3. Singular reflection geometry in the vicinity of EBW

The deduced equations (5) represent just exact relations which describe the discussed resonance reflection only in some definite narrow region of orientations of surface and sagittal plane. Let us demonstrate that such singular region really arises close to the geometry of EBW propagation related to the pair {n0, m0}. Indeed, wave superposition (1) in this exceptional geometry is decomposed into two independent solutions—EBW and three-partial reflection—satisfying the boundary conditions, which are fragments of Eq. (2):

$$\mathbf{L}\_{02} = \mathbf{0},\tag{6}$$

(the intersection line between P and P<sup>0</sup> planes) is specified by the angle χ, counting from reference line 1||m<sup>0</sup> and lying in the range of [0, π]. The direction m in the P plane is specified by the angle φ measured from line 2 along the projection of the

(a) Perturbations of the crystal surface P<sup>0</sup> ! P, its normal orientation n<sup>0</sup> ! n and the propagation direction m<sup>0</sup> ! m; (b) the scheme of the resonance reflection (only outer and middle sheets of the slowness surface are

Below, to compact equations, we will also use the alternative notation for the

In fact, as we shall see, the squares of these angles (in units of radians) rather than

As is seen in Figure 2b, where the scheme of resonance reflection is shown, the

, A <sup>¼</sup> <sup>v</sup>

^<sup>1</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> i

∂pi ∂v 

v¼v^

the angles themselves are small parameters in the theory developed below.

v = ω/k of the wave field close to the limiting velocity v

controlling reflection resonance is directly related to the difference

4. Analytical description of resonance characteristics

½ � L4L1L<sup>3</sup> ≈ ϕ<sup>i</sup>

angle α of incidence must be chosen near the threshold angle of total internal reflection α^: α ¼ α^ þ δα. This configuration is characterized by the tracing speed

where κ is the radius of curvature of the cross section of the middle sheet of the slowness surface by the sagittal plane at the limiting point corresponding to v ¼ v

In this section, we shall obtain approximate expressions for the coefficients K<sup>1</sup> and K<sup>2</sup> defined by Eq. (8) in the general case of arbitrary anisotropy. Reflectivities R<sup>1</sup> and R<sup>2</sup> are expressed in (5) in terms of vectors Lα, which will be considered as functions of the parameters δp, φ, ψ, and χ (see Figure 2). The last (not small) angle χ, which determines the axis for n rotation, is considered fixed at this stage. Its influence on the effect will be studied numerically. Expressions (5) can be expanded

Given (7), the numerator in Eq. (5) for R<sup>2</sup> can be approximately represented as:

∂ ∂ϕi

½ � L4L1L<sup>3</sup> 

0

, (11)

ϕ<sup>1</sup> ¼ φ, ϕ<sup>2</sup> ¼ ψ: (9)

^. The adjusting angle δα

, (10)

^.

unperturbed vector m<sup>0</sup> on the boundary P.

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

introduced small angles:

^ ≈ δp<sup>2</sup>

in the other small parameters.

=2κ [8]:

δα <sup>¼</sup> <sup>δ</sup><sup>v</sup> v ^ <sup>A</sup> <sup>¼</sup> <sup>A</sup> 2κv ^ <sup>δ</sup>p<sup>2</sup>

δv ¼ v � v

163

Figure 2.

shown).

$$\mathbf{C}\_{r1}\mathbf{L}\_{01} + \mathbf{C}\_{l}\mathbf{L}\_{03} + \mathbf{C}\_{i}\mathbf{L}\_{04} = \mathbf{0},\tag{7}$$

where the subscript 0 indicates the initial unperturbed configuration. Of course, near this geometry, the vector L<sup>2</sup> should be small and the other vectors L1, L3, and L<sup>4</sup> nearly coplanar. As a result, both the numerator and denominator of the expression for R<sup>2</sup> in (5) should independently approach zero; i.e., R<sup>2</sup> is singular. This singularity is responsible for the resonance character of reflection and, therefore, for the discussed effect. The coefficient R<sup>1</sup> in (5) is regular because the small vector L<sup>2</sup> appears in both the denominator and the numerator of the expression for this coefficient.

