Analytical solutions of the anisotropic equation

To solve equation (18), we use the plane wave,

$$F(x,y,z,t)= A(t) \exp{i(k_x x+k_y y+k_z z)},$$

as a trial solution. Substituting the trial solution into the partial differential equation (18), we obtain the following linear equation for A,

$$\begin{eqnarray} \frac{d^6 A}{dt^6}+\left({{{k_z}}^2}\,{{{v_v}}^2} + {{{k_x}}^2}... ...t( 1 - 4\,{{\eta }_1}\,{{\eta }_2} \right) \right) \right) A =0.\end{eqnarray}$$ (23)

The fact that equation (23) includes only even order derivates of A implies that we have three sets of complex-conjugate solutions. These solutions are

$$\begin{eqnarray} A_1(t) = e^{\pm{{\sqrt{\frac{a_1}{6}} }\, t }},\end{eqnarray}$$ (24)

where

$$\begin{eqnarray} a_1= 2\,a + {\frac{{2^{{\frac{4}{3}}}}\,\left( {a^2} + 3\,b \ri... ...a\,b + 27\,c \right) }^2}}} \right) }^ {{\frac{1}{3}}}}.\nonumber\end{eqnarray}$$

$$\begin{eqnarray} A_2(t) = e^{\pm{{\sqrt{\frac{a_2}{12}} }\, t }},\end{eqnarray}$$ (25)

where

$$\begin{eqnarray} a_2 = 4\,a - {\frac{i\,{2^{{\frac{4}{3}}}}\, \left( -i + {\sqr... ...a\,b + 27\,c \right) }^2}}} \right) }^ {{\frac{1}{3}}}}.\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray} A_3(t) = e^{\pm{{\sqrt{\frac{a_3}{12}} }\, t }},\end{eqnarray}$$ (26)

where

$$\begin{eqnarray} a_3 = 4\,a + {\frac{i\,{2^{{\frac{4}{3}}}}\, \left( i + {\sqrt... ...\,b + 27\,c \right) }^2}}} \right) }^ {{\frac{1}{3}}}}. \nonumber\end{eqnarray}$$

The above are simply the three roots of the following cubic polynomial

x³+a x²+b x+ c=0

with

$$a= -\left( {{{k_z}}^2}\,{{{v_v}}^2} \right) - {{{k_x}}^2}\,{{{V_1}}^2} - {{{k_y}}^2}\,{{{V_2}}^2},$$

$$b=\left( {{\gamma }^2}\,{{{k_x}}^2}\,{{{k_y}}^2}\,{{{V_1}}^4... ...{k_y}}^2}\,{{{k_z}}^2}\,{{{v_2}}^2}\,{{{v_v}}^2}\,{{\eta }_2},$$

and

$$c= {{{k_x}}^2}\,{{{k_y}}^2}\,{{{k_z}}^2}\,{{{v_1}}^2}\,{{{v_... ...}}^2}\,\left( 1 - 4\,{{\eta }_1}\,{{\eta }_2} \right) \right),$$

as it relates to our problem.

Solution (25) reduces to the isotropic medium solution when $\eta$=0. Solutions (24) and (26) are additional waves that reduces in the isotropic limit ($\eta_1 \rightarrow 0$, $\eta_2 \rightarrow 0$, and $\delta \rightarrow 0$) to 1 and, with the proper initial condition, its coefficient to zero. In other words, solutions (24) and (26) become independent of time for $\eta_1=\eta_2=\delta=0$. However, these waves will prove to be harmful in orthorhombic case.

The main concern here is the sign of a₁, a₂ and a₃. A negative sign will result in an imaginary exponential term which corresponds to wave propagation behavior. A positive sign will result in a real exponential that is either decaying or growing depending on the sign of the exponential term. Considering we have conjugate solutions, at least one the solutions will be growing exponentially and causing serious instability problems. I will leave the analysis of a₁, a₂, and a₃ to a follow up paper.