delta\_{34}}{c\_{14} \Delta\_{14}}.\tag{38}
$$

In turn, the small tuning angle δα corresponds to the interval δv ¼ v � v ^ and, consequently, to the parameter δp:

$$
\delta v = \hat{v}\_0 p \delta a = \frac{1}{2} \hat{v}\_0 \delta p^2, \qquad \delta a = \delta p^2 / 2p. \tag{39}
$$

Substituting expressions (32)–(34) for the vectors L<sup>α</sup> into Eq. (5), we obtain reflectances R<sup>1</sup> and R<sup>2</sup> as functions of the moduli of elasticity and perturbation parameters ϕ, ψ, and δp:

$$R\_1 = \frac{\rho^2 + i a \rho^2 - (\lambda' + i \lambda'') \left(\delta p + \stackrel{\frown}{\beta} \,\,\rho \mu\right)}{-\rho^2 + i a \rho^2 - (\lambda' - i \lambda'') \left(\delta p + \stackrel{\frown}{\beta} \,\,\rho \mu\right)},\tag{40}$$

$$R\_2 = \frac{\mu \rho + i \ddot{\mu} \varphi}{\rho^2 - i a \psi^2 + (\lambda' - i \lambda'') \left(\delta p + \stackrel{\frown}{\beta} \,\,\rho \psi\right)},\tag{41}$$

where

$$\begin{split} a &= \frac{p}{q\beta^2}, \quad \lambda' = p \left(\frac{2c\_{66}}{\beta c\_{44}}\right)^2, \quad \lambda'' = \frac{1}{q} \left(\frac{2c\_{66} - c\_{44}}{\beta c\_{44}}\right)^2, \\ \mu &= \frac{4pc\_{66}}{\beta c\_{44}} \sqrt{\frac{c\_{66}}{c\_{44}}}, \quad \tilde{\mu} = \frac{2p(2c\_{66} - c\_{44})}{q\beta^2 c\_{44}} \sqrt{\frac{c\_{66}}{c\_{44}}}. \end{split} \tag{42}$$

Taking into account Eq. (40), we can reduce the conversion condition R<sup>1</sup> = 0 to the system of equations:

Resonance Compression of Acoustic Beams in Crystals DOI: http://dx.doi.org/10.5772/intechopen.82364

A<sup>3</sup> ¼ f g 1; iq;ð Þ �φ þ iqψ d=Δ<sup>14</sup>

The following designations are introduced in formulas (32)–(34):

<sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ46=c<sup>66</sup>

k<sup>3</sup> ¼ kð Þ 1; iq; �φ :

Acoustics of Materials

<sup>β</sup> <sup>¼</sup> <sup>c</sup>12<sup>d</sup> � <sup>Δ</sup>13Δ<sup>14</sup> c44Δ<sup>14</sup>

from all crystals (say, in a half of them).

be directly related to the EBW speed v

consequently, to the parameter δp:

parameters ϕ, ψ, and δp:

the system of equations:

where

170

v ^<sup>2</sup> <sup>¼</sup> <sup>v</sup> ^2 <sup>0</sup> 1� β \_ <sup>φ</sup><sup>2</sup> � �

δv ¼ v

R<sup>1</sup> ¼

<sup>a</sup> <sup>¼</sup> <sup>p</sup>

<sup>q</sup>β<sup>2</sup> , <sup>λ</sup><sup>0</sup> <sup>¼</sup> <sup>p</sup>

<sup>μ</sup> <sup>¼</sup> <sup>4</sup>pc<sup>66</sup> βc<sup>44</sup>

<sup>L</sup><sup>3</sup> <sup>¼</sup> f g <sup>2</sup>ic66q;c<sup>44</sup> � <sup>2</sup>c66; �c44ð Þ <sup>ψ</sup> <sup>þ</sup> <sup>i</sup>κq<sup>φ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>p</sup> , q <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

, <sup>β</sup><sup>~</sup> <sup>¼</sup> <sup>c</sup>13<sup>d</sup> � <sup>Δ</sup>34Δ<sup>14</sup> c44Δ<sup>14</sup>

We assume the parameters q and p to be real, which holds true at c<sup>11</sup> . c<sup>44</sup> . c66.

