Next: One-way wave-equation Up: Riemannian wavefield extrapolation Previous: Introduction

Acoustic wave-equation

The acoustic propagation of a monochromatic wave is governed by the Helmholtz equation:
 (1)
where is temporal frequency, v is the spatially variable wave propagation velocity, and represents a wavefield.

The Laplacian operator acting on a scalar function in an arbitrary Riemannian space with coordinates takes the form
 (2)
where gij is a component of the associated metric tensor, and is its determinant (106). The differential geometry of any coordinate system is fully represented by the metric tensor gij.

The expression simplifies if one of the coordinates, e.g. the coordinate of one-way wave extrapolation ,is orthogonal to the other coordinates . The metric tensor reduces to
 (3)
where E, F, G, and are differential forms that can be found from mapping Cartesian coordinates to general Riemannian coordinates , as follows:

E& = & _k _k _k ,
F& = & _k _k _k ,
G& = & _k _k _k ,
Å^2 & = & _k _k _k .

The associated metric tensor has the matrix
 (4)
where . The metric determinant takes the form
 (5)

Substituting ametric and det into laplac, and making the notations , , and , with orthogonal to both and ,we obtain the Helmholtz wave helm for propagating waves in a 3D semi-orthogonal Riemannian space:  1ÅJ

JÅ + GÅJ - FÅJ + EÅJ - FÅJ

= - ^2v^2 .

In weqrc.3d, is the wave propagation velocity mapped to Riemannian coordinates.

For the special case of two dimensional spaces (F=0 and G=1), the Helmholtz wave equation reduces to the simpler form:
 (6)
which corresponds to a curvilinear orthogonal coordinate system.

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

Cartesian (propagation in depth):
, , ,