Appendix 1: Characteristic Polynomial of the Christoffel Equation
In order to explain the vanishing terms with odd components in the conditional equation of the determinant [cf. Eq. (3.54)] for the elasticity tensor of a monoclinic material, the coefficient matrix of the Christoffel equation (3.50) is determined
$$\displaystyle{ \begin{array}{rl} \overbrace{\left (\lambda _{11} -\rho c_{\mathrm{p}}^{2}\right )\left (\lambda _{22} -\rho c_{\mathrm{p}}^{2}\right )\left (\lambda _{33} -\rho c_{\mathrm{p}}^{2}\right )}^{(1)} +\overbrace{ 2\lambda _{12}\lambda _{13}\lambda _{23}}^{(2)} & \\ -\mathop{\underbrace{\left (\lambda _{11} -\rho c_{\mathrm{p}}^{2}\right )\lambda _{23}^{2}}}\limits _{(3)} -\mathop{\underbrace{\left (\lambda _{22} -\rho c_{\mathrm{p}}^{2}\right )\lambda _{13}^{2}}}\limits _{(4)}-&\mathop{\underbrace{\left (\lambda _{33} -\rho c_{\mathrm{p}}^{2}\right )\lambda _{12}^{2}}}\limits _{(5)} = 0.\end{array} }$$
(3.135)
The values of λ
ac
are computed with the help of Eq. (3.49). Using Voigt’s notation [16], and considering the allocation of the elasticity tensor (3.53), element λ
11 of Eq. (3.135) is calculated exemplary.
$$\displaystyle\begin{array}{rcl} \lambda _{ac}& =& C_{abcd}n_{d}n_{b} \\ \lambda _{11}& =& C_{1b1d}n_{d}n_{b} =\sum _{ b,d=1}^{3}C_{ 1b1d}n_{d}n_{b} \\ & =& C_{1111} + C_{1311}\alpha + C_{1113}\alpha + C_{1313}\alpha ^{2} \\ & =& C_{11} + C_{51}\alpha + C_{15}\alpha + C_{55}\alpha ^{2} \\ & =& C_{11} + C_{55}\alpha ^{2} {}\end{array}$$
(3.136)
This way, the other elements of λ
ac
result in
$$\displaystyle{ \begin{array}{rl} \lambda _{12} & = C_{16} + C_{45}\alpha ^{2}, \\ \lambda _{13} & = (C_{13} + C_{55})\alpha, \\ \lambda _{22} & = C_{66} + C_{44}\alpha ^{2}, \\ \lambda _{23} & = (C_{36} + C_{45})\alpha, \\ \lambda _{33} & = C_{55} + C_{33}\alpha ^{2}. \end{array} }$$
(3.137)
Substituting the elements of Eq. (3.135) by (3.136) and (3.137) yields the following terms
$$\displaystyle\begin{array}{rcl} (1): \quad & & \left [\left (C_{11} + C_{55}\alpha ^{2}\right ) -\rho c_{\mathrm{ p}}^{2}\right ]\left [\left (C_{ 66} + C_{44}\alpha ^{2}\right ) -\rho c_{\mathrm{ p}}^{2}\right ]\left [\left (C_{ 55} + C_{33}\alpha ^{2}\right ) -\rho c_{\mathrm{ p}}^{2}\right ], {}\\ (2): \quad & & 2\left (C_{16} + C_{45}\alpha ^{2}\right )\left (C_{ 55} + C_{13}\right )\left (C_{11} + C_{55}\right )\alpha ^{2}, {}\\ (3): \quad & & \left [\left (C_{11} + C_{55}\alpha ^{2}\right ) -\rho c_{\mathrm{ p}}^{2}\right ]\left (C_{ 45} + C_{36}\right )^{2}\alpha ^{2}, {}\\ (4): \quad & & \left [\left (C_{66} + C_{44}\alpha ^{2}\right ) -\rho c_{\mathrm{ p}}^{2}\right ]\left (C_{ 55} + C_{13}\right )^{2}\alpha ^{2}, {}\\ (5): \quad & & \left [\left (C_{55} + C_{33}\alpha ^{2}\right ) -\rho c_{\mathrm{ p}}^{2}\right ]\left (C_{ 16} + C_{45}\alpha ^{2}\right )^{2}. {}\\ \end{array}$$
Here, α exhibits only even exponents and therewith leads to Eq. (3.54).
