Acoustic wave-equation in 3-D Riemannian spaces

The Laplacian operator of a scalar function $\mathcal{U}$ in an arbitrary Riemannian space with coordinates $\{\xi_1,\xi_2,\xi_3\}$ has the form  

$$\Delta \mathcal{U}= \sum_{i=1}^{3}\frac{1}{\sqrt{\vert\math... ...g}\vert}\, \frac{\partial \mathcal{U}}{\partial \xi_j}\right),$$ (1)

where g^ij is a component of the associated metric tensor, and $\vert\mathbf{g}\vert$ is its determinant Synge and Schild (1978).

The expression simplifies if one of the coordinates (e.g. the coordinate of one-way wave extrapolation) is orthogonal to the other coordinates. Let $\xi_1=\xi$,$\xi_2=\eta$, and $\xi_3=\zeta$, with $\zeta$ orthogonal to both $\xi$ and $\eta$. Then the metric tensor has the matrix  

$$\left[g_{ij}\right] = \left[\begin{array} {ccc} E& F& 0 \\ F& G& 0 \\ 0 & 0 & \AA^2 \end{array}\right]\;,$$ (2)

where E, F, G, and $\AA$ are differential forms that can be found from mapping Cartesian coordinates $\bf{x}$ to the general coordinates $\{\xi,\eta,\zeta\}$, as follows:

$$\begin{eqnarray} E& = & \bf{x}_{\xi} \cdot \bf{x}_{\xi}\;, \nonumber \\ F& = & \... ...;, \nonumber \\ \AA^2 & = & \bf{x}_{\zeta} \cdot \bf{x}_{\zeta}\;.\end{eqnarray}$$ (3)

The associated metric tensor has the matrix  

$$\left[g^{ij}\right] = \left[\begin{array} {ccc} +G/J^2 & ... ... -F/J^2 & +E/J^2 & 0 \\ 0 & 0 & 1/\AA^2 \end{array}\right]\;,$$ (4)

where $J^2 = E\,G-F^2$. The metric determinant takes the form  

$$\vert\mathbf{g}\vert = \AA^2\,J^2\;.$$ (5)

Substituting equations (4) and (5) into (1), we can modify the Helmholtz wave equation  

 $$\Delta\mathcal{U}= - \frac{\omega^2}{v^2 \left(\mathbf{x} \right)} \mathcal{U}$$

for propagating waves in a 3-D Riemannian space:

$$\begin{eqnarray} \frac{1}{\AA\,J} \left[ \frac{\partial } {\partial \zeta } \lef... ...tial \xi } \right) \right] = - \frac{\omega^2}{v^2} \mathcal{U}\;.\end{eqnarray}$$ (6)

In equation (6), $\omega$ is temporal frequency, $v\left[\bf{x}\left(\xi,\eta,\zeta\right)\right]$ is the wave propagation velocity, and $\mathcal{U}$ represents a propagating wave.

For the special case of two dimensional spaces (F=0 and G=1), the Helmholtz wave equation reduces to the simpler form:  

$$\frac{1}{\AA J} \left[\frac{\partial } {\partial \zeta } \le... ...l \xi } \right) \right] = - \frac{\omega^2}{v^2} \mathcal{U}\;,$$ (7)

which corresponds to a curvilinear orthogonal coordinate system.

Particular examples of coordinate systems for one-way wave propagation are:

Cartesian (propagation in depth):

$x_1=\xi$, $x_2=\eta$, $x_3=\zeta$,

Cylindrical (propagation in radius):

$x_1=\zeta\,\cos{\xi}$, $x_2=\zeta\,\sin{\xi}$, $x_3=\eta$,

Spherical (propagation in radius):

$x_1=\zeta\,\sin{\xi}\,\cos{\eta}$, $x_2=\zeta\,\sin{\xi}\,\sin{\eta}$, $x_3=\zeta\,\cos{\xi}$,

Ray family (propagation along rays):

$\xi$ and $\eta$ represent parameters defining a particular ray in the family (i.e. the ray take-off angles), J is the geometrical spreading factor, related to the cross-sectional area of the ray tube Cervený (2001). The coefficients E, F, G, and J are easily computed by finite-difference approximations with the Huygens wavefront tracing technique Sava and Fomel (2001). If the propagation parameter $\zeta$ is taken to be time along the ray, then $\AA$ equals the propagation velocity v.