Christoffel tensor

Once again, our problem of determining the wave solutions associated with our medium has been reduced to a problem of matrix diagonalization. The eigenvalues of the acoustic tensor or Christoffel tensor, Aij = (A + p)qiqj + p-q Sij, are found to be... 

In this problem, deduce the eigenvalues of the Christoffel tensor and also recreate the vibrational density of states for the Debye model given in eqn (5.61). In light of these results, fill in the various steps culminating in eqn (5.66). 

The first term, known as the Ricci tensor, is obtained from the 4-index Riemann-Christoffel tensor on contraction with the mixed fundamental tensor ... 

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... 

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