Appendix 2: Summary of Stresses and Displacements of a Single Anisotropic Layer in a System of Equations
Once the pairs of values of α (depending on k) are computed using Eq. (3.54), the polarization vector p
a is determined with the help of
$$\displaystyle{ \left [\begin{array}{*{10}c} \lambda _{11} -\rho c_{\mathrm{p}}^{2} & \lambda _{12} & \lambda _{13} \\ \lambda _{12} & \lambda _{22} -\rho c_{\mathrm{p}}^{2} & \lambda _{23} \\ \lambda _{13} & \lambda _{23} & \lambda _{33} -\rho c_{\mathrm{p}}^{2} \end{array} \right ]\left [\begin{array}{*{10}c} \,p_{1} \\ \,p_{2} \\ \,p_{3}\end{array} \right ] = 0. }$$
(3.50)
At this, the polarization vectors are any nonzero scalar multiples of
$$\displaystyle{ \mathbf{p} = \left [\begin{array}{*{10}c} 1\\ v\\ w\\ \end{array} \right ] }$$
(3.138)
with
$$\displaystyle{ v = \frac{p_{2}} {p_{1}} = \frac{\lambda _{12}\lambda _{13} -\lambda _{23}(\lambda _{11} -\rho c_{\mathrm{p}}^{2})} {\lambda _{12}\lambda _{23} -\lambda _{13}(\lambda _{22} -\rho c_{\mathrm{p}}^{2})}, }$$
(3.139)
$$\displaystyle{ w = \frac{p_{3}} {p_{1}} = \frac{\lambda _{12}\lambda _{13} -\lambda _{23}(\lambda _{11} -\rho c_{\mathrm{p}}^{2})} {\lambda _{13}\lambda _{23} -\lambda _{12}(\lambda _{33} -\rho c_{\mathrm{p}}^{2})}. }$$
(3.140)
For monoclinic material behavior, the values of v and w are computed according to
$$\displaystyle{ v(\alpha ) = \frac{p_{2}} {p_{1}} = \frac{(C_{16} + C_{45}\alpha ^{2})(C_{13} + C_{55}) - (C_{36} + C_{45})(C_{11} + C_{55}\alpha ^{2} -\rho c_{\mathrm{p}}^{2})} {(C_{16} + C_{45}\alpha ^{2})(C_{36} + C_{45}) - (C_{13} + C_{55})(C_{66} + C_{44}\alpha ^{2} -\rho c_{\mathrm{p}}^{2})}, }$$
(3.141)
$$\displaystyle{ w(\alpha ) = \frac{p_{3}} {p_{1}} = \frac{(C_{16} + C_{45}\alpha ^{2})(C_{13} + C_{55})\alpha - (C_{36} + C_{45})\alpha (C_{11} + C_{55}\alpha ^{2} -\rho c_{\mathrm{p}}^{2})} {(C_{13} + C_{55})(C_{36} + C_{45})\alpha ^{2} - (C_{16} + C_{45}\alpha ^{2})(C_{55} + C_{33}\alpha ^{2} -\rho c_{\mathrm{p}}^{2})}. }$$
(3.142)
Since the values of α occur in pairs, the influence of a changing sign on v and w is examined subsequently. Inserting −α into Eqs. (3.141) and (3.142) results in
$$\displaystyle{ v(-\alpha ) =\phantom{ -}v(\alpha ) }$$
(3.143)
$$\displaystyle{ w(-\alpha ) = -w(\alpha ). }$$
(3.144)
As can be seen, a varying sign of α changes the sign of w but not of v. Consequently, for the coefficients v
m
and w
m
one has
$$\displaystyle{ v_{m}^{+} = v_{ m}^{-}\quad \text{and}\quad w_{ m}^{+} = -w_{ m}^{-}\qquad \text{with}\quad m = 1,2,3. }$$
(3.145)
The values (u, w)
m
+ denote the coefficients resulting from α
m
+ and (u, w)
m
− labels the coefficients computed with −α
m
−.
