Lateral velocity variation again

Having approximated the square root by a polynomial ratio, Table .4 or Table .5 can be inverse transformed from the horizontal wavenumber domain k_x to the horizontal space domain x by substituting $\ (ik_x )^2 = \partial^2 / \partial x^2$.The resulting extrapolation equations have a wide range of validity for v = v(x,z) even though the derivation would not seem to permit any change of velocity with x.

Ordinarily $\bar v(z)$ will be chosen to be some kind of horizontal average of v(x,z). Permitting $\bar v$ to become a function of x generates many new terms. The terms are awkward to implement and ignoring them introduced unknown hazards. So $\bar v$ is usually taken to depend on z but not x.