The acoustic wave equation for orthorhombic media

In the text we derived an acoustic dispersion relation [equation 17] for orthorhombic anisotropy. This relation can be used to derive the acoustic wave equation following the same steps I took in deriving such an equation for VTI media Alkhalifah (1997b).

First, we cast the dispersion relation in a polynomial form in terms of the slownesses and substitute these slownesses with wavenumbers as follows,

$$\begin{eqnarray} \omega^6= V_1^2 \omega^4 k_x^2 + V_2^2 \omega^4 k_y^2+v_v^2 \om... ...{\eta }_1}\,{{\eta }_2} \right) \right) \right) k_x^2 k_y^2 k_z^2\end{eqnarray}$$ (27)

where $p_x = \frac{k_x}{\omega}$, $p_y = \frac{k_y}{\omega}$, and $p_z = \frac{k_z}{\omega}$.As a reminder, V₁ and V₂ are the horizontal velocities along the x-axis and the y-axis, respectively. Multiplying both sides of equation (27) by the wavefield in the Fourier domain, $F(k_x,k_y,k_z,\omega)$,as well as using inverse Fourier transform on k_x, k_y, and k_z ($k_x \rightarrow -i\frac{d}{dx}$, $k_y \rightarrow -i\frac{d}{dy}$, and $k_z \rightarrow -i\frac{d}{dz}$) yields a wave equation in the space-frequency domain, given by

$$\begin{eqnarray} -\omega^6 F= V_1^2 \omega^4 \frac{\partial^2 F}{\partial x^2}+ ... ...ight) \frac{\partial^6 F}{\partial x^2 \partial y^2 \partial z^2}.\end{eqnarray}$$ (28)

Finally, applying inverse Fourier transform on $\omega$ ($\omega \rightarrow i\frac{\partial}{\partial t}$), the acoustic wave equation for VTI media is given by

$$\begin{eqnarray} \frac{\partial^6 F}{\partial t^6}= V_1^2 \frac{\partial^6 F}{\p... ...ight) \frac{\partial^6 F}{\partial x^2 \partial y^2 \partial z^2}.\end{eqnarray}$$ (29)

Unlike the acoustic wave equation for VTI media Alkhalifah (1997b) which is fourth order in time, equation (29) is sixth order in time, and thus can provide us with 6 independent solutions.