Additionally, the coefficients of (d
a
)
m
are investigated concerning a varying sign of ±α
m
. Therefore, subjected to α and its corresponding polarization vector p
a the elements of d
a
are computed using Eq. (3.61). For element d
1
$$\displaystyle\begin{array}{rcl} d_{1}& =& \sum _{c,d=1}^{3}C_{ 13cd}n_{d}p_{c} {}\\ & =& C_{13c1}p_{c} + C_{13c3}\alpha p_{c} {}\\ & =& C_{1311}p_{1} + C_{1321}p_{2} + C_{1331}p_{3} + (C_{1313}p_{1} + C_{1323}p_{2} + C_{1333}p_{3})\alpha {}\\ & =& C_{15}p_{1} + C_{56}p_{2} + C_{55}p_{3} + (C_{55}p_{1} + C_{45}p_{2} + C_{35}p_{3})\alpha {}\\ & =& C_{55}(\,p_{3} +\alpha p_{1}) + C_{45}p_{2}. {}\\ \end{array}$$
is obtained. Inserting p from Eq. (3.138) d
1 results in
$$\displaystyle{ d_{1} = C_{55}(w+\alpha ) + C_{45}\alpha v. }$$
(3.146)
Using the same procedure for d
2 and d
3 leads to
$$\displaystyle{ d_{2} = C_{45}(w+\alpha ) + C_{44}\alpha v, }$$
(3.147)
$$\displaystyle{ d_{3} = C_{13} + C_{33}\alpha w + C_{36}v. }$$
(3.148)
As accomplished for the parameters v
m
and w
m
, the influence of a changing sign of α is analyzed relating to the elements of vector d
a
. For that reason, −α is substituted in Eqs. (3.146)– (3.148) and results in
$$\displaystyle{ d_{1}(-\alpha ) = -d_{1}(\alpha ), }$$
(3.149)
$$\displaystyle{ d_{2}(-\alpha ) = -d_{2}(\alpha ), }$$
(3.150)
$$\displaystyle{ d_{3}(-\alpha ) =\phantom{ -}d_{3}(\alpha ). }$$
(3.151)
This connectivity is valid for all pairs of values ±α
m
, in order that the relation of the elements of the vectors (d
a
)
m
is given by
$$\displaystyle{ \begin{array}{r} (d_{1})_{m}^{+} = -(d_{1})_{m}^{-},\quad (d_{2})_{m}^{+} = -(d_{2})_{m}^{-},\quad (d_{3})_{m}^{+} = (d_{3})_{m}^{-}\quad \text{with}\quad m = 1,2,3, \end{array} }$$
(3.152)
where the parameters (d
a
)
m
, cf. Eqs. (3.146)– (3.148), are
$$\displaystyle{ d_{1m} = C_{55}(w_{m} +\alpha _{m}) + C_{45}\alpha _{m}v_{m}, }$$
(3.153)
$$\displaystyle{ d_{2m} = C_{45}(w_{m} +\alpha _{m}) + C_{44}\alpha _{m}v_{m}, }$$
(3.154)
$$\displaystyle{ d_{3m} = C_{13} + C_{33}\alpha _{m}w_{m} + C_{36}v_{m}. }$$
(3.155)
Applying Eqs. (3.141), (3.142), and (3.153)– (3.155), the displacements and stresses in a single anisotropic layer are summarized in a system of equations as follows
$$\displaystyle{ \left [\begin{array}{*{10}c} u_{1} \\ u_{2} \\ u_{3} \\ \sigma _{1}^{{\ast}} \\ \sigma _{2}^{{\ast}} \\ \sigma _{3}^{{\ast}}\end{array} \right ] = \left [\begin{array}{*{10}c} 1 & 1 & 1 & 1 & 1 & 1 \\ v_{1}^{+} & v_{2}^{+} & v_{3}^{+} & v_{1}^{-} & v_{2}^{-} & v_{3}^{-} \\ w_{1}^{+} & w_{2}^{+} & w_{3}^{+} & w_{1}^{-}& w_{2}^{-}& w_{3}^{-} \\ d_{11}^{+} & d_{12}^{+} & d_{13}^{+} & d_{11}^{-}&d_{12}^{-}&d_{13}^{-} \\ d_{21}^{+} & d_{22}^{+} & d_{23}^{+} & d_{21}^{-}&d_{22}^{-}&d_{23}^{-} \\ d_{31}^{+} & d_{32}^{+} & d_{33}^{+} & d_{31}^{-}&d_{32}^{-}&d_{33}^{-} \end{array} \right ]\left [\begin{array}{*{10}c} A_{1}^{+}e^{i\left (kx_{1}+k\alpha _{1}x_{3}-\omega t\right )} \\ A_{2}^{+}e^{i\left (kx_{1}+k\alpha _{2}x_{3}-\omega t\right )} \\ A_{3}^{+}e^{i\left (kx_{1}+k\alpha _{3}x_{3}-\omega t\right )} \\ A_{1}^{-}e^{i\left (kx_{1}-k\alpha _{1}x_{3}-\omega t\right )} \\ A_{2}^{-}e^{i\left (kx_{1}-k\alpha _{2}x_{3}-\omega t\right )} \\ A_{3}^{-}e^{i\left (kx_{1}-k\alpha _{3}x_{3}-\omega t\right )} \end{array} \right ], }$$
(3.156)
where
$$\displaystyle{ \sigma _{a}^{{\ast}} = \frac{\sigma _{a}} {ik}. }$$
(3.157)
With the help of the relations specified in Eq. (3.145) and (3.152), a rearrangement of the columns leads to the system of equations given in Sect. 3.3.2
$$\displaystyle{ \left [\begin{array}{*{10}c} u_{1} \\ u_{2} \\ u_{3} \\ \sigma _{1}^{{\ast}} \\ \sigma _{2}^{{\ast}} \\ \sigma _{3}^{{\ast}}\end{array} \right ] = \left [\begin{array}{*{10}c} 1 & 1 & 1 & 1 & 1 & 1\\ v_{ 1} & v_{1} & v_{2} & v_{2} & v_{3} & v_{3} \\ w_{1} & -w_{1} & w_{2} & -w_{2} & w_{3} & -w_{3} \\ d_{11} & -d_{11} & d_{12} & -d_{12} & d_{13} & -d_{13} \\ d_{21} & -d_{21} & d_{22} & -d_{22} & d_{23} & -d_{23} \\ d_{31} & d_{31} & d_{32} & d_{32} & d_{33} & d_{33} \end{array} \right ]\left [\begin{array}{*{10}c} A_{1}^{+}e^{ik\alpha _{1}x_{3}} \\ A_{1}^{-}e^{-ik\alpha _{1}x_{3}} \\ A_{2}^{+}e^{ik\alpha _{2}x_{3}} \\ A_{2}^{-}e^{-ik\alpha _{2}x_{3}} \\ A_{3}^{+}e^{ik\alpha _{3}x_{3}} \\ A_{3}^{-}e^{-ik\alpha _{3}x_{3}} \end{array} \right ]e^{i(kx_{1}-\omega t)}. }$$
(3.96)
Appendix 3: Separated Dispersion Relations for the Symmetric and Antisymmetric Wave Modes
Based on Eq. (3.65), separate dispersion relations for the symmetric and antisymmetric modes of LAMB and SH-waves shall be derived. First of all, Eq. (3.65) is formulated without shifting the point of origin to the top or bottom surface of the plate
$$\displaystyle{ \left [\begin{array}{*{10}c} \sigma _{a}^{t} \\ \sigma _{a}^{b}\\ \end{array} \right ] = ik\left [\begin{array}{*{10}c} d_{am}^{+}e^{ik\alpha _{m}\frac{h} {2} } & d_{am}^{-}e^{-ik\alpha _{m}\frac{h} {2} } \\ d_{am}^{+}e^{-ik\alpha _{m}\frac{h} {2} } & d_{am}^{-}e^{ik\alpha _{m}\frac{h} {2} }\\ \end{array} \right ]\left [\begin{array}{*{10}c} A_{m}^{+} \\ A_{m}^{-}\\ \end{array} \right ]e^{i\left (kx_{1}-\omega t\right )} = 0, }$$
(3.158)
where d
am
± is the simplified spelling of term (d
a
)
m
±. The coefficient matrix reads as follows
