Complex slowness vector for generalised propagation of harmonic plane waves at the boundaries of real materials

Abstract

This study considers the propagation of harmonic plane waves in general anisotropic dissipative media. This propagation is governed by a complex slowness vector, which is resolved into a propagation vector and an attenuation vector. The attenuation part is decomposed into homogeneous attenuation and evanescent attenuation. This makes a generalised specification of slowness vector to represent (in)homogeneous propagation in (an)isotropic (an)elastic media. This specification applies to incident waves, scattered waves as well as surface/interface waves at the plane boundary of the medium. With the choice of involved parameters, this specification can represent the corresponding wave-fields in the absence of anisotropy and/or dissipation. This specification has been used to formulate a corrected procedure for the reflection of plane waves in a transversely isotropic piezothermoelastic medium.

Appendices

Appendix A: Phase velocities

Consider a general anisotropic medium represented by elastic tensor $c_{ijkl}$, and density $\rho$. The Christoffel equations (3) govern the harmonic propagation of bulk waves, for ${\textbf{p}}={\textbf{N}}/V,~{\textbf{N}}\cdot {\textbf{N}}=1.$ The resulting characteristic equation ($\det \{c_{ijkl}N_jN_l-\rho V^2\delta _{ik}\}=0$) is solved into a cubic equation, given by

$$\begin{aligned}{} & {} V^{6}+c_1V^{4}+c_2V^{2}+c_3=0; \nonumber \\{} & {} \qquad c_1=-(\alpha +\beta +\gamma ),\nonumber \\{} & {} \qquad c_2=\alpha \beta +\alpha \gamma +\beta \gamma -\delta ^{2}-\eta ^{2}-\zeta ^{2},\nonumber \\{} & {} \qquad c_3=\alpha \zeta ^{2}+\gamma \delta ^{2}+\beta \eta ^{2}-\alpha \beta \gamma -2\eta \delta \zeta . \end{aligned}$$

(18)

Roots of this cubic equation define the velocities ($V_k,~k=1,2,3)$ of three bulk waves (say, qP, qS1, qS2) in general anisotropic media. Various unknowns in equation (10) are defined as follows.

$$\begin{aligned} \alpha= & {} {\textbf{NAN}}^{T},~~~~~~~~~\beta ={\textbf{NBN}}^{T},~~~~~~~~~\gamma ={\textbf{NCN}}^{T},\nonumber \\ \delta= & {} {\textbf{NDN}}^{T},~~~~~~~~~\eta ={\textbf{NEN}}^{T},~~~~~~~~~\zeta ={\textbf{NFN}}^{T}; \end{aligned}$$

(19)

for unit tensor $\textbf{N}$ as row matrix ${\textbf{N}}=(N_1,N_2,N_3)$ and ${\textbf{N}}^{T}$ being its transpose. The $3\times 3$ matrices, $\textbf{A,B,C,D,E}$ and $\textbf{F}$, are given by

$$\begin{aligned}{} & {} {\textbf{A}}=\{a_{11},a_{16},a_{15};a_{16},a_{66},a_{56};a_{15},a_{56},a_{55}\};\\{} & {} {\textbf{B}}=\{a_{66},a_{26},a_{46};a_{26},a_{22},a_{24};a_{46},a_{24},a_{44}\};\\{} & {} {\textbf{C}}=\{a_{55},a_{45},a_{35};a_{45},a_{44},a_{34};a_{35},a_{34},a_{33}\};\\{} & {} {\textbf{D}}=\{a_{16},a_{12},a_{14};a_{66},a_{26},a_{46};a_{56},a_{25},a_{45}\};\\{} & {} {\textbf{E}}=\{a_{15},a_{14},a_{13};a_{56},a_{46},a_{36};a_{55},a_{45},a_{35}\};\\{} & {} {\textbf{F}}=\{a_{56},a_{46},a_{36};a_{25},a_{24},a_{23};a_{45},a_{44},a_{34}\}. \end{aligned}$$

The elements $a_{IJ}=c_{IJ}/\rho ,~(I,J=1,2,...,6)$ make a $6\times 6$ matrix of density-normalized elastic coefficients, where $c_{IJ}$ are two-suffix notations for fourth-order elastic tensor ($c_{ijkl}$). As a convenient alternative, a symbol function in MATLAB can be used to resolve the determinant $\det \{c_{ijkl}N_jN_l-\rho V^2\delta _{ik}\}$ into a cubic polynomial in $V^2$.

