Eikonal function ($S$)

The rapidly varying real phase of a short-wavelength field written $a(\mathbf{r},t)e^{iS(\mathbf{r},t)}$. Its derivatives define the local wavevector $\mathbf{k}=\nabla S$ and angular frequency $\omega =-{\partial }_{t}S$. The WKB approximation assumes that $a$ and background parameters vary little across one wavelength.

The accumulated phase along a one-dimensional ray. In an oscillatory WKB approximation, the two travelling waves have phases $\pm \int kdx$. In an evanescent region, the positive local decay rate $\kappa =\sqrt{-k^2}$ replaces $k$, producing exponents $\pm \int \kappa dx$.