# Snell's law

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When a wave crosses a boundary between two isotropic media, the wave changes direction such that

${\displaystyle {\frac {\sin i}{V_{1}}}={\frac {\sin r}{V_{2}}}}$,

where i is the angle of the incident wave, Vi is the velocity of the incident medium, r is the angle of refraction, and V2 is the velocity of the second medium. If sinr exceeds 1 as given by this equation, a head wave will be generated. Snell's law is also called Descartes' law. In anisotropic media (where wavefronts are not necessarily perpendicular to raypaths), Snell's law holds for the angles measured between an interface and the wavefronts, using phase velocities. The law is named for Willebrord Snellius (1580-1626), Dutch mathematician.

Snell's law relations when a wave crosses a boundary between two isotropic solid media; the wave changes direction such that for either incident P-wave or S-wave,

${\displaystyle {\frac {\sin i}{V_{1}}}={\frac {\sin \theta _{p1}}{V_{p1}}}={\frac {\sin \theta _{s1}}{V_{s1}}}={\frac {\sin \theta _{p2}}{V_{p2}}}={\frac {\sin \theta _{s2}}{V_{s2}}}=p}$,

where i is the angle of the incident wave with a velocity Vi=VP1 if a P-wave or Vi=VS1 if an S-wave; ${\displaystyle \theta _{P1}}$ and ${\displaystyle \theta _{S1}}$ are the angles of reflection of the P- or S-waves in medium 1, which have velocities VP1 and VS1, respectively; ${\displaystyle \theta _{P2}}$ and ${\displaystyle \theta _{S2}}$ are the angles of refraction of the P- and S-waves in medium 2 which have velocities VP2 and VS2, respectively; p is the raypath parameter (which will be a constant along a raypath through parallel layering). If ${\displaystyle \sin \theta _{P2}}$ or ${\displaystyle \sin \theta _{S2}}$ exceeds 1 as given by this equation, a head wave will be generated.

## Derivation of Snell's Law

Snell’s Law for refraction describes one of the most important concepts of seismic wave propagation. A simple derivation of this law uses Huygens’ Principle and the so-called plane wave approximation. Huygens’ Principle states that every point on an advancing wavefront can be regarded as the source of a secondary wave, and a later wavefront is the envelope tangent to all of the secondary wavefronts, as shown in the following figure.

Building on this principle, you can consider a wavefront to be planar, as though its source is very far away (this approximation is useful, but is not strictly true in all circumstances). Given Huygens’ Principle and the plane wave approximation, you can derive Snell’s Law for refraction using the following two-layer model, in which the compressional wave velocity V in the upper layer is greater than in the lower layer (the results of the derivation are the same for the case where V in the lower layer is greater than in the upper layer). In this model the incident and refracted plane waves are color-coded in red and green, respectively, and sources of secondary wavefronts on the horizontal refracting boundary are represented by red dots. Secondary wavefronts propagating away from these sources into the lower layer are shown in gray and black.

Two-layer model for derivation of Snell’s Law for refraction.
Enlargement of a portion of the refracting boundary from the previous figure.

In the following derivation ${\displaystyle \Delta t}$ is a unit of travel time for a propagating wave. For a given ${\displaystyle \Delta t}$ the distance D2 in the lower layer is less than the distance D1 in the upper layer because ${\displaystyle V_{2}. The derivation hinges on recognition of the trigonometric relationships between the distance X along the refracting boundary and the distances D1 and D2.