Reflection and refraction laws and Fermat’s principle

From SEG Wiki
Jump to: navigation, search

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 , [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.

Figure 6.3a.  Deriving Snell’s law.

Solution

In Figure 6.3a, the source and the receiver have coordinates and . The traveltime for a wave from to with reflecting point is

To find the point for which the value of is stationary, we differentiate with respect to and set the result equal to zero. Thus,

The two terms in the brackets are the sines of the angles and ; hence,

Problem 6.3b

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

Solution

The traveltime for the path SMQ is


Differentiation gives

that is, .

Problem 6.3c

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

Solution

If we replace the angles with the angles and use the S-wave velocities , the foregoing proofs are otherwise unchanged.

Continue reading

Previous section Next section
Horizontal resolution Effect of reflector curvature on a plane wave
Previous chapter Next chapter
Geometry of seismic waves Characteristics of seismic events

Table of Contents (book)

Also in this chapter

External links

find literature about
Reflection and refraction laws and Fermat’s principle
SEG button search.png Datapages button.png GeoScienceWorld button.png OnePetro button.png Schlumberger button.png Google button.png AGI button.png