Let us introduce practically important characteristics of the investigated resonance, specifically, the gain of excited wave intensity (K2) and energy loss (K1) in a parasitic wave:

$$K\_2 = \frac{P\_2}{P\_4} = |R\_2|^2 \frac{\mathfrak{s}\_2}{\mathfrak{s}\_4}, \qquad K\_1 = \frac{P\_1}{P\_4} = |R\_1|^2 \frac{\mathfrak{s}\_1}{\mathfrak{s}\_4}.\tag{8}$$

Here, Pα are the Poynting vector lengths, which are products of the energy density in the corresponding partial wave ( ∝ C<sup>2</sup> <sup>α</sup>) and its ray speed sα. Another useful characteristic of the resonance is the excitation efficiency η = 1�K1, equal to the fraction of energy transferred from the incident to excited wave.

Thus, a small perturbation of the crystal orientation in the vicinity of the geometry of the EBW propagation may provide a resonance intensification of the reflected wave r2. The control parameters of resonance are the optimized characteristics of geometry of reflection {n, m} (Figure 1a), i.e., the orientations of the surface and sagittal plane, as well as the angle of incidence α of the pump wave i related to these characteristics. The perturbation {n0, m0} ! {n, m} is shown in Figure 2a. The new plane boundary P of the crystal is specified by the unit normal n = n(ψ, χ) rotated by a small angle ψ = ∠(n, n0) with respect to the normal n<sup>0</sup> to the initial boundary P<sup>0</sup> along which the EBW can propagate. The axis of rotation

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

Figure 2.

the nonphysical inhomogeneous component were naturally excluded (C<sup>5</sup> = C<sup>6</sup> = 0)

The amplitude Ci of the incident wave is assumed to be known, while the remaining amplitudes may be expressed in terms of Ci through scalar multiplication of Eq. (2) by the vector products L<sup>2</sup> � L3, L<sup>1</sup> � L3, or L<sup>1</sup> � L2. As a result, we arrive at the following reflection coefficients in the form of the ratios of mixed

The deduced equations (5) represent just exact relations which describe the discussed resonance reflection only in some definite narrow region of orientations of surface and sagittal plane. Let us demonstrate that such singular region really arises close to the geometry of EBW propagation related to the pair {n0, m0}. Indeed, wave superposition (1) in this exceptional geometry is decomposed into two independent solutions—EBW and three-partial reflection—satisfying the

where the subscript 0 indicates the initial unperturbed configuration. Of course, near this geometry, the vector L<sup>2</sup> should be small and the other vectors L1, L3, and L<sup>4</sup> nearly coplanar. As a result, both the numerator and denominator of the expression for R<sup>2</sup> in (5) should independently approach zero; i.e., R<sup>2</sup> is singular. This singularity is responsible for the resonance character of reflection and, therefore, for the discussed effect. The coefficient R<sup>1</sup> in (5) is regular because the small vector L<sup>2</sup> appears in both the denominator and the numerator of the expression for this

Let us introduce practically important characteristics of the investigated resonance, specifically, the gain of excited wave intensity (K2) and energy loss (K1) in a

Here, Pα are the Poynting vector lengths, which are products of the energy density

Thus, a small perturbation of the crystal orientation in the vicinity of the geom-

characteristic of the resonance is the excitation efficiency η = 1�K1, equal to the

etry of the EBW propagation may provide a resonance intensification of the reflected wave r2. The control parameters of resonance are the optimized characteristics of geometry of reflection {n, m} (Figure 1a), i.e., the orientations of the surface and sagittal plane, as well as the angle of incidence α of the pump wave i related to these characteristics. The perturbation {n0, m0} ! {n, m} is shown in Figure 2a. The new plane boundary P of the crystal is specified by the unit normal n = n(ψ, χ) rotated by a small angle ψ = ∠(n, n0) with respect to the normal n<sup>0</sup> to the initial boundary P<sup>0</sup> along which the EBW can propagate. The axis of rotation

