Christoffel equation

This is indeed a system of three second-order differential equations. The tensor elements Cyki may be complex-valued in case of viscoelasticity. Analysis shows that the propagation can be split into three orthogonally polarized planar waves propagating along a wave vector k. Those three waves may have different propagating celerities. Phase celerity and polarization ilj are connected through Christoffel equation ... 

All crystals are anisotropic many other structures also have elastic anisotropy. The propagation of elastic waves in anisotropic media is described by the Christoffel equation. This still depends on Newton s law and Hooke s law, but it is expressed in tensor form so that elastic anisotropy may be included. The tensor description of elastic stiffness was summarized in 6.2, especially eqns (6.23)—(6.29). The Christoffel equation is... 

The solutions to the Christoffel equation can be found by rewriting it in the form... 

Like the isotropic wave equation the Christoffel equation has three solutions, although in general there is no degeneracy except along symmetry directions. The motions of the particles are orthogonal for the three solutions, but not necessarily exactly parallel or perpendicular to the propagation direction, and so the waves are described as quasi-longitudinal or quasi-shear. 

The solution to the Christoffel equation for a given material can be plotted in k-space as surfaces of k/w. These are known as slowness surfaces, because they represent the reciprocal of phase velocity. Figure 11.1 is a representation of slowness surfaces of nickel in three-dimensional k-space. Because longitudinal waves have the greatest velocity and therefore... 

In order to calculate the reflectance function for waves from a fluid incident on an anisotropic surface, the Christoffel equation must be solved with the... 

The Christoffel equation yields a sextic in lz. For each of these solutions the normalized particle velocity vector v can be found. Hence the traction is... 

The phonon frequency Vphonon= < /27i is determined by the Christoffel equation, which relates the phase ve-... 

When the equation of motion and the wave equation are combined, the following relationship, known as the Christoffel equation, results ... 

When the matrix multiplication represented by the Christoffel equation is performed, a set of simultaneous equations is obtained. These equations are expressions for all allowed modes of plane wave propagation in an unbounded... 

For an isotropic pol3mer the h3qDersonic sound velocity is determined as a function of direction, and the elastic constants are obtained by fitting the data to a set of equations known as the Christoffel equations, which essentially relate the values of Vs to the stiffness constants Cs through Cs = pV, where p is the density [24]. For a detailed discussion of this technique the reader is referred to papers by Kruger, Pietralla and co-workers [25,26]. 

The Christoffel symbols (Spiegel, 1974), appearing in the second term on the right-hand side of Equation (5.57), are defined as... 

We denote projective differentiation with respect to H and P by and respectively. On solving the equations (20) one obtains the Christoffel formulae... 

In particular, from the third equations of (3.29) together with (3.27), the coefficients are shown to be equal to the Christoffel symbols given in (2.7) ... 

The Christoffel symbols can be calculated from the base vectors in equations (4) and (5) and are given by ... 

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