$$\displaystyle{ \left [\begin{array}{*{10}c} d_{11}^{+}e^{ik\alpha _{1} \frac{h} {2} } & d_{12}^{+}e^{ik\alpha _{2} \frac{h} {2} } & d_{13}^{+}e^{ik\alpha _{3} \frac{h} {2} } & d_{11}^{-}e^{-ik\alpha _{1} \frac{h} {2} } & d_{12}^{-}e^{-ik\alpha _{2} \frac{h} {2} } & d_{13}^{-}e^{-ik\alpha _{3} \frac{h} {2} } \\ d_{21}^{+}e^{ik\alpha _{1} \frac{h} {2} } & d_{22}^{+}e^{ik\alpha _{2} \frac{h} {2} } & d_{23}^{+}e^{ik\alpha _{3} \frac{h} {2} } & d_{21}^{-}e^{-ik\alpha _{1} \frac{h} {2} } & d_{22}^{-}e^{-ik\alpha _{2} \frac{h} {2} } & d_{23}^{-}e^{-ik\alpha _{3} \frac{h} {2} } \\ d_{31}^{+}e^{ik\alpha _{1} \frac{h} {2} } & d_{32}^{+}e^{ik\alpha _{2} \frac{h} {2} } & d_{33}^{+}e^{ik\alpha _{3} \frac{h} {2} } & d_{31}^{-}e^{-ik\alpha _{1} \frac{h} {2} } & d_{32}^{-}e^{-ik\alpha _{2} \frac{h} {2} } & d_{33}^{-}e^{-ik\alpha _{3} \frac{h} {2} } \\ d_{11}^{+}e^{-ik\alpha _{1} \frac{h} {2} } & d_{12}^{+}e^{-ik\alpha _{2} \frac{h} {2} } & d_{13}^{+}e^{-ik\alpha _{3} \frac{h} {2} } & d_{11}^{-}e^{ik\alpha _{1} \frac{h} {2} } & d_{12}^{-}e^{ik\alpha _{2} \frac{h} {2} } & d_{13}^{-}e^{ik\alpha _{3} \frac{h} {2} } \\ d_{21}^{+}e^{-ik\alpha _{1} \frac{h} {2} } & d_{22}^{+}e^{-ik\alpha _{2} \frac{h} {2} } & d_{23}^{+}e^{-ik\alpha _{3} \frac{h} {2} } & d_{21}^{-}e^{ik\alpha _{1} \frac{h} {2} } & d_{22}^{-}e^{ik\alpha _{2} \frac{h} {2} } & d_{23}^{-}e^{ik\alpha _{3} \frac{h} {2} } \\ d_{31}^{+}e^{-ik\alpha _{1} \frac{h} {2} } & d_{32}^{+}e^{-ik\alpha _{2} \frac{h} {2} } & d_{33}^{+}e^{-ik\alpha _{3} \frac{h} {2} } & d_{31}^{-}e^{ik\alpha _{1} \frac{h} {2} } & d_{32}^{-}e^{ik\alpha _{2} \frac{h} {2} } & d_{33}^{-}e^{ik\alpha _{3} \frac{h} {2} } \end{array} \right ]. }$$
Taking account of the relations in Eq. (3.152), the coefficient matrix becomes
$$\displaystyle{ \left [\begin{array}{*{10}c} d_{11}e^{ik\alpha _{1} \frac{h} {2} } & d_{12}e^{ik\alpha _{2} \frac{h} {2} } & d_{13}e^{ik\alpha _{3} \frac{h} {2} } & -d_{11}e^{-ik\alpha _{1} \frac{h} {2} } & -d_{12}e^{-ik\alpha _{2} \frac{h} {2} } & -d_{13}e^{-ik\alpha _{3} \frac{h} {2} } \\ d_{21}e^{ik\alpha _{1} \frac{h} {2} } & d_{22}e^{ik\alpha _{2} \frac{h} {2} } & d_{23}e^{ik\alpha _{3} \frac{h} {2} } & -d_{21}e^{-ik\alpha _{1} \frac{h} {2} } & -d_{22}e^{-ik\alpha _{2} \frac{h} {2} } & -d_{23}e^{-ik\alpha _{3} \frac{h} {2} } \\ d_{31}e^{ik\alpha _{1} \frac{h} {2} } & d_{32}e^{ik\alpha _{2} \frac{h} {2} } & d_{33}e^{ik\alpha _{3} \frac{h} {2} } & d_{31}e^{-ik\alpha _{1} \frac{h} {2} } & d_{32}e^{-ik\alpha _{2} \frac{h} {2} } & d_{33}e^{-ik\alpha _{3} \frac{h} {2} } \\ d_{11}e^{-ik\alpha _{1} \frac{h} {2} } & d_{12}e^{-ik\alpha _{2} \frac{h} {2} } & d_{13}e^{-ik\alpha _{3} \frac{h} {2} } & -d_{11}e^{ik\alpha _{1} \frac{h} {2} } & -d_{12}e^{ik\alpha _{2} \frac{h} {2} } & -d_{13}e^{ik\alpha _{3} \frac{h} {2} } \\ d_{21}e^{-ik\alpha _{1} \frac{h} {2} } & d_{22}e^{-ik\alpha _{2} \frac{h} {2} } & d_{23}e^{-ik\alpha _{3} \frac{h} {2} } & -d_{21}e^{ik\alpha _{1} \frac{h} {2} } & -d_{22}e^{ik\alpha _{2} \frac{h} {2} } & -d_{23}e^{ik\alpha _{3} \frac{h} {2} } \\ d_{31}e^{-ik\alpha _{1} \frac{h} {2} } & d_{32}e^{-ik\alpha _{2} \frac{h} {2} } & d_{33}e^{-ik\alpha _{3} \frac{h} {2} } & d_{31}e^{ik\alpha _{1} \frac{h} {2} } & d_{32}e^{ik\alpha _{2} \frac{h} {2} } & d_{33}e^{ik\alpha _{3} \frac{h} {2} } \end{array} \right ]. }$$