, K<sup>1</sup> <sup>¼</sup> <sup>P</sup><sup>1</sup>

P4

¼ j j R<sup>1</sup>

<sup>2</sup> s<sup>1</sup> s4

<sup>α</sup>) and its ray speed sα. Another useful

: (8)

, R<sup>2</sup> <sup>¼</sup> Cr2

Ci

<sup>¼</sup> ½ � <sup>L</sup>4L1L<sup>3</sup> ½ � L1L2L<sup>3</sup>

L<sup>02</sup> ¼ 0, (6)

Cr1L<sup>01</sup> þ ClL<sup>03</sup> þ CiL<sup>04</sup> ¼ 0, (7)

: (5)

¼ � ½ � <sup>L</sup>4L2L<sup>3</sup> ½ � L1L2L<sup>3</sup>

3. Singular reflection geometry in the vicinity of EBW

boundary conditions, which are fragments of Eq. (2):

<sup>K</sup><sup>2</sup> <sup>¼</sup> <sup>P</sup><sup>2</sup> P4

in the corresponding partial wave ( ∝ C<sup>2</sup>

¼ j j R<sup>2</sup>

fraction of energy transferred from the incident to excited wave.

<sup>2</sup> s<sup>2</sup> s4

from the sums in Eqs. (1) and (2).

<sup>R</sup><sup>1</sup> <sup>¼</sup> Cr<sup>1</sup> Ci

products:

Acoustics of Materials

coefficient.

162

parasitic wave:

(a) Perturbations of the crystal surface P<sup>0</sup> ! P, its normal orientation n<sup>0</sup> ! n and the propagation direction m<sup>0</sup> ! m; (b) the scheme of the resonance reflection (only outer and middle sheets of the slowness surface are shown).

(the intersection line between P and P<sup>0</sup> planes) is specified by the angle χ, counting from reference line 1||m<sup>0</sup> and lying in the range of [0, π]. The direction m in the P plane is specified by the angle φ measured from line 2 along the projection of the unperturbed vector m<sup>0</sup> on the boundary P.

Below, to compact equations, we will also use the alternative notation for the introduced small angles:

$$
\phi\_1 = \varphi, \quad \phi\_2 = \varphi. \tag{9}
$$

In fact, as we shall see, the squares of these angles (in units of radians) rather than the angles themselves are small parameters in the theory developed below.

As is seen in Figure 2b, where the scheme of resonance reflection is shown, the angle α of incidence must be chosen near the threshold angle of total internal reflection α^: α ¼ α^ þ δα. This configuration is characterized by the tracing speed v = ω/k of the wave field close to the limiting velocity v ^. The adjusting angle δα controlling reflection resonance is directly related to the difference δv ¼ v � v ^ ≈ δp<sup>2</sup> =2κ [8]:

$$
\delta a = \frac{\delta v}{\hat{v}} A = \frac{A}{2\kappa \hat{v}} \delta p^2, \qquad A = \frac{\hat{v}}{1 + p\_i^2} \left(\frac{\partial p\_i}{\partial v}\right)\_{v = \hat{v}}, \tag{10}
$$

where κ is the radius of curvature of the cross section of the middle sheet of the slowness surface by the sagittal plane at the limiting point corresponding to v ¼ v ^.

## 4. Analytical description of resonance characteristics

In this section, we shall obtain approximate expressions for the coefficients K<sup>1</sup> and K<sup>2</sup> defined by Eq. (8) in the general case of arbitrary anisotropy. Reflectivities R<sup>1</sup> and R<sup>2</sup> are expressed in (5) in terms of vectors Lα, which will be considered as functions of the parameters δp, φ, ψ, and χ (see Figure 2). The last (not small) angle χ, which determines the axis for n rotation, is considered fixed at this stage. Its influence on the effect will be studied numerically. Expressions (5) can be expanded in the other small parameters.

Given (7), the numerator in Eq. (5) for R<sup>2</sup> can be approximately represented as:

$$\left[\mathbf{L}\_{4}\mathbf{L}\_{1}\mathbf{L}\_{3}\right] \approx \phi\_{i} \left(\frac{\partial}{\partial\phi\_{i}}\left[\mathbf{L}\_{4}\mathbf{L}\_{1}\mathbf{L}\_{3}\right]\right)\_{0},\tag{11}$$

where the subscript 0 means that after differentiation, one should put ϕ = ψ = 0, δp = 0.

