On Fermat’s principle and Snell’s law in lossy anisotropic media

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. T107-T116 ◽  
Author(s):  
José M. Carcione ◽  
Bjorn Ursin

Fermat’s principle of least action is one of the methods used to trace rays in inhomogeneous media. Its form is the same in anisotropic elastic and anelastic media, with the difference that the velocity depends on frequency in the latter case. Moreover, the ray, envelope, and energy velocities replace the group velocity because this concept has no physical meaning in anelastic media. We have first considered a lossy (anelastic) anisotropic medium and established the equivalence between Fermat’s principle and Snell’s law in homogeneous media. Then, we found that the different ray velocities defined in the literature were the same for stationary rays in homogeneous media, with phase and inhomogeneity angles satisfying the principle and the law. We considered an example of a transversely isotropic medium with a vertical symmetry axis and wavelike and diffusionlike properties. In the first case, the differences were negligible, which was the case of real rocks having a quality factor greater than five. Strictly, ray tracing should be based on the so-called stationary complex slowness vector to obtain correct results, although the use of homogeneous viscoelastic waves (zero inhomogeneity angle) is acceptable as an approximation for earth materials. However, from a rigorous point of view, the three velocities introduced in the literature to define the rays present discrepancies in heterogeneous media, although the differences are too small to be measured in earth materials. The findings are also valid for electromagnetic waves by virtue of the acoustic-electromagnetic analogy.

2003 ◽  
Vol 26 (6) ◽  
pp. 779-781
Author(s):  
Hans-Martin Gaertner

I will present evidence that nature does not optimize in the sense of Fermat's principle of least time, contrary to what Schoemaker's unintentionally ambiguous exposition might suggest. First, Huygens' principle, an alternative nonteleological account of Snell's law, is outlined. Second, I confront Fermat's principle with a substantive conceptual problem.


On Purpose ◽  
2019 ◽  
pp. 129-152
Author(s):  
Michael Ruse

This chapter explains occurrences during and after the Scientific Revolution, in which the personification of nature that is at the heart of the Aristotelian philosophy had a nasty way of reappearing in the most orthodox of machine-metaphor- influenced places. Even more than mechanics, optics was riddled with final-cause thinking. Pierrre de Fermat's “principle of least time” explains Snell's law of refraction, the connection between the angle of incidence and the angle of refraction. Since light going from a less dense to a denser medium is bent toward the normal, it is not going from beginning to end by the shortest distance. But assuming that light travels less quickly in a more dense than less dense medium, one can show that it does travel in the shortest time.


2019 ◽  
Vol 54 (5) ◽  
pp. 055019
Author(s):  
Sushil Kumar Singh ◽  
Jaya Shivangani Kashyap ◽  
Priyanka Rajwani ◽  
Savinder Kaur

Geophysics ◽  
1976 ◽  
Vol 41 (6) ◽  
pp. 1126-1132 ◽  
Author(s):  
John W. Clough

Electromagnetic waves refracted at the critical angle according to Snell's law give rise to the lateral wave. The low amplitude lateral wave is usually obscured by other waves when continuous wave sources are used. Using a pulsed source (radar) and continuously recording echoes reflected from within dielectric earth materials as a function of angle of incidence, records are produced which clearly show the lateral wave. In some earth‐probing applications, the lateral wave may predominate and proper identification of its characteristics is important.


Geophysics ◽  
1979 ◽  
Vol 44 (5) ◽  
pp. 987-990 ◽  
Author(s):  
K. Helbig

Levin treats the subject concisely and exhaustively. Nevertheless, I feel a few comments to be indicated. My first point is rather general: of the three surfaces mentioned in the Appendix, the phase velocity surface (or normal surface) is easiest to calculate, since it is nothing but the graphical representation of the plane‐wave solutions for each direction. The wave surface has the greatest intuitive appeal, since it has the shape of the far‐field wavefront generated by an impulsive point source. The slowness surface, though apparently an insignificant transformation of the phase‐velocity surface, has the greatest significance for two reasons: (1) The projection of the slowness vector on a plane (the “component” of the slowness vector) is the apparent slowness, a quantity directly observed in seismic measurement. Continuity of wave‐fronts across an interface—the idea on which Snell’s law is based—is synonymous with continuity of apparent (or trace) slownesses; and (2) the slowness surface is the polar reciprocal of the wave surface; that is to say, not only has the radius vector of the slowness surface the direction of the normal to the wave surface (which follows from the definition of the two surfaces), but the inverse is also true. That is, the normal to the slowness surface has the direction of the corresponding ray (the radius vector of the wave surface). The fact that this surface so conveniently embodies all relevant information—direction of wave normal and ray, inverse phase velocity, inverse ray velocity (projection of the slowness vector on the ray direction), and the trace slowness along an interface—was the main reason for its introduction by Hamilton (1837) and McCullagh (1837). It is true that this information also can be obtained from the other surfaces, but only in a somewhat roundabout way, which can lead to serious complications. That only few of these complications are apparent in Levin’s article is a consequence of the fact that the polar reciprocal of a surface of second degree is another surface of second degree, in this case an ellipsoid. For more complicated and realistic types of anisotropy, one has to expect much more complicated surfaces. For transverse anisotropy, the slowness surface consists of one ellipsoid (SH‐waves) and a two‐leaved surface of fourth degree, the wave surface of an ellipsoid and a two‐leaved surface of degree 36. More general types of elastic anisotropy can lead to wave surfaces of up to degree 150, while the slowness surface is at most of degree six. It is, therefore, in the interest of a unified theory of wave propagation in anisotropic media to use, wherever possible, the slowness surface. The advantages of this are exemplified by Snell’s law in its general form. While it is impossible to base a concise formulation on the wave surface (reflected and refracted rays do not always lie in the plane containing the incident ray and the normal to the interface), the use of the slowness surface allows the following simple statement (Helbig 1965): “The slowness vectors of all waves in a reflection/refraction process have their end points on a common normal to the interface; the direction of the rays is parallel to the corresponding normals to the slowness surfaces”. A method to interpret refraction seismic data with an anisotropic overburden based on this form of Snell’s law has been described in Helbig (1964).


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