This paper reports the eigenstructure of a set of first-order hyperbolic partial differential equations for modeling waves in solids with a trigonal 32 symmetry. The governing equations include the equation of motion and partial differentiation of the elastic constitutive relation with respect to time. The result is a set of nine, first-order, fully coupled, hyperbolic partial differential equations with velocity and stress components as the unknowns. Shown in the vector form, the model equations have three $9×9$ coefficient matrices. The wave physics are fully described by the eigenvalues and eigenvectors of these matrices; i.e., the nontrivial eigenvalues are the wave speeds, and a part of the corresponding left eigenvectors represents wave polarization. For a wave moving in a certain direction, three wave speeds can be identified by calculating the eigenvalues of the coefficient matrix in a rotated coordinate system. In this process, without using the plane-wave solution, we recover the Christoffel matrix and thus validate the formulation. To demonstrate this approach, two- and three-dimensional slowness profiles of quartz are calculated. Wave polarization vectors for wave propagation in several compression directions as well as noncompression directions are discussed.

1.
,
W. G.
, 1946,
Piezoelectricity: An Introduction to the Theory and Application of Electromechanical Phenomena in Crystals
,
McGraw-Hill
,
New York
.
2.
Mason
,
W. P.
, 1966,
Crystal Physics of Interaction Processes
,
,
New York
.
3.
Fedorov
,
F. I.
, 1968,
Theory of Elastic Waves in Crystals
, translated by J. E. S. Bradley,
Plenum
,
New York
.
4.
Tiersten
,
H. F.
, 1969,
Linear Piezoelectric Plate Vibrations
,
Plenum
,
Troy, NY
.
5.
Musgrave
,
M. J. P.
, 1970,
Crystal Acoustics: Introduction to the Study of Elastic Waves and Vibrations in Crystals
,
Holden-Day
,
San Francisco
6.
Auld
,
B. A.
, 1990,
Acoustic Fields and Waves in Solids
, 2nd ed.,
Krieger
,
Malabar, FL
, Vols.
1
and 2.
7.
Ting
,
T. T.-C.
, 1996,
Anisotropic Elasticity: Theory and Applications
,
Oxford University Press
,
New York
.
8.
Royer
,
D.
, and
Dieulesaint
,
E.
, 2000,
Elastic Waves in Solids I: Free and Guided Propagation
, translated by D. P. Morgan,
Springer-Verlag
,
New York
.
9.
Kyame
,
J. J.
, 1949, “
Wave Propagation in Piezoelectric Crystals
,”
J. Acoust. Soc. Am.
0001-4966,
21
, pp.
159
167
.
10.
Musgrave
,
M. J. P.
, 1954, “
On the Propagation of Elastic Waves in Anisotropic Media. I. Media of Cubic Symmetry
,”
Proc. R. Soc. London, Ser. A
0950-1207,
226
, pp.
339
355
.
11.
Musgrave
,
M. J. P.
, 1954, “
On the Propagation of Elastic Waves in Anisotropic Media. II. Media of Hexagonal Symmetry
,”
Proc. R. Soc. London, Ser. A
0950-1207,
226
, pp.
356
366
.
12.
Borgnis
,
F. E.
, 1955, “
Specific Directions of Longitudinal Wave Propagation in Anisotropic Media
,”
Phys. Rev.
0096-8250,
98
, pp.
1000
1005
.
13.
Buchwald
,
V. T.
, 1959, “
Elastic Waves in Anisotropic Media
,”
Proc. R. Soc. London, Ser. A
0950-1207,
253
, pp.
563
580
.
14.
Duff
,
G. F. D.
, 1960, “
The Cauchy Problem for Elastic Waves in an Anisotropic Medium
,”
Philos. Trans. R. Soc. London, Ser. A
0962-8428,
252
, pp.
249
273
.
15.
Hutson
,
A. R.
, and
White
,
D. L.
, 1962, “
Elastic Wave Propagation in Piezoelectric Semiconductors
,”
J. Appl. Phys.
0021-8979,
33
, pp.
40
47
.
16.
Brugger
,
K.
, 1965, “
Pure Modes for Elastic Waves in Crystals
,”
J. Appl. Phys.
0021-8979,
36
, pp.
759
768
.
17.
Truesdell
,
C.
, 1966, “
Existence of Longitudinal Waves
,”
J. Acoust. Soc. Am.
0001-4966,
40
(
3
), pp.
729
730
.
18.
Kolodner
,
I. I.
, 1966, “
Existence of Longitudinal Waves in Anisotropic Media
,”
J. Acoust. Soc. Am.
0001-4966,
40
(
3
), pp.
730
731
.
19.
Wilson
,
L. O.
, and
Morrison
,
J. A.
, 1977, “
Wave Propagation in Piezoelectric Rods of Hexagonal Crystal Symmetry
,”
Q. J. Mech. Appl. Math.
0033-5614,
30
, pp.
387
395
.
20.
Every
,
A. G.
, 1980, “
General Close-Form Expressions for Acoustic Waves in Elastically Anisotropic Solids
,”
Phys. Rev. B
0163-1829,
22
, pp.
1746
1760
.
21.
,
E. L.
, 1989, “
Analysis of Anisotropic Multilayer Bulk-Acoustic-Wave Transducers
,”
Electron. Lett.
0013-5194,
25
(
1
), pp.
57
59
.
22.
Wang
,
C. -Y.
, and
Achenbach
,
J. D.
, 1994, “
Elastodynamic Fundamental Solutions for Anisotropic Solids
,”
Geophys. J. Int.
0956-540X,
118
, pp.
384
392
.
23.
Zuo
,
Q. H.
, and
Schreyer
,
H. L.
, 1995, “
A Note on Pure-Longitudinal and Pure-Shear Waves in Cubic Crystals
,”
J. Acoust. Soc. Am.
0001-4966,
98
(
1
), pp.
580
583
.
24.
Wang
,
L.
, and
Rokhlin
,
S. I.
, 2001, “
Stable Reformulation of Transfer Matrix Method for Wave Propagation in Layered Anisotropic Media
,”
Ultrasonics
0041-624X,
39
, pp.
413
424
.
25.
Ostrosablin
,
N. I.
, 2003, “
Elastic Anisotropic Material With Purely Longitudinal and Transverse Waves
,”
J. Appl. Mech. Tech. Phys.
0021-8944,
44
, pp.
271
278
.
26.
Declercq
,
N. F.
,
Polikarpova
,
N. V.
,
Voloshinov
,
V. B.
,
Leroy
,
O.
, and
Degrieck
,
J.
, 2006, “
Enhanced Anisotropy in Paratellurite for Inhomogeneous Waves and Its Possible Importance in the Future Development of Acousto-Optic Devices
,”
Ultrasonics
0041-624X,
44
, pp.
e833
e837
.
27.
Ting
,
T. C. T.
, 2006, “
Longitudinal and Transverse Waves in Anisotropic Elastic Materials
,”
Acta Mech.
0001-5970,
185
, pp.
147
164
.
28.
Ting
,
T. C. T.
, 2006, “
On Anisotropic Elastic Materials for Which One Sheet of the Slowness Surface is a Sphere or a Cross-Section of a Slowness Sheet in a Circle
,”
Wave Motion
0165-2125,
43
, pp.
287
300
.
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