To calculate the denominator in the expressions for reflection coefficients R<sup>1</sup> and R<sup>2</sup> (5), let us expand the two cofactors separately:

$$\mathbf{L}\_3 \times \mathbf{L}\_1 \approx \mathbf{L}\_{03} \times \mathbf{L}\_{01} + \phi\_i \left( \frac{\partial}{\partial \phi\_i} [\mathbf{L}\_3 \times \mathbf{L}\_1] \right)\_0,\tag{12}$$

Gð Þ <sup>α</sup>

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

> <sup>K</sup><sup>1</sup> <sup>¼</sup> <sup>s</sup><sup>1</sup> s4

tends to zero, and at δp ≪ ϕiϕ<sup>j</sup> ≪ 1, it diverges.

5. Optimization of control parameters of reflection

λ″ ð Þ 4

(10), the incidence angle δα(φ).

effect which is under consideration.

compact form:

165

ij <sup>=</sup>Fð Þ <sup>α</sup> <sup>¼</sup> <sup>λ</sup>

ð Þ α ij � <sup>λ</sup>0ð Þ <sup>α</sup>

The substitution of Eq. (17) in terms of (20) into Eq. (8) gives

 

<sup>2</sup> <sup>δ</sup><sup>p</sup> <sup>þ</sup> <sup>λ</sup>0ð Þ <sup>4</sup>

<sup>K</sup><sup>2</sup> <sup>¼</sup> <sup>μ</sup>iϕ<sup>i</sup> ð Þ<sup>2</sup>

<sup>δ</sup><sup>p</sup> <sup>þ</sup> <sup>λ</sup>0ð Þ<sup>1</sup>

reducing the criterion of conversion K<sup>1</sup> = 0 to the system of equations:

ij <sup>ϕ</sup>iϕ<sup>j</sup> <sup>¼</sup> <sup>0</sup>, <sup>δ</sup><sup>p</sup> ¼ �λ0ð Þ <sup>4</sup>

where the first equation determines the relation between the angles of rotation of the sagittal plane ð Þ ϕ<sup>1</sup> � φ and the normal to the surface ð Þ ϕ<sup>2</sup> � ψ at a fixed position (χ) of the axis of rotation of the vector n (see Figure 2a). The second equation in (23) at the found relation between φ and ψ specifies the dependence δp(φ) and, by

The first requirement in (23) is reduced to a quadratic equation with respect to the ratio ψ/φ. The existence of real roots of this equation (and, therefore, mode conversion) is generally not guaranteed. However, numerical calculations for a number of crystals of different symmetry systems did not give us examples of the absence of such roots. Furthermore, as is shown in the next section, for hexagonal crystals, this equation always has real roots for the case c<sup>44</sup> > c66. Thus, in many crystals, the consistent variation of orientations of the surface and sagittal plane really can provide the mode conversion near the total internal reflection, i.e., the

The general conditions of mode conversion (23) can be represented in the

<sup>ψ</sup> <sup>¼</sup> <sup>γ</sup> � <sup>φ</sup>, <sup>δ</sup><sup>p</sup> <sup>¼</sup> <sup>λ</sup> � <sup>φ</sup><sup>2</sup>

These conditions with real roots γ � specify two variants of the orientations of the surface, sagittal plane, and angle of incidence (for each angle χ, see Figure 2a) that

As was expected, the gain K<sup>2</sup> given by Eq. (22) should obviously be large because

ensure the energy concentrating in the reflected beam r2.

it is inversely proportional to the square of the small parameter:

<sup>δ</sup><sup>p</sup> <sup>þ</sup> <sup>λ</sup>0ð Þ<sup>1</sup>

ij ϕiϕ<sup>j</sup> <sup>2</sup>

As could be expected, loss coefficient K<sup>1</sup> (21) is regular in the control parameters φ, ψ, and δp, whereas gain K<sup>2</sup> (22) is singularly dependent on them: at ϕiϕ<sup>j</sup> ≪ δp ≪ 1, K<sup>2</sup>

