Substituting the definitions (60) into equation (65) et. seq. gives dispersion relationships for comparison to the exact expression (59).

(65) | ||

Identification of *i k*_{z} with converts
the dispersion relations (65)
into the differential equations

(66) | ||

The differential equations above
in Table .4
were based on a dispersion relation
that in turn was based on an assumption of constant velocity.
Surprisingly, these equations also have validity and great utility
when the velocity is depth-variable, *v* = *v*(*z*).
The limitation is that the velocity be constant over each
depth ``slab'' of width over which the downward-continuation
is carried out.

12/26/2000