# Reflection and refraction laws and Fermat’s principle

Series | Geophysical References Series |
---|---|

Title | Problems in Exploration Seismology and their Solutions |

Author | Lloyd P. Geldart and Robert E. Sheriff |

Chapter | 6 |

Pages | 181 - 220 |

DOI | http://dx.doi.org/10.1190/1.9781560801733 |

ISBN | ISBN 9781560801153 |

Store | SEG Online Store |

## Problem 6.3a

Use Fermat’s principle of stationary time to derive the law of reflection.

### Background

In the solution of problem 3.1a we showed that the angle of incidence equals the angle of reflection and that, for the angle of refraction **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta_{2}}**
, **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \; sin\; }\theta_{2} =(V_{2}/V_{1}){\rm \; sin\; }\theta_{1}}**
[see also equation (3.1a)].

These are the laws of reflection and refraction. *Fermat’s principle of least time* (more accurately, *of stationary time*) states that wave travel between any two points is along the path for which the traveltime is either a maximum or a minimum value (i.e., the derivative of the traveltime equals zero) compared with the traveltimes along adjacent paths.

### Solution

In Figure 6.3a, the source **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S}**
and the receiver **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R}**
have coordinates **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (0,\; h_{1} )}**
and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (a,\; h_{2} )}**
. The traveltime for a wave from **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S}**
to **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R}**
with reflecting point **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M(x,\; 0)}**
is

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} t=(1/V)\{ (x^{2} +h_{1}^{2} )^{1/2} +[(a-x)^{2} +h_{2}^{2} ]^{1/2} \} . \end{align} }**

To find the point **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M}**
for which the value of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t}**
is stationary, we differentiate **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t}**
with respect to **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x}**
and set the result equal to zero. Thus,

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{{\rm d}t}{{\rm d}x} =(1/V)\left\{ \frac{x}{(x^{2} +h_{1}^{2} )^{1/2} } -\frac{(a-x)}{[(a-x)^{2} +h_{2}^{2} ]^{1/2} } \right\} = 0. \end{align} }**

The two terms in the brackets are the sines of the angles **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta _{1}}**
and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta _{1}^{'}}**
; hence, **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \; sin\; }\theta _{1} ={\rm \; sin\; }\theta _{2}^{'}.}**

## Problem 6.3b

Repeat part (a) for the refracted path *SMQ*, in Figure 5.3a.

### Solution

The traveltime for the path *SMQ* is

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} t=(x^{2} +h_{1}^{2} )^{1/2} /V_{1} +[(b-x)^{2} +h_{3}^{2} ]^{1/2} /V_{2} . \end{align} }**

Differentiation gives

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{{\rm d}t}{{\rm d}x} =\frac{x}{V_{1} (x^{2} +h_{1}^{2} )^{1/2} } -\frac{(b-x)}{V_{2} [(b-x)^{2} +h_{3}^{2} ]^{1/2} } =0, \end{align} }**

that is, **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \; sin\; }\theta_{1} /V_{1} ={\rm \; sin\; }\theta_{2} /V_{2}}**
.

## Problem 6.3c

Repeat parts (a), (b) for reflected and refracted converted S-waves.

### Solution

If we replace the angles **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta}**
with the angles **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta}**
and use the S-wave velocities , the foregoing proofs are otherwise unchanged.

## Continue reading

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Horizontal resolution | Effect of reflector curvature on a plane wave |

Previous chapter | Next chapter |

Geometry of seismic waves | Characteristics of seismic events |

## Also in this chapter

- Characteristics of different types of events and noise
- Horizontal resolution
- Reflection and refraction laws and Fermat’s principle
- Effect of reflector curvature on a plane wave
- Diffraction traveltime curves
- Amplitude variation with offset for seafloor multiples
- Ghost amplitude and energy
- Directivity of a source plus its ghost
- Directivity of a harmonic source plus ghost
- Differential moveout between primary and multiple
- Suppressing multiples by NMO differences
- Distinguishing horizontal/vertical discontinuities
- Identification of events
- Traveltime curves for various events
- Reflections/diffractions from refractor terminations
- Refractions and refraction multiples
- Destructive and constructive interference for a wedge
- Dependence of resolvable limit on frequency
- Vertical resolution
- Causes of high-frequency losses
- Ricker wavelet relations
- Improvement of signal/noise ratio by stacking