Until now, we were free with a choice of geometry of the considered resonance reflection. It looks natural to choose the parameters φ, ψ, and δp so that the loss coefficient K<sup>1</sup> would be as small as possible, i.e., the efficiency η would be close to 100%. In terms of Figure 1, this means an exclusion of the parasitic reflected beam r1 which is equivalent to a realization of the mode conversion. Formula (21) allows

Fð Þ <sup>4</sup> Fð Þ<sup>1</sup>

  ij <sup>þ</sup> <sup>i</sup>λ″

ij ϕiϕ<sup>j</sup> <sup>2</sup>

ij ϕiϕ<sup>j</sup> <sup>2</sup>

s2=s<sup>4</sup>

<sup>þ</sup> <sup>λ</sup>″ ð Þ1 ij ϕiϕ<sup>j</sup>

ð Þ α

<sup>þ</sup> <sup>λ</sup>″ ð Þ 4 ij ϕiϕ<sup>j</sup> <sup>2</sup>

<sup>þ</sup> <sup>λ</sup>″ ð Þ1 ij ϕiϕ<sup>j</sup>

ij ϕiϕ<sup>j</sup>

ij , Hi=Fð Þ<sup>1</sup> <sup>¼</sup> <sup>μ</sup>i: (20)

<sup>2</sup> , (21)

, (23)

: (24)

<sup>2</sup> : (22)

$$\mathbf{L}\_2 \approx \phi\_i \left(\frac{\partial \mathbf{L}\_2}{\partial \phi\_i}\right)\_0 + \frac{1}{2} \phi\_i \phi\_j \left(\frac{\partial^2 \mathbf{L}\_2}{\partial \phi\_i \partial \phi\_j}\right)\_0 + \delta p \left(\frac{\partial \mathbf{L}\_2}{\partial p\_2}\right)\_0. \tag{13}$$

In writing (13), we used the fact that L<sup>02</sup> = 0 (6). Substituting Eqs. (12) and (13) into the denominators of ratios (5), it is easy to verify that the term linear in φ and ψ drops out of the result. This term is proportional to the mixed product, which is zero,

$$\left[\mathbf{L}\_{03}\mathbf{L}\_{01}\left(\frac{\partial \mathbf{L}\_2}{\partial \phi\_i}\right)\_0\right] = \mathbf{0},\tag{14}$$

because, as we will see, it is composed of coplanar vectors. Let us prove that they are all perpendicular to the same real vector A02. From orthogonality condition (4), one has

$$\mathbf{A}\_{02} \cdot \mathbf{L}\_{0a} + \mathbf{L}\_{02} \cdot \mathbf{A}\_{0a} = \mathbf{A}\_{02} \cdot \mathbf{L}\_{0a} = \mathbf{0}.\tag{15}$$

for all α 6¼ 2 where the fact that L<sup>02</sup> ¼ 0 was again used.

In order to prove that the derivative <sup>∂</sup>L2=∂ϕ<sup>i</sup> ð Þ<sup>0</sup> is also orthogonal to <sup>A</sup>02, one can use identity A2ð Þ� v^ L2ð Þ¼ v^ 0 valid for any transonic states of arbitrary geometry {m, n} [13]. Let us differentiate this identity and then set ϕ = ψ = 0:

$$\left(\mathbf{A}\_2 \cdot \frac{\partial \mathbf{L}\_2}{\partial \phi\_i} + \mathbf{L}\_2 \cdot \frac{\partial \mathbf{A}\_2}{\partial \phi\_i}\right)\_0 = \mathbf{A}\_{02} \cdot \left(\frac{\partial \mathbf{L}\_2}{\partial \phi\_i}\right)\_0 = 0. \tag{16}$$

