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Book Chapter

On Application of Fermat’s Principle to Anisotropic Media1

M.P. Rudzki
M.P. Rudzki
Memoir presented on May 5, 1913.
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January 01, 2001


Following Fermat's principle, the time during which the light covers its trajectory from a given point A to another given point B satisfies the condition

where N denotes the index of refraction. By transformation, which is unnecessary to restate here3, one writes the above equation in this form:

from which results the well-known differential equations4

All of the above relate to isotropic media. In anisotropic media, N no longer denotes the index of refraction. In such media, one must distinguish between the speed of the propagation of light in the direction of the ray s and the speed in the direction normal to the wave surface q5. The index of refraction is inversely proportional to q, whereas N is inversely proportional to s. There is more; N depends also on direction, and it is a function of not only x, y, z but also of direction cosines

If, for convenience, we denote

the total variation of N becomes

The three direction cosines λ, μ, v are subject to the condition

however, it is unnecessary to introduce this additional condition, since the above relationship results simply from the equality

which we use in the transformation of the integral (I).

Let us calculate the variation of this integral under the assumption that N depends not only on coordinates, but also on direction cosines. With fixed limits, we obtain

We shall transform the second (sic)6 integral in the same fashion as in the case of isotropic media, namely, by virtue of equalities

The second integral (sic) becomes

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Society of Exploration Geophysicists Open File

Anisotropy 2000: Fractures, Converted Waves, and Case Studies

L. Ikelle
L. Ikelle
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A. Gangi
A. Gangi
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Society of Exploration Geophysicists
ISBN electronic:
Publication date:
January 01, 2001




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