Complex velocity in the 5 degree equation

The 5$^\circ$ equation, namely,

$$\begin{eqnarray} {\partial P \over \partial z}&=&{1 \over v }\ {\partial P \over \partial t} \\ i\,k_z &=& {-i \omega \over v }\end{eqnarray}$$ (2) (3)

states that a wavefront will take some time to get from one depth to another. With velocity v being a real constant, waves controlled by (2) and (3) propagate without change in form. In practice waveform changes are observed. So v should not be a real constant. An imaginary part of the velocity would cause attenuation. A frequency-dependent velocity would cause frequency dispersion.