Thus, Eqs. (15) and (16) prove vanishing in (14), which means that indeed the denominator of both reflection coefficients (5) does not contain terms linear in φ and ψ. The numerator of R<sup>1</sup> is found from the same relations (12) and (13) after replacing in them indices 1 ! 4, and the numerator of R<sup>2</sup> is given by Eq. (11). After some straightforward calculations, one obtains

$$R\_1 = -\frac{F^{(4)}\delta p + G\_{ij}^{(4)}\phi\_i\phi\_j}{F^{(1)}\delta p + G\_{ij}^{(1)}\phi\_i\phi\_j}, \qquad R\_2 = \frac{H\_i\phi\_i}{F^{(1)}\delta p + G\_{ij}^{(1)}\phi\_i\phi\_j},\tag{17}$$

where the repeated subscripts imply summation, and the new notations are introduced:

$$F^{(a)} = \left[\mathbf{L}\_a \frac{\partial \mathbf{L}\_2}{\partial p\_2} \mathbf{L}\_3\right]\_0, \qquad H\_i = \left(\frac{\partial}{\partial \phi\_i} [\mathbf{L}\_4 \mathbf{L}\_1 \mathbf{L}\_3] \right)\_0,\tag{18}$$

$$G\_{ij}^{(a)} = \left[\frac{\partial \mathbf{L}\_2}{\partial \phi\_i} \cdot \frac{\partial (\mathbf{L}\_3 \times \mathbf{L}\_a)}{\partial \phi\_j} + \frac{1}{2} \left(\mathbf{L}\_a \frac{\partial^2 \mathbf{L}\_2}{\partial \phi\_i \partial \phi\_j} \mathbf{L}\_3\right)\right]\_0. \tag{19}$$

For further compactness of expressions, let us also denote:

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

where the subscript 0 means that after differentiation, one should put ϕ = ψ = 0,

In writing (13), we used the fact that L<sup>02</sup> = 0 (6). Substituting Eqs. (12) and (13) into the denominators of ratios (5), it is easy to verify that the term linear in φ and ψ drops out of the result. This term is proportional to the mixed product, which is

> ∂L<sup>2</sup> ∂ϕi � �

because, as we will see, it is composed of coplanar vectors. Let us prove that they are all perpendicular to the same real vector A02. From orthogonality condition (4), one

In order to prove that the derivative <sup>∂</sup>L2=∂ϕ<sup>i</sup> ð Þ<sup>0</sup> is also orthogonal to <sup>A</sup>02, one can use identity A2ð Þ� v^ L2ð Þ¼ v^ 0 valid for any transonic states of arbitrary geometry

0

Thus, Eqs. (15) and (16) prove vanishing in (14), which means that indeed the denominator of both reflection coefficients (5) does not contain terms linear in φ and ψ. The numerator of R<sup>1</sup> is found from the same relations (12) and (13) after replacing in them indices 1 ! 4, and the numerator of R<sup>2</sup> is given by Eq. (11). After

� �

0

R<sup>2</sup> (5), let us expand the two cofactors separately:

L<sup>2</sup> ≈ ϕ<sup>i</sup>

L<sup>3</sup> � L<sup>1</sup> ≈ L<sup>03</sup> � L<sup>01</sup> þ ϕ<sup>i</sup>

0 þ 1 2 ϕiϕ<sup>j</sup>

L03L<sup>01</sup>

{m, n} [13]. Let us differentiate this identity and then set ϕ = ψ = 0:

∂A<sup>2</sup> ∂ϕi

ij ϕiϕ<sup>j</sup>

ij ϕiϕ<sup>j</sup>

0

<sup>∂</sup>ð Þ <sup>L</sup><sup>3</sup> � <sup>L</sup><sup>α</sup> ∂ϕj

where the repeated subscripts imply summation, and the new notations are intro-

, Hi <sup>¼</sup> <sup>∂</sup>

þ 1 2 Lα ∂2 L2 ∂ϕi∂ϕ<sup>j</sup>

" # !

þ L<sup>2</sup> �

� �

∂L<sup>2</sup> ∂ϕi � �

for all α 6¼ 2 where the fact that L<sup>02</sup> ¼ 0 was again used.

