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Acoustic wave-equation in 3D generalized Riemannian spaces

The acoustic wave-equation for wavefield, $\mathcal{U}$, in a generalized Riemannian space is,  
 \Delta\mathcal{U}= - \omega^2\ss^2 \left(\mathbf{x} \right)\mathcal{U},\end{displaymath} (5)
where the $\omega$ is frequency, $\ss$ is the propagation slowness, and $\Delta$ is the Laplacian operator,  
 \Delta \mathcal{U}= \frac{1}{\sqrt{\vert\mathbf{g}\vert}}\,...
 ...t(\,m^{ij}\,\frac{\partial \mathcal{U}}{\partial \xi_j}\right).\end{displaymath} (6)
Substituting equation 6 into 5 generates a Helmholtz equation appropriate for propagating waves through a 3D space,  
\frac{1}{\sqrt{\left\vert \mathbf{g} \right\vert}} \frac{\pa...
 ...tial \W}{\partial \xi_j } \right)= - \omega^2\ss^2 \mathcal{U}.\end{displaymath} (7)
Expanding the derivative terms and multiplying through by $\sqrt{\left\vert \mathbf{g} \right\vert}$yields ,  
\frac{\partial m^{\ii\jj} }{\partial \xi_i } \frac{\partial ...
 ...t{\left\vert \mathbf{g} \right\vert}\omega^2 \ss^2 \mathcal{U}.\end{displaymath} (8)
Defining nj as,
n_j=\frac{\partial m^{\ii\jj} }{\partial \xi_i } = 
 ...}}{\partial \xi_2 } +
\frac{\partial m^{3j}}{\partial \xi_3 } ,\end{displaymath} (9)
leads to a more compact notation of equation 8,  
n_j \frac{\partial \W}{\partial \xi_j } + m^{\ii\jj} \frac{\...
 ...t{\left\vert \mathbf{g} \right\vert}\omega^2 \ss^2 \mathcal{U}.\end{displaymath} (10)
Developing a wave-equation dispersion relation is achieved by replacing the partial differential operators acting on wavefield $\mathcal{U}$ with their Fourier domain duals,  
\left(m^{ij} k_\xi_i-i n_j \right)k_\xi_j= \sqrt{\left\vert \mathbf{g} \right\vert}\omega^2\ss^2,\end{displaymath} (11)
where $k_\xi_i$ is the Fourier domain dual of differential operator $\frac{\partial}{\partial \xi_i}$. Equation 11 represents the dispersion relationship for wavefield propagation on a generalized 3-D Riemannian space.