For =0, the VTI acoustic wave equation reduces to the simple elliptically anisotropic wave
equation (9), which is second order in *t*, and therefore has only two solutions.
Assuming that is small, let us insert, when possible, the 2-D elliptical wave equation

in place of one in each term of the 2-D version of equation (9),

The resultant formula after some manipulation is

This equation is second order in time, and therefore has two solutions. To arrive at this form
we had to ignore the difference between *P* and *F*, which is valid since our main
objective is to treat the kinematics of wave propagation.

To understand this equation kinematically, we Fourier transform equation (39)
in *x*, *z*, and *t*, and obtain the following dispersion relation:

(41) |

Figure 10 compares *k*_{z} given by
this dispersion relation with that given by equation (4), which is the
exact eikonal equation for acoustic VTI media. The difference
is small for , which implies that equation (39) can serve as a
valid substitute, kinematically, for equation (11).

Figure 10

10/9/1997