Reflection and refraction laws and Fermat’s principle

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Problem 6.3a

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


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.


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.


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.


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

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