Here, the superscripted ” + ” of d
am
+ is relinquished, because there are solely vectors of d
am
belonging to a positive α
m
. A rearranging of rows and columns results in
$$\displaystyle{ \left [\begin{array}{*{10}c} d_{11}e^{ik\alpha _{1} \frac{h} {2} } & -d_{11}e^{-ik\alpha _{1} \frac{h} {2} } & d_{12}e^{ik\alpha _{2} \frac{h} {2} } & -d_{12}e^{-ik\alpha _{2} \frac{h} {2} } & d_{13}e^{ik\alpha _{3} \frac{h} {2} } & -d_{13}e^{-ik\alpha _{3} \frac{h} {2} } \\ d_{11}e^{-ik\alpha _{1} \frac{h} {2} } & -d_{11}e^{ik\alpha _{1} \frac{h} {2} } & d_{12}e^{-ik\alpha _{2} \frac{h} {2} } & -d_{12}e^{ik\alpha _{2} \frac{h} {2} } & d_{13}e^{-ik\alpha _{3} \frac{h} {2} } & -d_{13}e^{ik\alpha _{3} \frac{h} {2} } \\ d_{21}e^{ik\alpha _{1} \frac{h} {2} } & -d_{21}e^{-ik\alpha _{1} \frac{h} {2} } & d_{22}e^{ik\alpha _{2} \frac{h} {2} } & -d_{22}e^{-ik\alpha _{2} \frac{h} {2} } & d_{23}e^{ik\alpha _{3} \frac{h} {2} } & -d_{23}e^{-ik\alpha _{3} \frac{h} {2} } \\ d_{21}e^{-ik\alpha _{1} \frac{h} {2} } & -d_{21}e^{ik\alpha _{1} \frac{h} {2} } & d_{22}e^{-ik\alpha _{2} \frac{h} {2} } & -d_{22}e^{ik\alpha _{2} \frac{h} {2} } & d_{23}e^{-ik\alpha _{3} \frac{h} {2} } & -d_{23}e^{ik\alpha _{3} \frac{h} {2} } \\ d_{31}e^{ik\alpha _{1} \frac{h} {2} } & d_{31}e^{-ik\alpha _{1} \frac{h} {2} } & d_{32}e^{ik\alpha _{2} \frac{h} {2} } & d_{32}e^{-ik\alpha _{2} \frac{h} {2} } & d_{33}e^{ik\alpha _{3} \frac{h} {2} } & d_{33}e^{-ik\alpha _{3} \frac{h} {2} } \\ d_{31}e^{-ik\alpha _{1} \frac{h} {2} } & d_{31}e^{ik\alpha _{1} \frac{h} {2} } & d_{32}e^{-ik\alpha _{2} \frac{h} {2} } & d_{32}e^{ik\alpha _{2} \frac{h} {2} } & d_{33}e^{-ik\alpha _{3} \frac{h} {2} } & d_{33}e^{ik\alpha _{3} \frac{h} {2} } \end{array} \right ]. }$$
From the pairwise addition and subtraction of the columns following the pattern
the coefficient matrix composed of summands and differences may be written as
$$\displaystyle{ \left [\begin{array}{*{10}c} (a_{11}) - (a_{12})&(a_{11}) + (a_{12})&(a_{13}) - (a_{14})&(a_{13}) + (a_{14})&(a_{15}) - (a_{16})&(a_{15}) + (a_{16}) \\ (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) \\ (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) \\ (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) \\ (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) \\ (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) & (\ldots ) \end{array} \right ], }$$
exemplary, with the following entries of the first row