Appendix B: Vertical slownesses

For known horizontal slowness $(p_1,p_2)$, the determinantal equation $\det \{c_{ijkl}p_jp_l-\rho \delta _{ik}\}=0$ is solved into a sixth-degree algebraic equation in vertical slowness $p_3$, given by

$$\begin{aligned} \sum _{n=0}^{6}d_{n}p_3^{n}=0. \end{aligned}$$

(20)

The coefficients $d_n$ are calculated as:

$$\begin{gathered} d_{6} = |XXX|;\,\,\,d_{5} = |XXY| + |XYX| + |YXX|; \hfill \\ d_{4} = |XXZ| + |XZX| + |ZXX| + |XYY| + |YXY| + |YYX|; \hfill \\ d_{3} = |XYZ| + |XZY| + |YXZ| + |YZX| + |ZXY| + |ZYX| + |YYY|; \hfill \\ d_{2} = |XZZ| + |ZXZ| + |ZZX| + |YYZ| + |YZY| + |ZYY|; \hfill \\ d_{1} = |YZZ| + |ZYZ| + |ZZY|;~d_{0} = |ZZZ|, \hfill \\ \end{gathered}$$

where the symbol |XYZ| denotes the determinant of a matrix obtained by selecting first row from matrix $\textbf{X}$, second row from matrix $\textbf{Y}$ and third row from matrix $\textbf{Z}$. The $3\times 3$ matrices $\mathbf{X,~Y,~Z}$ are defined as:

$$\begin{gathered} {\mathbf{X}} = \{ c_{{56}} ,~c_{{45}} ,~c_{{35}} ;~c_{{45}} ,c_{{44}} ,~c_{{34}} ;~c_{{35}} ,~c_{{34}} ,~c_{{33}} \} ; \hfill \\ Y_{{11}} = 2(c_{{15}} p_{1} + c_{{56}} p_{2} ); \hfill \\ Y_{{12}} = Y_{{21}} = (c_{{14}} + c_{{56}} )p_{1} + (c_{{25}} + c_{{46}} )p_{2} ; \hfill \\ Y_{{13}} = Y_{{31}} = (c_{{13}} + c_{{55}} )p_{1} + (c_{{36}} + c_{{45}} )p_{2} ; \hfill \\ Y_{{22}} = 2(c_{{46}} p_{1} + c_{{24}} p_{2} );Y_{{23}} = Y_{{32}} = (c_{{36}} + c_{{45}} )p_{1} + (c_{{23}} + c_{{44}} )p_{2} ; \hfill \\ Y_{{33}} = 2(c_{{35}} p_{1} + c_{{34}} p_{2} ).Z_{{11}} = c_{{11}} p_{1}^{2} + c_{{66}} p_{2}^{2} + 2c_{{16}} p_{1} p_{2} - \rho ;~ \hfill \\ Z_{{12}} = Z_{{21}} = c_{{16}} p_{1}^{2} + c_{{26}} p_{2}^{2} + (c_{{12}} + c_{{66}} )p_{1} p_{2} ; \hfill \\ Z_{{13}} = Z_{{31}} = c_{{15}} p_{1}^{2} + c_{{46}} p_{2}^{2} + (c_{{14}} + c_{{56}} )p_{1} p_{2} ;~ \hfill \\ Z_{{22}} = c_{{66}} p_{1}^{2} + c_{{22}} p_{2}^{2} + 2c_{{26}} p_{1} p_{2} - \rho ; \hfill \\ Z_{{23}} = Z_{{32}} = c_{{56}} p_{1}^{2} + c_{{24}} p_{2}^{2} + (c_{{25}} + c_{{46}} )p_{1} p_{2} ;~ \hfill \\ Z_{{33}} = c_{{55}} p_{1}^{2} + c_{{44}} p_{2}^{2} + 2c_{{45}} p_{1} p_{2} - \rho , \hfill \\ \end{gathered}$$

where $c_{IJ}~(I,J=1,2,...,6)$ are two-suffix notations for elastic tensor $c_{ijkl}$. A symbol function in MATLAB can be used to resolve the determinant $\det \{c_{ijkl}p_jp_l-\rho \delta _{ik}\}$ into a polynomial of degree six.