Dispersion relations

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

$$\begin{eqnarray} 5^\circ : \quad\quad\quad & k_z \eq & \displaystyle {\strut\om... ...2\,{\omega\over v} - {\strut v k_x^2\over 2\omega}}} \nonumber\end{eqnarray}$$ (65)

Identification of i k_z with $\partial / \partial z$ converts the dispersion relations (65) into the differential equations

$$\begin{eqnarray} 5^\circ : \quad\quad& \displaystyle {\strut \partial P\over \p... ...a\over v} - {\strut v k_x^2\over 2\omega}}} \right) P \nonumber\end{eqnarray}$$ (66)

which are extrapolation equations for when velocity depends only on depth.

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 $\Delta z$ over which the downward-continuation is carried out.