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Fresnel's Elasticity Surface


The envelope of the plane

 lx+my+nz=c,
(1)

where c is the speed of propagation of a wave in the direction (l,m,n) (i.e., l, m, and n are the direction cosines) is known as the wave surface of a given medium (Love 1944, p. 299).

In an isotropic medium, c is independent of l, m, and n and is given by

 (lambda+2mu-rhoc^2)(mu-rhoc^2)^2=0
(2)

where rho is the medium density and lambda and mu are the so-called Lamé constants of the solid. The wave surface is then two-sheeted and both sheets are spheres (Love 1944, p. 299).

In the case of anisotropy, the surface in consists of three sheets corresponding to the values of c^2 that are roots of

 |lambda_(11)-rhoc^2 lambda_(12) lambda_(31); lambda_(12) lambda_(22)-rhoc^2 lambda_(23); lambda_(31) lambda_(23) lambda_(33)-rhoc^2|=0,
(3)

where the lambda_(ij) are functions of l, m, and n in terms of the coefficients of the strain-energy function (Christoffel 1877, Love 1944, p. 299). Green (1839) showed that the wave surface for the most general case of an elastic solid that allows the propagation of purely transverse plane waves consists of a sphere and the two sheets that are the envelope of the plane (1) subject to the condition

 (l^2)/(c^2-L/rho)+(m^2)/(c^2-M/rho)+(n^2)/(c^2-N/rho)=0,
(4)

where L, M, and N are constants characteristic of the surface. The two sheets correspond to what is known as Fresnel's wave-surface (Love 1944, p. 299).

FresnelsElasticitySurface

The images above shows the two sheets of Fresnel's wave surface for particular elasticity parameters (JavaView).

von Seggern (1993, p. 304) defines Fresnel's elasticity surface as the quartic surface given by

 r=sqrt(a^2x^2+b^2y^2+c^2z^2),
(5)

where

 r^2=x^('2)+y^('2)+z^('2),
(6)

though this appears to be distinct from the surface described by Love (1944) and on the JavaView website.


See also

Quartic Surface

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References

Christoffel, E. B. Ann. di Mat. (Ser. 2), t. 8, 1877. Reprinted in Ges. Math. Abhandlungen, Vol. 2. Leipzig, Germany: p. 81, 1910.Fischer, G. (Ed.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband. Braunschweig, Germany: Vieweg, p. 17, 1986.Fischer, G. (Ed.). Plates 38-39 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 38-39, 1986.Green, G. "On the Propagation of Light in Crystallized Media." Cambridge Philos. Soc. Trans. 7, 121-140, 1839. Reprinted in Mathematical Papers. London, p. 293, 1871.JavaView. "Classic Surfaces from Differential Geometry: Fresnel (Single Eigenvalue)." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_Fresnel.html.Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity, 4th ed. New York: Dover, pp. 299-300, 1944.von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 304, 1993.

Referenced on Wolfram|Alpha

Fresnel's Elasticity Surface

Cite this as:

Weisstein, Eric W. "Fresnel's Elasticity Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FresnelsElasticitySurface.html

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