$$\displaystyle\begin{array}{rcl} (a_{11}) - (a_{12}):& \qquad d_{11}\left [e^{ik\alpha _{1} \frac{h} {2} } + e^{-ik\alpha _{1} \frac{h} {2} }\right ],& {}\\ (a_{11}) + (a_{12}):& \qquad d_{11}\left [e^{ik\alpha _{1} \frac{h} {2} } - e^{-ik\alpha _{1} \frac{h} {2} }\right ],& {}\\ (a_{13}) - (a_{14}):& \qquad d_{12}\left [e^{ik\alpha _{2} \frac{h} {2} } + e^{-ik\alpha _{2} \frac{h} {2} }\right ],& {}\\ (a_{13}) + (a_{14}):& \qquad d_{12}\left [e^{ik\alpha _{2} \frac{h} {2} } - e^{-ik\alpha _{2} \frac{h} {2} }\right ],& {}\\ (a_{15}) - (a_{16}):& \qquad d_{13}\left [e^{ik\alpha _{3} \frac{h} {2} } + e^{-ik\alpha _{3} \frac{h} {2} }\right ],& {}\\ (a_{15}) + (a_{16}):& \qquad d_{13}\left [e^{ik\alpha _{3} \frac{h} {2} } - e^{-ik\alpha _{3} \frac{h} {2} }\right ].& {}\\ \end{array}$$
Using Euler’s formula
$$\displaystyle{ \sin x = \frac{1} {2i}\left (e^{ix} - e^{-ix}\right )\;\;\text{and}\;\;\cos x = \frac{1} {2}\left (e^{ix} + e^{-ix}\right ) }$$
(3.159)
the terms in squared brackets are further transformed and one obtains the matrix
$$\displaystyle{ \left [\begin{array}{*{10}c} 2d_{11}\cos (k\alpha _{1} \frac{h} {2} ) & 2id_{11}\sin (k\alpha _{1} \frac{h} {2} ) & 2d_{12}\cos (k\alpha _{2} \frac{h} {2} ) & -2id_{12}\sin (k\alpha _{2} \frac{h} {2} ) & 2d_{13}\cos (k\alpha _{3} \frac{h} {2} ) & 2id_{13}\sin (k\alpha _{3} \frac{h} {2} ) \\ 2d_{11}\cos (k\alpha _{1} \frac{h} {2} ) & -2id_{11}\sin (k\alpha _{1} \frac{h} {2} ) & 2d_{12}\cos (k\alpha _{2} \frac{h} {2} ) & 2id_{12}\sin (k\alpha _{2} \frac{h} {2} ) & 2d_{13}\cos (k\alpha _{3} \frac{h} {2} ) & -2id_{13}\sin (k\alpha _{3} \frac{h} {2} ) \\ 2d_{21}\cos (k\alpha _{1} \frac{h} {2} ) & 2id_{21}\sin (k\alpha _{1} \frac{h} {2} ) & 2d_{22}\cos (k\alpha _{2} \frac{h} {2} ) & -2id_{22}\sin (k\alpha _{2} \frac{h} {2} ) & 2d_{23}\cos (k\alpha _{3} \frac{h} {2} ) & 2id_{23}\sin (k\alpha _{3} \frac{h} {2} ) \\ 2d_{21}\cos (k\alpha _{1} \frac{h} {2} ) & -2id_{21}\sin (k\alpha _{1} \frac{h} {2} ) & 2d_{22}\cos (k\alpha _{2} \frac{h} {2} ) & 2id_{22}\sin (k\alpha _{2} \frac{h} {2} ) & 2d_{23}\cos (k\alpha _{3} \frac{h} {2} ) & -2id_{23}\sin (k\alpha _{3} \frac{h} {2} ) \\ 2id_{31}\sin (k\alpha _{1} \frac{h} {2} ) & 2d_{31}\cos (k\alpha _{1} \frac{h} {2} ) & 2id_{32}\sin (k\alpha _{2} \frac{h} {2} ) & 2d_{32}\cos (k\alpha _{2} \frac{h} {2} ) & 2id_{33}\sin (k\alpha _{3} \frac{h} {2} ) & 2id_{33}\sin (k\alpha _{3} \frac{h} {2} ) \\ -2id_{31}\sin (k\alpha _{1} \frac{h} {2} ) & 2d_{31}\cos (k\alpha _{1} \frac{h} {2} ) & -2id_{32}\sin (k\alpha _{2} \frac{h} {2} ) & 2d_{32}\cos (k\alpha _{2} \frac{h} {2} ) & -2id_{33}\sin (k\alpha _{3} \frac{h} {2} ) & 2d_{33}\cos (k\alpha _{3} \frac{h} {2} ) \end{array} \right ]. }$$
