International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.3, pp. 8689
Section 1.3.4. Propagation of elastic waves in continuous media – dynamic elasticity^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France, and ^{b}Laboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France 
The elastic properties of materials have been considered in the preceding section in the static state and the elastic constants have been defined in terms of the response of the material to particular static forces. It is effectively the way the elastic constants have been measured in the past, although the measurements could not be very precise. A way of proceeding frequently used now is to excite a mechanical wave in the crystal and measure its propagation velocity or the wavelength associated with a particular frequency. One method consists in sending a train of ultrasonic waves through the crystal; one uses a pulse generator and a piezoelectric transducer glued to the crystal. The elapsed time between the emission of the train of waves and its reception after reflection from the rear face of the sample is then measured. Another method involves producing a system of standing waves after reflection at the inner surface of the crystal and determining the set of resonance frequencies. The experimental techniques will be described in Section 1.3.4.6.
The purpose of the next sections is to establish relations between the wavelength – or the velocity of propagation – and the elastic constants.
Consider the propagation of a wave in a continuous medium. The elongation of each point will be of the form where ν is the frequency and q is the wavevector. The velocity of propagation of the wave is
We saw in Section 1.3.3.6 that the equilibrium condition is Here the only volume forces that we must consider are the inertial forces:
The position vector of the point under consideration is of the form where only depends on the time and defines the mean position. Equation (1.3.4.3) is written therefore
Replacing u by its value in (1.3.4.1), dividing by and using orthonormal coordinates, we get
It can be seen that, for a given wavevector, appears as an eigenvalue of the matrix 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 be the direction cosines of the wavevector q. The components of the wavevector are With this relation and (1.3.4.2), the system (1.3.4.5) becomes Putting in (1.3.4.6), the condition that the Christoffel determinant is zero can be written
On account of the intrinsic symmetry of the tensor of elastic stiffnesses, the matrix 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 and
The expression for the effective value, , of the `stiffened' elastic stiffness in the case of piezoelectric crystals is given in Section 2.4.2.2 .
Equation (1.3.4.7) may be written
This shows that in a dynamic process only the sums can be measured and not and separately. On the contrary, can be measured directly. In the cubic system therefore, for instance, is determined from the measurement of on the one hand and from that of on the other hand.
The Christoffel determinant has three roots and the Christoffel matrix, being Hermitian with real coefficients, has three real eigenvalues and three orthogonal eigenvectors. The wavevector q, therefore, encompasses three waves with vibration vectors , , which are perpendicular to one another. In the general case, there is no particular angular relationship between the vibration vectors (or polarization vectors). However, if the latter are parallel to certain symmetry directions in the crystal, one of the vibration vectors is along this direction. The corresponding wave is called longitudinal. The two other waves have their polarization direction perpendicular to the wavevector and are thus transverse. If one of the polarization vectors is almost parallel to the wavevector, which often happens, then one speaks of the vibration as being quasilongitudinal.
We shall limit ourselves to cubic, hexagonal and tetragonal crystals and consider particular cases.

In hexagonal crystals, there are five independent elastic stiffnesses, , , , , and (Section 1.1.4.10.4 ).

In tetragonal crystals, there are six independent elastic stiffnesses, , , , , and (Section 1.1.4.10.4 ).

As mentioned in Section 1.3.4.1, the elastic constants of a material can be obtained by the elastic response of the material to particular static forces; however, such measurements are not precise and the most often used approach nowadays consists of determining the velocity of ultrasonic waves propagating along different directions of the crystal and calculating the elastic constants from the Christoffel determinants (1.3.4.8). The experimental values are often accurate enough to justify the distinction between static and dynamic values of the elastic constants and between phase and group velocities, and the careful consideration of the frequency range of the experiments.
The use of the resonance technique is a well established approach for determining the velocity of sound in a gas by observing nodes and antinodes of a system of standing waves produced in the socalled Kund tube. In the case of transparent solids, optical means allow us to visualize the standing waves and to measure the wavelength directly (Zarembowitch, 1965). An easier procedure can be used: let us consider a transparent crystal in the shape of a parallelepiped (Fig. 1.3.4.1). A piezoelectric transducer is glued to the crystal and excited at varying frequencies. If the bonding between the transducer and the crystal is loose enough, the crystal can be considered as free from stress and the sequence of its resonance frequencies is given bywhere n is an integer, V the phase velocity of the wave in the direction orthogonal to the parallel faces and l the distance between these faces.
The looseness of the bonding can be checked by the regularity of the arithmetic ratio, . On account of the elastooptic coupling, a phase grating is associated with the elastic standingwave system and a light beam can be diffracted by this grating. The intensity of the diffraction pattern is maximum when resonance occurs. A large number of resonance frequencies can be detected, usually more than 100, sometimes 1000 for nonattenuating materials. Consequently, in favourable cases the absolute value of the ultrasonic velocity can be determined with an uncertainty less than .
Pulseecho techniques are valid for transparent and opaque materials. They are currently used for measuring ultrasonic velocities in solids and can be used in very simple as well as in sophisticated versions according to the required precision (McSkimmin, 1964). In the simplest version (Fig. 1.3.4.2), an electronic pulse generator excites the mechanical vibrations of a piezoelectric transducer glued to one of two planeparallel faces of a specimen. An ultrasonic pulse whose duration is of the order of a microsecond is generated and transmitted through the specimen. After reflection at the opposite face, it returns and, when it arrives back at the transducer, it gives rise to an electronic signal, or echo. The whole sequence of such echos is displayed on the screen of an oscilloscope and it is possible to measure from them the time interval for transit. Usually, Xcut quartz crystals or ferroelectric ceramics are used to excite longitudinal waves and Ycut quartz is used to excite transverse waves. In many cases, a circulator, or gate, is used to protect the receiver from saturation following the main `bang'. This method is rough because the beginning and the end of a pulse are not well characterized. Several improvements have therefore been made, mainly based on interferometric techniques (pulsesuperposition method, `sing around' method etc.). Nevertheless, if the absolute value of the ultrasonic velocity is not determined with a high accuracy by using pulseecho techniques, this approach has proved valuable when relative values of ultrasonic velocities are needed, e.g. temperature and pressure dependences of ultrasonic velocities.
References
Brillouin, L. (1932). Propagation des ondes électromagnétiques dans les milieux matériels. Congrès International d'Électricité, Vol. 2, Section 1, pp. 739–788. Paris: GauthierVillars.McSkimmin, H. J. (1964). Ultrasonic methods for measuring the mechanical properties of solids and fluids. Physical Acoustics, Vol. IA, edited by W. P. Mason, pp. 271–334. New York: Academic Press.
Michard, F., Zarembowitch, A., Vacher, R. & Boyer, L. (1971). Premier son et son zéro dans les nitrates de strontium, barium et plomb. Phonons, edited by M. A. Nusimovici, pp. 321–325. Paris: Flammarion.
Zarembowitch, A. (1965). Etude théorique et détermination optique des constantes élastiques de monocristaux. Bull. Soc. Fr. Minéral. Cristallogr. 28, 17–49.