Note that the inequality c<sup>11</sup> . c<sup>66</sup> is always satisfied (this is the crystal stability condition [14]) and the inequality c<sup>11</sup> . c<sup>44</sup> is almost always satisfied (we do not know exclusions). Meanwhile, the condition c<sup>44</sup> . c66, which indicates that EBW (29) belongs to the middle sheet of the slowness surface (Figure 2b), is valid in far

As before, the angle of incidence is chosen near the angle α^ of total internal reflection (Figure 2b). This angle corresponds to the limiting speed, which now can

^<sup>0</sup> (29):

In turn, the small tuning angle δα corresponds to the interval δv ¼ v � v

2 v ^0δp<sup>2</sup>

Substituting expressions (32)–(34) for the vectors L<sup>α</sup> into Eq. (5), we obtain reflectances R<sup>1</sup> and R<sup>2</sup> as functions of the moduli of elasticity and perturbation

<sup>φ</sup><sup>2</sup> <sup>þ</sup> iaψ<sup>2</sup> � <sup>λ</sup><sup>0</sup> ð Þ <sup>þ</sup> <sup>i</sup>λ″ <sup>δ</sup>p<sup>þ</sup> <sup>β</sup>

�φ<sup>2</sup> <sup>þ</sup> iaψ<sup>2</sup> � <sup>λ</sup><sup>0</sup> ð Þ � <sup>i</sup>λ″ <sup>δ</sup>p<sup>þ</sup> <sup>β</sup>

<sup>φ</sup><sup>2</sup> � iaψ<sup>2</sup> <sup>þ</sup> <sup>λ</sup><sup>0</sup> ð Þ � <sup>i</sup>λ″ <sup>δ</sup>p<sup>þ</sup> <sup>β</sup>

, <sup>λ</sup>″ <sup>¼</sup> <sup>1</sup>

, <sup>μ</sup><sup>~</sup> <sup>¼</sup> <sup>2</sup>pð Þ <sup>2</sup>c<sup>66</sup> � <sup>c</sup><sup>44</sup> qβ<sup>2</sup> c<sup>44</sup>

Taking into account Eq. (40), we can reduce the conversion condition R<sup>1</sup> = 0 to

q

^0pδα <sup>¼</sup> <sup>1</sup>

<sup>R</sup><sup>2</sup> <sup>¼</sup> μφ <sup>þ</sup> <sup>i</sup>μψ<sup>~</sup>

2c<sup>66</sup> βc<sup>44</sup> � �<sup>2</sup>

ffiffiffiffiffiffi c<sup>66</sup> c<sup>44</sup> r

, β \_

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi c11=c<sup>44</sup> p ,

Δ14=c<sup>11</sup>

Δij ¼ cii � cjj, d ¼ c<sup>44</sup> þ c13: (37)

<sup>¼</sup> <sup>d</sup><sup>2</sup> � <sup>Δ</sup>14Δ<sup>34</sup> c14Δ<sup>14</sup>

\_ φψ � �

> \_ φψ

\_ φψ

2c<sup>66</sup> � c<sup>44</sup> βc<sup>44</sup> � �<sup>2</sup>

> ffiffiffiffiffiffi c<sup>66</sup> c<sup>44</sup> r

:

, δα <sup>¼</sup> <sup>δ</sup>p<sup>2</sup>

, κ ¼ 1 þ

c11=c<sup>44</sup> p ,

p ; (35)

d Δ<sup>14</sup> (34)

; (36)

: (38)

=2p: (39)

� � , (40)

� � , (41)

,

(42)

^ and,

$$\begin{aligned} \lambda'' \rho^2 - a \lambda' \mu^2 &= 0, \\ \rho^2 - \lambda' \left( \delta p + \rho \mu \stackrel{\frown}{\beta} \right) &= 0. \end{aligned} \tag{43}$$

The first equation yields two versions of the mode conversion relationship between the rotation angles of the boundary (ψ) and sagittal plane (ϕ):

$$
\varphi\_{\rm con} = \pm \chi \rho\_{\rm con}, \quad \gamma = \sqrt{\frac{\lambda''}{a\lambda'}}.\tag{44}
$$

Substituting the parameter values from Eq. (42) into the radicand of Eq. (44), we can easily see that for с<sup>44</sup> > с<sup>66</sup> in the case under consideration, the γ value is always real:

$$\gamma = \left| \frac{(c\_{44} - 2c\_{66})[c\_{44}(c\_{11} - 2c\_{66}) + c\_{13}(c\_{44} - 2c\_{66})]}{2c\_{44}(c\_{11} - c\_{44})\sqrt{c\_{66}(c\_{44} - c\_{66})}} \right|. \tag{45}$$

