Acoustic wave-equation

The acoustic propagation of a monochromatic wave is governed by the Helmholtz equation:  

$$\LAP \UU = - \frac{\ww^2}{v^2} \UU \;,$$ (1)

where $\ww$ is temporal frequency, v is the spatially variable wave propagation velocity, and $\UU$ represents a wavefield.

The Laplacian operator acting on a scalar function $\UU$ in an arbitrary Riemannian space with coordinates $\qvec=\{\qone,\qtwo,\qthr\}$ takes the form  

$$\LAP \UU = \sum_{i=1}^{3}\frac{1}{\sqrt{\vert\gvec\vert}}\,... ...qrt{\vert\gvec\vert}\,\frac{\partial \UU}{\partial \qx_j} \rp,$$ (2)

where g^ij is a component of the associated metric tensor, and $\vert\gvec\vert$ is its determinant (106). The differential geometry of any coordinate system is fully represented by the metric tensor g^ij.

The expression simplifies if one of the coordinates, e.g. the coordinate of one-way wave extrapolation $\qone$,is orthogonal to the other coordinates $(\qtwo,\qthr)$. The metric tensor reduces to  

$$\left[g_{ij}\right] = \lb\bea{ccc} E& F& 0 \\ F& G& 0 \\ 0 & 0 & \AA^2 \eea\rb\;,$$ (3)

where E, F, G, and $\AA$ are differential forms that can be found from mapping Cartesian coordinates $\xvec=\lc \xone,\xtwo,\xthr \rc$ to general Riemannian coordinates $\qvec=\lc \qone,\qtwo,\qthr \rc$, as follows:

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

The associated metric tensor $\lb g^{ij} \rb = \lb g_{ij} \rb ^{-1}$has the matrix  

$$\lb g^{ij} \rb = \lb \bea{ccc} +G/J^2 & -F/J^2 & 0 \\ -F/J^2 & +E/J^2 & 0 \\ 0 & 0 & 1/\AA^2 \eea\rb\;,$$ (4)

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

$$\vert\gvec\vert = \AA^2\,J^2\;.$$ (5)

Substituting ametric and det into laplac, and making the notations $\qone=\qx$, $\qtwo=\qy$, and $\qthr=\qz$, with $\qz$ orthogonal to both $\qx$ and $\qy$,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, $v \lp \qx,\qy,\qz \rp$ 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:  

$$\frac{1}{\AA J} \lb \eone{\lp \frac{J}{\AA} \done{\UU}{\qz} ... ...J} \done{\UU}{\qx} \rp}{\qx} \rb = - \frac{\ww^2}{v^2} \UU \;,$$ (6)

which corresponds to a curvilinear orthogonal coordinate system.

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

Cartesian (propagation in depth):

$\xone=\qx$, $\xtwo=\qy$, $\xthr=\qz$,

Cylindrical (propagation in radius):

$\xone=\qz\,\cos{\qx}$, $\xtwo=\qz\,\sin{\qx}$, $\xthr=\qy$,

Spherical (propagation in radius):

$\xone=\qz\,\sin{\qx}\,\cos{\qy}$, $\xtwo=\qz\,\sin{\qx}\,\sin{\qy}$, $\xthr=\qz\,\cos{\qx}$,

Ray family (propagation along rays):

$\qx$ and $\qy$ 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 (19). The coefficients E, F, G, and J are easily computed by finite-difference approximations with the Huygens wavefront tracing technique (92). If the propagation parameter $\qz$ is taken to be time along the ray, then $\AA$ equals the propagation velocity v.