Eikonal equations including wavenumber

The vector form of Eikonal equations can be rewritten in order to include the wavenumber as[14]:

$$\frac{d}{ds}\left( k\frac{d\mbox{$\mathbf{r}$}}{ds} \right) = \mbox{\boldmath$\nabla$}k .$$ (A.3)

The previous equation can be written as the system:

$$\frac{d\mbox{$\mathbf{r}$}}{ds} = \mbox{$\mathbf{e}$}_s \hsk... ...nabla$}k - \frac {1}{k}\frac{dk}{ds}\mbox{$\mathbf{e}$}_s .$$ (A.4)