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

International Tables for Crystallography (2013). Vol. D, ch. 2.4, pp. 349-350

Section 2.4.2. Elastic waves

R. Vachera* and E. Courtensa

aLaboratoire des Verres, Université Montpellier 2, Case 069, Place Eugène Bataillon, 34095 Montpellier CEDEX, France
Correspondence e-mail:

2.4.2. Elastic waves

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The fundamental equation of dynamics (see Section[link] ), applied to the displacement u of an elementary volume at r in a homogeneous material is [\rho \ddot u_i = {{\partial T _{ij}}\over {\partial x_j }}. \eqno (]Summation over repeated indices will always be implied, and T is the stress tensor. In non-piezoelectric media, the constitutive equation for small strains S is simply [T _{ij} = c_{ijk\ell }S_{k\ell }. \eqno (]The strain being the symmetrized spatial derivative of u, and c being symmetric upon interchange of k and [\ell], the introduction of ([link] in ([link] gives (see also Section[link] ) [\rho \ddot u_i = c_{ijk\ell }{{\partial ^2 u_k }\over {\partial x_j \partial x_\ell }}. \eqno (]One considers harmonic plane-wave solutions of wavevector Q and frequency ω,[{\bf u}({\bf r},t) = {\bf u}_0 \exp i({\bf Q}\cdot {\bf r}- \omega t). \eqno (]For [{\bf u}{_0}] small compared with the wavelength [2\pi/Q], the total derivative [\ddot u] can be replaced by the partial [\partial ^2 u/\partial t^2] in ([link]. Introducing ([link] into ([link], one obtains [c_{ijk\ell }\hat Q_j \hat Q_\ell u_{0k} = C\delta _{ik}u_{0k}, \eqno (]where [\hat{\bf Q}= {\bf Q}/| {\bf Q}|] is the unit vector in the propagation direction, [\delta _{ik}] is the unit tensor and [C \equiv \rho V^2], where [V = \omega / | {\bf Q} |] is the phase velocity of the wave. This shows that [u_0] is an eigenvector of the tensor [c_{ijk\ell }\hat Q_j \hat Q_\ell]. For a given propagation direction [\hat{\bf Q}], the three eigenvalues [C^{(s)}] are obtained by solving  [\left| {c_{ijk\ell }\hat Q_j \hat Q_\ell - C\delta _{ik}}\right| = 0. \eqno (]To each [C^{(s)}] there is an eigenvector [{\bf u}^{(s)}] given by ([link] and an associated phase velocity [V^{(s)} = \sqrt {C^{(s)}/\rho }. \eqno (]

The tensor [c_{ijk\ell }\hat Q_j \hat Q_\ell] is symmetric upon interchange of the indices ([i,k]) because [c_{ijk\ell } = c_{k\ell ij}]. Its eigenvalues are real positive, and the three directions of vibration [\hat{\bf u}^{(s)}] are mutually perpendicular. The notation [\hat{\bf u}^{(s)}] indicates a unit vector. The tensor [c_{ijk\ell }\hat Q_j \hat Q_\ell] is also invariant upon a change of sign of the propagation direction. This implies that the solution of ([link] is the same for all symmetry classes belonging to the same Laue class.

For a general direction [\hat{\bf Q}], and for a symmetry lower than isotropic, [\hat{\bf u}^{(s)}] is neither parallel nor perpendicular to [\hat{\bf Q}], so that the modes are neither purely longitudinal nor purely transverse. In this case ([link] is also difficult to solve. The situation is much simpler when [\hat{\bf Q}] is parallel to a symmetry axis of the Laue class. Then, one of the vibrations is purely longitudinal (LA), while the other two are purely transverse (TA). A pure mode also exists when [\hat{\bf Q}] belongs to a symmetry plane of the Laue class, in which case there is a transverse vibration with [\hat{\bf u}] perpendicular to the symmetry plane. For all these pure mode directions, ([link] can be factorized to obtain simple analytical solutions. In this chapter, only pure mode directions are considered. Piezoelectric media

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In piezoelectric crystals, a stress component is also produced by the internal electric field E, so that the constitutive equation ([link] has an additional term (see Section[link] ), [T _{ij} = c_{ijk\ell }S_{k\ell } - e_{mij}E_m, \eqno (]where e is the piezoelectric tensor at constant strain.

The electrical displacement vector D, related to E by the dielectric tensor [\boldvarepsilon], also contains a contribution from the strain, [D_m = \varepsilon _{mn}E_n + e_{mk\ell }S_{k\ell }, \eqno (]where [\boldvarepsilon] is at the frequency of the elastic wave.

In the absence of free charges, [{\rm div}\,{\bf D}= 0], and ([link] provides a relation between E and S, [\varepsilon _{mn}Q_n E_m + e_{mk\ell }Q_mS_{k\ell } = 0. \eqno (]For long waves, it can be shown that E and Q are parallel. ([link] can then be solved for E, and this value is replaced in ([link] to give [T _{ij} = \left [{c_{ijk\ell } + {{e_{mij}e_{nk\ell }\hat Q_m \hat Q_n }\over {\varepsilon _{gh}\hat Q_g \hat Q_h }}}\right]S_{k\ell } \equiv c_{ijk\ell }^{(e)}S_{k\ell }. \eqno (]Comparing ([link] and ([link], one sees that the effective elastic tensor [{\bf c}^{(e)}] now depends on the propagation direction [\hat{\bf Q}]. Otherwise, all considerations of the previous section, starting from ([link], remain, with c simply replaced by [{\bf c}^{(e)}].

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