In other words, at с<sup>44</sup> > с66, there are always two crystal cuts, i.e., two versions of coupled orientations of the boundary and sagittal planes, which ensure mode conversion near angle α^ of total internal reflection. Certainly, each set of chosen angles <sup>ψ</sup> <sup>=</sup> �γφ corresponds to its own definite angle of incidence: δα <sup>¼</sup> <sup>δ</sup>p<sup>2</sup> =2p (39). The second equation in (43) yields:

$$
\delta p\_{\rm con}^{\pm} = \rho\_{\rm con}^2 \left( \frac{\mathbf{1}}{\boldsymbol{\lambda}'} \mp \boldsymbol{\chi} \,\widehat{\boldsymbol{\beta}} \right), \qquad \delta \boldsymbol{a}\_{\rm con}^{\pm} = \frac{\rho\_{\rm con}^4}{2p} \left( \frac{\mathbf{1}}{\boldsymbol{\lambda}'} \mp \boldsymbol{\chi} \,\widehat{\boldsymbol{\beta}} \right)^2. \tag{46}
$$

Thus, the resonance width with respect to the incidence angle is indeed very small: δα ∝ φ4. As we have seen, it is preferable to choose the sign corresponding to the maximum value δα � con (the upper sign at β \_ , 0 and the lower sign at β \_ . 0):

$$\boldsymbol{\Psi}\_{\rm con} = -\chi \boldsymbol{\rho}\_{\rm con} \text{sgn } \stackrel{\frown}{\boldsymbol{\beta}} \,, \quad \delta \boldsymbol{a}\_{\rm con} = \frac{\boldsymbol{\rho}\_{\rm con}^{4}}{2p} \left( \frac{\mathbf{1}}{\boldsymbol{\lambda}} + \boldsymbol{\gamma} \left| \stackrel{\frown}{\boldsymbol{\beta}} \right| \right)^{2} . \tag{47}$$

Given the found relations for reflectances (40) and (41), we can obtain the gain (K2) and loss (K1) coefficients (8) where one can put

$$
\mathfrak{s}\_1 \approx \mathfrak{s}\_4, \quad \mathfrak{s}\_2/\mathfrak{s}\_4 \approx \sqrt{\mathfrak{c}\_{44}/\mathfrak{c}\_{66}}.\tag{48}
$$

Here we took into account that for the studied reflection geometry with the sagittal plane close to transverse isotropy, the ray speeds s<sup>α</sup> are approximately equal to the phase speeds of the bulk waves involved in the reflection. Under the total conversion conditions (44)–(47), the coefficient K<sup>1</sup> is 0; therefore, the excitation efficiency of beam r2 is maximum: η = 1 = 100%. In this case, the gain K<sup>2</sup> can be written as:

$$K\_2^{\rm con} = \frac{G\_{\rm con}}{\rho\_{\rm con}^2}, \quad G\_{\rm con} = \frac{(\mu^2 + \bar{\mu}^2 \chi^2)\mathfrak{s}\_2/\mathfrak{s}\_4}{4(1 + a^2 \chi^4)}. \tag{49}$$

The above analysis based on expansion of the equations in small parameters φ<sup>2</sup> , ψ 2 , δp, and δα is approximate. Therefore, the range of applicability of the results obtained may differ, depending on the degree of crystal anisotropy and other factors. In Figure 6, the analytical linear relation between the conversion angles φ

true even when the moduli are not so close to each other. For instance, at |c44�2c66|/ <sup>c</sup><sup>44</sup> � 1/3, Eq. (50) gives the estimate <sup>K</sup><sup>1</sup> � <sup>10</sup>�<sup>2</sup> and efficiency <sup>η</sup> = 1�K<sup>1</sup> becomes � 99%. In accordance with reference book [16], there are quite a number of hexagonal crystals where K<sup>1</sup> ≪ 1. Below, we shall also give examples of crystals of

This motivates us to a short consideration of the simplified approach to the reflection resonance with unchanged crystal surface orientation. As was shown in [8], the description becomes especially compact if to choose the crystal boundary parallel to the plane of crystal symmetry. By the way, such planes exist in all crystals, except triclinic [14, 15]. In this case, expressions (21) and (22) acquire the

<sup>2</sup> <sup>þ</sup> ð Þ <sup>λ</sup>″δ<sup>p</sup>

<sup>2</sup> <sup>þ</sup> ð Þ <sup>λ</sup>″δ<sup>p</sup>

<sup>2</sup> <sup>þ</sup> ð Þ <sup>λ</sup>″δ<sup>p</sup>

<sup>2</sup> <sup>þ</sup> ð Þ <sup>λ</sup>″δ<sup>p</sup>

2 s2=s<sup>4</sup>

δpφ<sup>2</sup>

2

<sup>2</sup> , (51)

