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The acoustic wave-equation for wavefield, , in a generalized
Riemannian space is,
| |
(5) |

where the is frequency, is the propagation slowness, and
is the Laplacian operator,
| |
(6) |

Substituting equation 6 into 5 generates
a Helmholtz equation appropriate for propagating waves through a 3D
space,
| |
(7) |

Expanding the derivative terms and multiplying through by yields ,
| |
(8) |

Defining *n*_{j} as,
| |
(9) |

leads to a more compact notation of equation 8,
| |
(10) |

Developing a wave-equation dispersion relation is achieved by
replacing the partial differential operators acting on wavefield
with their Fourier domain duals,
| |
(11) |

where is the Fourier domain dual of differential operator
. Equation 11 represents
the dispersion relationship for wavefield propagation on a generalized
3-D Riemannian space.