Snell's Law (Seismic Refraction) Equation, Derivation & Rearrangements

Core idea

Overview

Snell's Law in seismology describes how seismic waves change direction and speed as they transition between geological layers with differing elastic properties. This relationship is fundamental for determining the ray paths of P-waves and S-waves as they refract across boundaries like the crust-mantle interface.

When to use: Use this equation when calculating the trajectory of a seismic wave encountering a distinct boundary between two rock types or layers. It assumes the layers are isotropic and that the seismic velocity is constant within each layer.

Why it matters: This principle is the foundation of seismic refraction surveys, which allow geologists to map subsurface structures for oil exploration and engineering projects. It is also used to locate the depth of the Mohorovičić discontinuity and other internal Earth layers.

Symbols

Variables

$θ_{2}$ = Refraction Angle, $θ_{1}$ = Incidence Angle, $v_{1}$ = Velocity 1, $v_{2}$ = Velocity 2

θ_{2}

Refraction Angle

°

θ_{1}

Incidence Angle

°

v_{1}

Velocity 1

km/s

v_{2}

Velocity 2

km/s

Walkthrough

Derivation

Formula: Snell's Law (Seismic Refraction)

Determines the angle of refraction at a seismic boundary.

Mediums are homogeneous.
Specular refraction.

1

Relate angles and velocities:

The ratio of the sine of the angle to the velocity is constant across the interface.

\frac{sin θ _{1}}{v _{1}} = \frac{sin θ _{2}}{v _{2}}

Result

\frac{sin θ _{1}}{v _{1}} = \frac{sin θ _{2}}{v _{2}}

Source: Geology — Seismology

Free formulas

Rearrangements

Solve for $θ_{2}$

Snell's Law (Seismic Refraction): Make $θ_{2}$ the subject

θ_{2} = arcsin (\frac{v _{2} sin θ _{1}}{v _{1}})

Rearrange Snell's Law, which describes the refraction of waves, to solve for the refraction angle, $θ_{2}$ . This involves isolating $sin$ $θ_{2}$ and then applying the inverse sine function.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Visualize a seismic wavefront as a line of marchers crossing a boundary at an angle; if the ground changes from pavement to sand, the marchers on the sand slow down first, causing the entire line to pivot and change

θ_{1}

Angle of incidence of the seismic wave

The angle at which the incoming seismic ray strikes the boundary, measured from the normal (perpendicular line) to the interface.

v_{1}

Velocity of the seismic wave in the first medium

How fast the seismic wave travels through the initial geological layer before encountering the boundary.

θ_{2}

Angle of refraction of the seismic wave

The angle at which the seismic ray travels after crossing the boundary, measured from the normal to the interface.

v_{2}

Velocity of the seismic wave in the second medium

How fast the seismic wave travels through the new geological layer after crossing the boundary.

Free study cues

Insight

Canonical usage

The velocities ( $v_{1}$ , $v_{2}$ ) must be expressed in the same units, and the angles (θ_1, θ_2) must be used consistently with the trigonometric function (e.g., both in degrees or both in radians).

Common confusion

Mixing units for velocity (e.g., one in m/s and the other in km/s) without conversion, or using inconsistent angle units (degrees vs. radians) in calculations without adjusting the calculator mode.

Unit systems

$θ_{1}$ degrees or radians · Angle of incidence. Must be consistent with θ_2 (both degrees or both radians) for trigonometric functions. The sine function itself is dimensionless.

$θ_{2}$ degrees or radians · Angle of refraction. Must be consistent with θ_1 (both degrees or both radians) for trigonometric functions. The sine function itself is dimensionless.

$v_{1}$ m/s or km/s · Velocity of the seismic wave in the first medium. Must be in consistent units with v_2.

$v_{2}$ m/s or km/s · Velocity of the seismic wave in the second medium. Must be in consistent units with v_1.

One free problem

Practice Problem

A seismic P-wave travels through a layer of weathered rock (v1 = 2500 m/s) and strikes a limestone basement (v2 = 4500 m/s) at an incident angle of 20°. Calculate the angle of refraction as the wave enters the limestone.

Velocity 12500 km/s

Velocity 24500 km/s

Incidence Angle20 °

Solve for: $t 2$

Hint: Rearrange the formula to solve for sin(t2) by multiplying v2 by the ratio of sin(t1) to v1.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In a physics application involving Snell's Law (Seismic Refraction), Snell's Law (Seismic Refraction) is used to calculate Refraction Angle from Incidence Angle, Velocity 1, and Velocity 2. The result matters because it helps predict motion, energy transfer, waves, fields, or circuit behaviour and check whether the answer is plausible.

Study smarter

Tips

Ensure your calculator is set to degree mode, as most geological angles are provided in degrees.
The angle is always measured between the ray path and the normal line perpendicular to the interface.
If the wave travels from a low-velocity layer to a high-velocity layer, it will refract away from the normal.

Avoid these traps

Common Mistakes

Using degrees in calculator when sin() expects radians.
Convert units and scales before substituting, especially when the inputs mix °, km/s.
Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Determines the angle of refraction at a seismic boundary.

Use this equation when calculating the trajectory of a seismic wave encountering a distinct boundary between two rock types or layers. It assumes the layers are isotropic and that the seismic velocity is constant within each layer.

This principle is the foundation of seismic refraction surveys, which allow geologists to map subsurface structures for oil exploration and engineering projects. It is also used to locate the depth of the Mohorovičić discontinuity and other internal Earth layers.

Using degrees in calculator when sin() expects radians. Convert units and scales before substituting, especially when the inputs mix °, km/s. Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.