Sub-determinants of $R^{(i)}$

The solution of the equation of motion for Rayleigh modes implies the computation of sub-determinants of matrix $R(z_0)$ (section 3.1.4, equation (3.34)). In this appendix, we present the complete analytical expressions of these terms. For doing so, we define the following adimensional real quantities:

$$h_n = \hat h_n / k$$ (A.1)

$$k_n = \hat k_n / k$$ (A.2)

$$\left. \begin{array}{ccc} SH & = & 0.5 \frac{1-e^{-2 d_n \hat h_n... ..._n \hat h_n}) \end{array}\right\} \textrm{if} \quad h_n \quad \textrm{is real.}$$ (A.3)

$$\left. \begin{array}{ccc} SH & = & \frac{sin(-i d_n \hat h_n)}{h_... ...hat h_n) \end{array}\right\} \textrm{if} \quad h_n \quad \textrm{is imaginary.}$$ (A.4)

$$\left. \begin{array}{ccc} SK & = & 0.5 \frac {1-e^{-2 d_n \hat k_... ..._n \hat k_n}) \end{array}\right\} \textrm{if} \quad k_n \quad \textrm{is real.}$$ (A.5)

$$\left. \begin{array}{ccc} SK & = & \frac{sin(-i d_n \hat k_n)}{k_... ...hat k_n) \end{array}\right\} \textrm{if} \quad k_n \quad \textrm{is imaginary.}$$ (A.6)

where $d_n=z_n-z_{n-1}$ is the thickness of layer $n$ .

$$\gamma_n=2k^2/(\omega/V_{pn})^2$$ (A.7)

$$a_1 = \gamma_n^2-2\gamma_n +1$$

$$a_2 = h_n^2 k_n^2$$

$$a_3 = \gamma_n^2+a_1$$ (A.8)

$$a_4 = 1-\gamma_n$$

$$a_5 = \gamma_n^2 a_2$$

$$expCorr=e^{-\hat h_n d_n- \hat k_n d_n}$$ (A.9)

And we also define the following two dimensional quantities:

$$c_1 = \rho_n \omega^2 / k$$

$$c_2 = 1/c_1$$ (A.10)

The sub-determinants of $G_{n}$ are detailed here below ( $G_{ijkl}= g_n \left\vert \begin{array}{cc}i & j \ k & l\end{array}\right\vert$ , $G$ is real, $i$ before $G$ means that this component is imaginary) :

$$G_{1212} = a3 CH CK - (a_1+a_5) SH SK - (a_3-1) expCorr \nonumber$$

$$G_{1213} = c_2 ( CH SK - h_n^2 SH CK)$$

$$iG_{1214} = i c_2 ((a1-\gamma_n^2) (expCorr-CH CK)+ (a_4 - \gamma_n a_2) SH SK)$$

$$iG_{1223} = iG_{1414}$$

$$G_{1224} = c_2 (k_n^2 CH SK - SH CK)$$

$$G_{1234} = c_2^2 (2 CH CK + (1+a_2) SH SK)$$

$$G_{1312} = c_1 (\gamma_n^2 k_n^2 CH SK - a_1 SH CK)$$

$$G_{1313} = CH CK$$

$$iG_{1314} = i (a_4 SH CK+\gamma_n k_n^2 CH SK)$$

$$iG_{1323} = iG_{1314}$$ (A.11)

$$G_{1324} = k_n^2 SH SK$$

$$G_{1334} = G_{1224}$$

$$iG_{1412} = i c_1 ((a1-a4) (a4-\gamma_n) (expCorr-CH CK)+(a_4 a_1 - \gamma_n a_5) SH SK)$$

$$iG_{1413} = i (\gamma_n h_n^2 SH CH+a_4 CH SK)$$

$$G_{1414} = expCorr + G_{1423}$$

$$G_{1423} = CH CK - G_{1212}$$

$$iG_{1424} = iG_{1314}$$

$$iG_{1434} = iG_{1214}$$

$$iG_{2312} = iG_{1412}$$

$$iG_{2313} = iG_{1413}$$

$$G_{2314} = G_{1423}$$

$$G_{2323} = G_{1414}$$

$$iG_{2324} = iG_{1314}$$

$$iG_{2334} = iG_{1214}$$

$$G_{2412} = c_1 (a_1 CH SK - \gamma_n^2 h_n^2 SH CK)$$

$$G_{2413} = h_n^2 SH SK$$

$$iG_{2414} = iG_{1314}$$

$$iG_{2423} = iG_{1413}$$

$$G_{2424} = G_{1313}$$

$$G_{2434} = G_{1213}$$

$$G_{3412} = c_1^2(2 \gamma_n^2 a_1 CH CK+(a_1^2+\gamma_n^2 a_5) SH SK)$$

$$G_{3413} = G_{2412}$$

$$iG_{3414} = iG_{1412}$$

$$iG_{3423} = iG_{1412}$$

$$G_{3424} = G_{1312}$$

$$G_{3434} = G_{1212}$$

$T_{ijkl}= t_n^-1 \left\vert \begin{array}{cc}i & j \ k & l\end{array}\right\vert$ and from equation (3.31), $T_{1214}$ and $T_{1223}$ are equal and imaginary. Using equation (3.34) and definition of $G$ (equations (A.11)), it follows:

$$R_{1212}(z_{n-1}) = T_{1212} G_{1212} + (T_{1213} G_{1312}-2 T_{1214} iG_{1412}+T_{1224} G_{2412} - T_{1234} G_{3412})/\omega^2$$ (A.12)

$$R_{1213}(z_{n-1}) = \omega^2 T_{1212} G_{1213} + T_{1213} CH CK-2 T_{1214} iG_{1413}-T_{1224} G_{2413} + T_{1234} G_{2412}$$

$$R_{1214}(z_{n-1}) = \omega^2 T_{1212} iG_{1214} + T_{1213} iG_{1314}+T_{1214} (2 G_{1423}+expCorr)-T_{1224} iG_{1413}+ T_{1234} iG_{1412}$$

$$R_{1223}(z_{n-1}) = \omega^2 T_{1212} iG_{1214} + T_{1213} iG_{1314}+T_{1214} (2 G_{1423}+expCorr)-T_{1224} iG_{1413}+ T_{1234} iG_{1412}$$

$$R_{1224}(z_{n-1}) = \omega^2 T_{1212} G_{1224} + T_{1213} G_{1324}-2 T_{1214} iG_{1314}+T_{1224} CH CK + T_{1234} G_{1312}$$

$$R_{1234}(z_{n-1}) = -\omega^2 T_{1212} G_{1234}+T_{1213} G_{1224}-2 T_{1214} iG_{1214}+T_{1224} G_{1213} + T_{1234} G_{1212}$$