Translations:Snell’s law/3/en

Ray vector AB in layer 1 has length $n_{\rm {1}}$ and makes angle ${\theta }_{\rm {1}}$ with the vertical. Ray vector BC in layer 2 has length $n_{\rm {2}}$ and makes angle ${\theta }_{\rm {2}}$ with the vertical. By Snell’s law, we have $n_{1}\mathrm {sin} {\theta }_{\rm {l}}{\rm {=}}n_{\rm {2}}{\rm {\ sin\ }}{\theta }_{\rm {2}}$. This equation says that the horizontal projections of the two ray vectors are the same. We expect this result because the slowness does not change in the horizontal direction. The slowness does E change at the interface in the vertical direction. The shallow medium has slowness $n_{\rm {l}}$, and the deep medium has a smaller slowness $n_{\rm {2}}$. Because the interface is level, it follows that slowness is constant along any horizontal line, so $\partial n/\partial x{\rm {=0}}$. However, slowness does change at the interface. The difference of the slowness at the interface is $n_{\rm {2}}-n_{\rm {1}}$, which is negative because $n_{\rm {2}}{\rm {<}}n_{\rm {1}}$. The gradient of slowness at the interface is proportional to $\left(n_{\rm {2}}-n_{\rm {l}}\right)\mathbf {j}$. This vector points straight into the low-velocity layer. The difference vector BD = BC – AB must be proportional to the gradient of the slowness, and hence vector BD must point straight down. Thus, $n_{\rm {1}}{\rm {\ cos\ }}{\theta }_{\rm {l}}{\rm {>}}n_{\rm {2}}{\rm {\ cos\ }}{\theta }_{\rm {2}}$, which says that the raypath bends to the right at the interface. In this section, we have given a preview of the ray equation.