<sup>2</sup> : (52)

<sup>2</sup> : (53)

<sup>2</sup> <sup>þ</sup> ð Þ <sup>λ</sup>″δ<sup>p</sup>

� �<sup>2</sup> " #: (55)

s4

<sup>2</sup> ð Þ <sup>φ</sup>; δα and <sup>η</sup>maxð Þ <sup>φ</sup>; δα . A dif-

: (56)

≪ 1. In this approximation, one

<sup>φ</sup><sup>2</sup> , G <sup>≈</sup> <sup>μ</sup>s<sup>2</sup>

<sup>2</sup> : (54)

φ2

φ<sup>2</sup> þ λ<sup>0</sup> ð Þ δp

1 2 λ″ λ0

<sup>K</sup><sup>1</sup> <sup>¼</sup> <sup>φ</sup><sup>2</sup> � <sup>λ</sup><sup>0</sup> ð Þ <sup>δ</sup><sup>p</sup>

<sup>K</sup><sup>2</sup> <sup>¼</sup> ð Þ μφ

<sup>η</sup> <sup>¼</sup> <sup>4</sup>λ<sup>0</sup>

φ<sup>2</sup> þ λ<sup>0</sup> ð Þ δp

Thus, for a fixed δp, the coefficients K<sup>2</sup> and η are determined by the same

δpFð Þ φ , Fð Þ¼ φ

Accordingly, their maximum magnitudes are determined by the same extremum

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>λ</sup>0<sup>2</sup> <sup>þ</sup> <sup>λ</sup>″<sup>2</sup>

, Kmax

ference between them depends on angle φ and crystal anisotropy. On the other hand, as was discussed above, the occurrence of those trajectories might be used for

Thus, one can conclude that the consistent variation of the sagittal plane orientation φ and the tuning incidence angle δα ∝ δp<sup>2</sup> (10) along one extremal trajectory (55) provides simultaneous optimization of both the gain and the efficiency of the resonance reflection. However, as was shown above, this coincidence is not an exact result but a consequence of our approximate calculations. The computer analysis based on exact formulas (5) without their expansion in small parameters leads to

≈ λ<sup>0</sup>

<sup>2</sup> ð Þ <sup>φ</sup>; δα <sup>≈</sup> <sup>G</sup>

δp 1 þ

q

2λ<sup>0</sup> � �<sup>2</sup>

And the efficiency of the resonance η = 1�K<sup>1</sup> is given by:

Fð Þ φ , η φð Þ¼ 4λ<sup>0</sup>

<sup>φ</sup><sup>2</sup> <sup>¼</sup> j j <sup>λ</sup> <sup>δ</sup><sup>p</sup> <sup>¼</sup> <sup>δ</sup><sup>p</sup>

The last estimate in (55) is valid when <sup>λ</sup>″=λ<sup>0</sup> ð Þ<sup>2</sup>

<sup>1</sup> ð Þ <sup>φ</sup>; δα <sup>≈</sup> <sup>λ</sup>″

distinct extremal trajectories for the functions Kmax

Kmin

φ<sup>2</sup> þ λ<sup>0</sup> ð Þ δp

φ<sup>2</sup> þ λ<sup>0</sup> ð Þ δp

monoclinic symmetry systems having got the same property.

Resonance Compression of Acoustic Beams in Crystals DOI: http://dx.doi.org/10.5772/intechopen.82364

structure:

function F(φ):

K2ð Þ¼ φ

condition:

obtains

173

μs<sup>2</sup> s4

#### Figure 6.

Correlations between the conversion angles ψcon and φcon (in radians) obtained by numerical calculation (solid lines) and approximate theoretical analysis (dashed lines) for the (1) Ti and (2) BeCu crystals.

and ψ (47) is compared with the results of numerical calculations based on formulas (5), without their expansion in small parameters, on the example of two (Ti, and BeCu) hexagonal crystals [16, 17].

In both cases, our computations not only practically confirm linearity of the relation between ϕ and ψ but also yield slopes of these dependences close to theoretical ones. In case of Ti, we have β \_ , 0 and coincident signs of angles ϕ and ψ, while for BeCu, β \_ . 0 and the angle signs are different.

We can only wonder why the predictions in the first order of the perturbation theory are confirmed so well by the exact numerical calculation in a wide range of angles ϕ, which are far from small. Anyhow, but the Ti crystal reveals one more "loyalty" with respect to our approximate theory: in its case, the geometries of the mode conversion and the extremal gain are almost identical.
