Substituting the definitions (60) into equation (65) et. seq. gives dispersion relationships for comparison to the exact expression (59).
Identification of i kz with converts the dispersion relations (65) into the differential equations
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.