Now, the rows are added and subtracted in the same way as the columns which leads to
$$\displaystyle{ \left [\begin{array}{*{10}c} 0 & 4id_{11}\sin (k\alpha _{1} \frac{h} {2} ) & 0 & 4id_{12}\sin (k\alpha _{2} \frac{h} {2} ) & 0 & 4id_{13}\sin (k\alpha _{3} \frac{h} {2} ) \\ 4d_{11}\cos (k\alpha _{1} \frac{h} {2} ) & 0 & 4d_{12}\cos (k\alpha _{2} \frac{h} {2} ) & 0 & 4d_{13}\cos (k\alpha _{3} \frac{h} {2} ) & 0 \\ 0 & 4id_{21}\sin (k\alpha _{1} \frac{h} {2} ) & 0 & 4id_{22}\sin (k\alpha _{2} \frac{h} {2} ) & 0 & 4id_{23}\sin (k\alpha _{3} \frac{h} {2} ) \\ 4d_{21}\cos (k\alpha _{1} \frac{h} {2} ) & 0 & 4d_{22}\cos (k\alpha _{2} \frac{h} {2} ) & 0 & 4d_{23}\cos (k\alpha _{3} \frac{h} {2} ) & 0 \\ 4id_{31}\sin (k\alpha _{1} \frac{h} {2} ) & 0 & 4id_{32}\sin (k\alpha _{2} \frac{h} {2} ) & 0 & 4id_{33}\sin (k\alpha _{3} \frac{h} {2} ) & 0 \\ 0 & 4d_{31}\cos (k\alpha _{1} \frac{h} {2} ) & 0 & 4d_{32}\cos (k\alpha _{2} \frac{h} {2} ) & 0 & 4d_{33}\cos (k\alpha _{3} \frac{h} {2} ) \end{array} \right ] }$$
A further rearrangement of rows and columns yields
$$\displaystyle{ \left [\begin{array}{*{10}c} 4d_{11}\cos (k\alpha _{1}\frac{h} {2} ) & 4d_{12}\cos (k\alpha _{2}\frac{h} {2} ) & 4d_{13}\cos (k\alpha _{3}\frac{h} {2} ) & 0 & 0 & 0 \\ 4d_{21}\cos (k\alpha _{1}\frac{h} {2} ) & 4d_{22}\cos (k\alpha _{2}\frac{h} {2} ) & 4d_{23}\cos (k\alpha _{3}\frac{h} {2} ) & 0 & 0 & 0 \\ 4id_{31}\sin (k\alpha _{1}\frac{h} {2} )&4id_{32}\sin (k\alpha _{2}\frac{h} {2} )&4id_{33}\sin (k\alpha _{3}\frac{h} {2} )& 0 & 0 & 0 \\ 0 & 0 & 0 &4id_{11}\sin (k\alpha _{1}\frac{h} {2} )&4id_{12}\sin (k\alpha _{2}\frac{h} {2} )&4id_{13}\sin (k\alpha _{3}\frac{h} {2} ) \\ 0 & 0 & 0 &4id_{21}\sin (k\alpha _{1}\frac{h} {2} )&4id_{22}\sin (k\alpha _{2}\frac{h} {2} )&4id_{23}\sin (k\alpha _{3}\frac{h} {2} ) \\ 0 & 0 & 0 & 4d_{31}\cos (k\alpha _{1}\frac{h} {2} ) & 4d_{32}\cos (k\alpha _{2}\frac{h} {2} ) & 4d_{33}\cos (k\alpha _{3}\frac{h} {2} ) \end{array} \right ]. }$$
(3.160)
The determinants of the two resulting submatrices provide the separated dispersion relations of the symmetric and antisymmetric wave modes. The conditional equations of the symmetric Lamb wave mode are written as
$$\displaystyle{ d_{31}G_{1}\cot (\alpha _{1}\gamma ) + d_{32}G_{2}\cot (\alpha _{2}\gamma ) + d_{33}G_{3}\cot (\alpha _{3}\gamma ) = 0 }$$
(3.69)
and for the antisymmetric Lamb wave mode one obtains
$$\displaystyle{ d_{31}G_{1}\tan (\alpha _{1}\gamma ) + d_{32}G_{2}\tan (\alpha _{2}\gamma ) + d_{33}G_{3}\tan (\alpha _{3}\gamma ) = 0 }$$
(3.70)
with
$$\displaystyle\begin{array}{rcl} G_{1}& =& d_{12}d_{23} - d_{22}d_{13}, {}\\ G_{2}& =& d_{13}d_{21} - d_{23}d_{11}, {}\\ G_{3}& =& d_{11}d_{22} - d_{21}d_{12}, {}\\ \gamma & =& \frac{kh} {2}. {}\\ \end{array}$$