A<sup>2</sup> � ∂L<sup>2</sup> ∂ϕi

some straightforward calculations, one obtains

<sup>F</sup>ð Þ <sup>α</sup> <sup>¼</sup> <sup>L</sup><sup>α</sup>

ij <sup>¼</sup> <sup>∂</sup>L<sup>2</sup> ∂ϕi �

Gð Þ <sup>α</sup>

<sup>F</sup>ð Þ <sup>4</sup> <sup>δ</sup><sup>p</sup> <sup>þ</sup> <sup>G</sup>ð Þ <sup>4</sup>

<sup>F</sup>ð Þ<sup>1</sup> <sup>δ</sup><sup>p</sup> <sup>þ</sup> <sup>G</sup>ð Þ<sup>1</sup>

∂L<sup>2</sup> ∂p2 L3 � �

For further compactness of expressions, let us also denote:

<sup>R</sup><sup>1</sup> <sup>¼</sup> ‒

To calculate the denominator in the expressions for reflection coefficients R<sup>1</sup> and

∂2 L2 ∂ϕi∂ϕ<sup>j</sup> !

∂ ∂ϕi

½ � L<sup>3</sup> � L<sup>1</sup> � �

> 0 þ δp

A<sup>02</sup> � L0<sup>α</sup> þ L<sup>02</sup> � A0<sup>α</sup> ¼ A<sup>02</sup> � L0<sup>α</sup> ¼ 0: (15)

<sup>¼</sup> <sup>A</sup><sup>02</sup> � <sup>∂</sup>L<sup>2</sup>

, R<sup>2</sup> <sup>¼</sup> Hiϕ<sup>i</sup>

∂ϕi

∂ϕi � �

0

<sup>F</sup>ð Þ<sup>1</sup> <sup>δ</sup><sup>p</sup> <sup>þ</sup> <sup>G</sup>ð Þ<sup>1</sup>

½ � L4L1L<sup>3</sup> � �

L3

ij ϕiϕ<sup>j</sup>

0

0

¼ 0: (16)

, (17)

, (18)

: (19)

0

∂L<sup>2</sup> ∂p2 � �

0

¼ 0, (14)

, (12)

: (13)

δp = 0.

Acoustics of Materials

zero,

has

duced:

164

$$\mathcal{G}^{(a)}\_{\vec{\eta}}/F^{(a)} = \lambda^{(a)}\_{\vec{\eta}} \equiv \lambda'^{(a)}\_{\vec{\eta}} + i\lambda''^{(a)}\_{\vec{\eta}}, \qquad H\_i/F^{(1)} = \mu\_i. \tag{20}$$

The substitution of Eq. (17) in terms of (20) into Eq. (8) gives

$$K\_{1} = \frac{s\_{1}}{s\_{4}} \left| \frac{F^{(4)}}{F^{(1)}} \right|^{2} \frac{\left(\delta p + \lambda\_{ij}^{\prime(4)} \phi\_{i} \phi\_{j}\right)^{2} + \left(\lambda\_{ij}^{\prime(4)} \phi\_{i} \phi\_{j}\right)^{2}}{\left(\delta p + \lambda\_{ij}^{\prime(1)} \phi\_{i} \phi\_{j}\right)^{2} + \left(\lambda\_{ij}^{\prime(1)} \phi\_{i} \phi\_{j}\right)^{2}},\tag{21}$$

$$K\_2 = \frac{\left(\mu\_i \phi\_i\right)^2 s\_2 / s\_4}{\left(\delta p + \lambda\_{\vec{\eta}}'^{(1)} \phi\_i \phi\_j\right)^2 + \left(\lambda\_{\vec{\eta}}''^{(1)} \phi\_i \phi\_j\right)^2}. \tag{22}$$

As could be expected, loss coefficient K<sup>1</sup> (21) is regular in the control parameters φ, ψ, and δp, whereas gain K<sup>2</sup> (22) is singularly dependent on them: at ϕiϕ<sup>j</sup> ≪ δp ≪ 1, K<sup>2</sup> tends to zero, and at δp ≪ ϕiϕ<sup>j</sup> ≪ 1, it diverges.
