Acoustic Velocity in Crystalline Materials

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The velocity of sound in anisotropic materials is directionally dependent. Three wave velocities may exist in any given direction: two transverse waves (primary oscillation perpendicular to the wave propagation direction) and one longitudinal wave (polarization in the same direction as the propagation). You can select the type of wave (transverse or longitudinal) and whether to view the velocity surface or the inverse velocity (slowness). The contribution can either be from elastic effects alone or from elastic and piezoelectric effects. You can choose several different materials.

Contributed by: Robert McIntosh (February 2013)
Open content licensed under CC BY-NC-SA



The velocity of sound in materials depends on the elastic constants and the density , and is expressed as . Velocity calculations can often be very simple, but because of the anisotropy of many materials the elastic constant is actually a 6×6 matrix of values. Solving for the velocity of the wave requires the Christoffel equation. This Demonstration also considers the contribution from the piezoelectric coupling. An excellent treatment of solving the Christoffel equation can be found in [4].

The elastic constants of , , , , , , , , are from [1].

The density of , , , , , , , , quartz, -quartz, , , and are from [1]. The density of and are from [2] and is from [3].

The piezoelectric constants of are from [6] and the constants of , , , quartz, , and are from [4]. The piezoelectric constants of are from [7], is from [8], and is from [3].

The permittivity of is from [9], while , , , quartz, , and are from [5]. The permittivity of is from [7], is from [8], and is from [3].


[1] M. J. Weber, Handbook of Optical Materials, Boca Raton, FL: CRC Press, 2003.

[2] Z. Rui, B. Jiang, W. Jiang, and W. Cao, "Anisotropy in Domain Engineered 0.92Pb()-0.08PbTi Single Crystal and Analysis of Its Property Fluctuations," Ultrasonics, Ferroelectrics and Frequency Control, 49(12), 2002, pp. 1622–1627. doi:10.1109/TUFFC.2002.1159841.

[3] R. Zhang, W. Jiang, B. Jiang, and W. Cao, "Elastic, Dielectric and Piezoelectric Coefficients Domain Engineered 0.70Pb()-0.30PbTi Single Crystal," in AIP Conference Proceedings, 626, 2002, pp. 188–197. doi:10.1063/1.1499567.

[4] R. E. Newnham, Properties of Materials, New York: Oxford University Press, 2005.

[5] COMSOL Multiphysics, Materials Database Version 4.3, Burlington, MA: COMSOL, Inc., 2012.

[6] L. Liang, Y. L. Li, S. Y. Hu, L-Q. Chen, and G-H. Lu, "Piezoelectric Anisotropy of a KNb Single Crystal," Journal of Applied Physics, 108(9), 2010 pp. 094111–094119. doi:10.1063/1.3511336.

[7] J. Yin, B. Jiang, and W. Cao, "Elastic, Piezoelectric, and Dielectric Properties of 0.955Pb()-0.45PbTi Single Crystal with Designed Multidomains," IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 47(1), 2000 pp. 285–291. doi:10.1109/58.818772.

[8] W. Jiang, R. Zhang, B. Jiang, and W. Cao, "Characterization of Piezoelectric Materials with Large Piezoelectric and Electromechanical Coupling Coefficients," Ultrasonics, 41(2), 2003 pp. 55–63. doi:10.1016/S0041-624X(02)00436-5.

[9] E. Wiesendanger, "Optical Properties of KNb," Ferroelectrics, 1(1), 1970, pp. 141–148. doi:10.1080/00150197008241478.

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