# Eikonal equation - Snell's law

I am interested in equations of the form $$|\nabla d|= F(x)$$, where $$F(x)$$ is piecewise constant and $$d(x) = 0$$ on $$\Gamma_D$$, a subset of the boundary. In particular, like in the figure, one can consider $$F(x)$$ taking two values, delimited by an interface $$\Sigma$$ (which can be considered smooth).

Snell's law gives a relation between the incidence angles of a ray passing through the interface: $$\sin \theta_+/\sin \theta_- = V_+/V_-$$. I cannot find any clear references proving that Snell's law holds for the eikonal equation described above. Therefore I arrive at my question:

1. Does Snell's law hold across the interface $$\Sigma$$?
1. If the answer to the first question is affirmative, can you indicate references where a proof can be found?

We substitute into the eikonal equation the definition of a ray, being a unit vector $${\bf v}$$ in the direction of $$\nabla d$$. Then the eikonal equation $$|\nabla d|=F$$ can be rewritten in vectorial form as $$\nabla d=F{\bf v}$$, and hence for any closed contour $$C$$ one has $$\oint_C F\,{\bf v}\cdot {\rm d}{\bf l}=\oint_C\nabla d\cdot {\rm d}{\bf l}=0.$$ If you take a contour as in the figure (with tangential unit vector $$\hat{\bf t}$$ and infinitesimal perpendicular thickness), you arrive at $$\hat{\bf t}\cdot(V^+{\bf v}_+-V^-{\bf v}_-)=0,$$ which implies Snell's law.
• The gradient $\nabla d$ might not be continuous across the interface where the speed changes. Is the contour integral of $\nabla d$ still equal to zero in this case? Sep 30 at 21:38
• the component of $\nabla d$ parallel to the interface must be continuous, because $F$ is only discontinuous perpendicularly to the interface, so yes, the contour integral of $\nabla d\cdot{\rm d}{\bf l}$ vanishes. Sep 30 at 22:07
• "The component of $\nabla d$ parallel to the interface must be continuous" is equivalent to Snell's law. To bring this argument in justifying the contour integral is zero leads to a circular argument. Oct 1 at 20:49