The book is derived from a series of lectures presented by the author in the fall of 1962 to graduate students and others specializing in crystal physics at Moscow University. During its preparation for publication, the material was significantly extended and supplemented. The theory of elastic waves in crystals is fundamental to ultrasonic technology, piezoelectronic devices, and various areas of solid-state physics that hold practical and theoretical significance. There has been a recent surge of interest and research in this field, with many papers published on the subject. While surveys have been published in other countries, they are typically review articles or single chapters in broader books on elasticity theory or solid mechanics. The author notes a lack of comprehensive, up-to-date books focused solely on the general theory of elastic waves in homogeneous crystalline solids.

The book aims to address this gap. The author’s approach differs from other works on the subject, relying heavily on the general and systematic use of direct (coordinate-free) methods from vector and tensor calculus. These methods have demonstrated their efficacy in crystal optics and are particularly useful for studying the complex laws governing the propagation of elastic waves in crystals. Since these methods are still relatively unfamiliar to most physicists, the second chapter is devoted entirely to a detailed exposition of them.

Much of the material represents the author’s original work, with a strong emphasis on general methods for solving various classes of problems. Specific features of particular crystals within different symmetry classes are given less space, serving primarily an illustrative purpose.

Translated from Russian by J. E. S. Bradley

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Contents

CHAPTER 1. General Equations of the Theory of Elasticity

1. Deformation tensor
2. Stress tensor
3. Equilibrium conditions and the equation of an elastic medium
4. Hooke’s law
5. Energy of a deformed elastic body
6. Tensor for the elastic moduli
7. Crystal symmetry
8. Elastic moduli of crystals of the lower systems
9. Elastic moduli of crystals of the higher systems

CHAPTER 2. Elements of Linear Algebra and Direct Tensor Calculus

10. Vectors and matrices in n-dimensional space
11. Three-dimensional tensors and dyads
12. The Levi-Civita tensor and its applications
13. Eigenvalues and eigenvectors of a second-rank tensor
14. Tensor relations in a plane

CHAPTER 3. General Laws of Propagation of Elastic Waves in Crystals

15. Plane waves and Christoffel’s equation
16. General properties of the A tensor and forms of plane elastic waves in crystals
17. Special directions for elastic waves in crystals
18. Longitudinal normals and acoustic axes
19. Form of the A tensor for various crystal systems
20. Convoluted tensor for the elastic moduli

CHAPTER 4. Energy Flux and Wave Surfaces

21. The energy-flux vector and the ray velocity
22. Energy vector with acoustic axes
23. Elliptical polarization in elastic waves and the instantaneous energy-flux vector
24. Wave surfaces
25. Sections of the wave surfaces by symmetry planes

CHAPTER 5. General Theory of Elastic Waves in Crystals Based on Comparison with an Isotropic Medium

26. Mean elastic anisotropy of a crystal
27. Comparison with an isotropic medium
28. Special directions
29. Approximate theory of quasilongitudinal waves
30. Another form of the approximate theory

CHAPTER 6. Elastic Waves in Transversely Isotropic Media

31. Covariant form of the A tensor
32. Phase velocities and displacements
33. Comparison of a hexagonal crystal with an isotropic medium
34. Mean transverse anisotropy
35. Comparison with a transversely isotropic medium

CHAPTER 7. Elastic Waves in Crystals of the Higher Systems

36. Cubic crystals
37. Approximate theory for cubic crystals
38. Tetragonal crystals
39. Comparison with a hexagonal crystal
40. Trigonal crystals

CHAPTER 8. Reflection and Refraction of Elastic Waves

41. Boundary conditions for plane elastic waves
42. Reflection of elastic waves at the free boundary of an isotropic medium
43. Reflection at the free boundary of a crystal
44. The complex refraction vector and inhomogeneous plane waves
45. Invariant characteristics of the polarization of plane waves
46. Inhomogeneous waves at a free boundary

CHAPTER 9. Elastic Waves and the Thermal Capacity of a Crystal

47. Statistical theory of the thermal capacity of a solid
48. Computation of the Debye temperature
49. Averaging of the products of components of unit vector
50. Debye temperatures of cubic crystals
51. Debye temperatures of hexagonal crystals

LITERATURE CITED

INDEX