Equation of propagation of a wave in a material

Authier, A.; Zarembowitch, A.

doi:10.1107/97809553602060000630

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.3, p. 86

Section 1.3.4.2. Equation of propagation of a wave in a material

A. Authier^a ^* and A. Zarembowitch^b

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and ^bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France
Correspondence e-mail: aauthier@wanadoo.fr

1.3.4.2. Equation of propagation of a wave in a material

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Consider the propagation of a wave in a continuous medium. The elongation of each point will be of the form $[{\bf u} = {\bf u_0}\exp(2\pi i\nu t) \exp(-2\pi i{\bf q \cdot r}), \eqno(1.3.4.1) ]$ where ν is the frequency and q is the wavevector. The velocity of propagation of the wave is $[V = \nu / q. \eqno(1.3.4.2) ]$

We saw in Section 1.3.3.6 that the equilibrium condition is $[c_{ijkl}{\partial^2 u_k \over \partial x_l \partial x_{j}}+ \rho F_{i} = 0. ]$ Here the only volume forces that we must consider are the inertial forces: $[c_{ijkl}{\partial^2 u_k \over \partial x_l \partial x_{j}} = \rho {\partial^2 x_i\over \partial t^2}. \eqno(1.3.4.3) ]$

The position vector of the point under consideration is of the form $[{\bf r} = {\bf r}_0 + {\bf u},]$ where only $[{\bf u}]$ depends on the time and $[{\bf r}_0]$ defines the mean position. Equation (1.3.4.3) is written therefore $[c_{ijkl }{\partial^2 u_k \over \partial x_l \partial x_{j}} = \rho {\partial^2 u_i\over \partial t^2 }. \eqno(1.3.4.4) ]$

Replacing u by its value in (1.3.4.1), dividing by $[-4 \pi^{2}]$ and using orthonormal coordinates, we get $[c_{ijkl }u_{k}q_{j}q_{l } = \rho \nu ^{2}u_{i}. \eqno(1.3.4.5) ]$

It can be seen that, for a given wavevector, $[\rho \nu^{2} ]$ appears as an eigenvalue of the matrix $[c_{ijkl } u_{k}q_{j}q_{l } ]$ of which the vibration vector u is an eigenvector. This matrix is called the dynamical matrix, or Christoffel matrix. In order that the system (1.3.4.5) has a solution other than a trivial one, it is necessary that the associated determinant be equal to zero. It is called the Christoffel determinant and it plays a fundamental role in the study of the propagation of elastic waves in crystals.

Let $[\alpha_{1}, \alpha_{2}, \alpha_{3}]$ be the direction cosines of the wavevector q. The components of the wavevector are $[q_{i} = q \alpha_{i}.]$ With this relation and (1.3.4.2), the system (1.3.4.5) becomes $[c_{ijkl} u_{k}\alpha_j\alpha_l = \rho \nu ^{2}u_{i}. \eqno(1.3.4.6) ]$ Putting $[\Gamma_{ik} = c_{ijkl}\alpha_{j}\alpha_{l} \eqno(1.3.4.7) ]$ in (1.3.4.6), the condition that the Christoffel determinant is zero can be written $[\Delta \left(\Gamma_{ik} - \rho \nu^{2}\delta _{ik}\right) = 0. \eqno(1.3.4.8) ]$

On account of the intrinsic symmetry of the tensor of elastic stiffnesses, the matrix $[\Gamma_{ik}]$ is symmetrical.

If we introduce into expression (1.3.4.7) the elastic stiffnesses with two indices [equation (1.3.3.6)], we find, for instance, for $[\Gamma_{11}]$ and $[\Gamma_{12}]$ $[\eqalign{\Gamma_{11} &= c_{11}(\alpha_{1}){}^{2}+ c_{66}(\alpha_{2}){}^{2}+ c_{55}(\alpha_{3}){}^{2} + 2c_{16}\alpha_{1}\alpha_{2}\cr&\quad+ 2c_{15}\alpha_{1}\alpha_{3} +2c_{56}\alpha_{2}\alpha_{3} \cr \Gamma_{12}&= c_{16}(\alpha_{1}){}^{2}+ c_{26}(\alpha_{2}){}^{2} + c_{45}(\alpha_{3}){}^{2}+ (c_{12}+c_{66})\alpha_{1}\alpha_{2} \cr&\quad+ (c_{14}+c_{56})\alpha_{1}\alpha_{3}+(c_{46} + c_{25})\alpha_{2}\alpha_{3}.\cr} ]$

The expression for the effective value, $[c^{e}_{ijkl}]$ , of the `stiffened' elastic stiffness in the case of piezoelectric crystals is given in Section 2.4.2.2 .

References

International Tables for Crystallography (2006). Vol. D. ch. 1.3, p. 86