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Reduced Cristoffel's tensor and acoustic axes: Necessary and sufficient condition
Classical Physics · Submitted 10 Oct 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the conditions for the existence of an acoustic axes in a given propagation direction are determined solely by the reduced acoustic tensor.
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The abstract, sentence by sentence
Acoustic axes are spatial directions in media (often crystals) where at least two of the three acoustic waves have the same phase velocity. Identification of such directions for materials with specific elasticity parameters is both theoretically fascinating and practical in acoustic applications. In this paper, we introduce the notion of the reduced acoustic tensor. In contrast to the ordinary Cristoffel's tensor, this tensor is traceless. Then it has only five independent components and fulfills the depressed cubic characteristic equation. We show that the conditions for the existence of an acoustic axes in a given propagation direction are determined solely by the reduced acoustic tensor. The necessary and sufficient conditions takes a simple compact invariant form. We also present the expression for the wave velocities along the acoustic axis. The conditions for the oblate/prolate types of the axis are derived in terms of reduced acoustic tensor.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Acoustic axes are spatial directions in media (often crystals) where at least two of the three acoustic waves have the same phase velocity. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s1 | Identification of such directions for materials with specific elasticity parameters is both theoretically fascinating and practical in acoustic applications. | background 0.99 | 0.05 | 0.02 | asserted 0.99 |
| s2 | In this paper, we introduce the notion of the reduced acoustic tensor. | contribution 0.99 | 0.95 | 0.06 | asserted 0.99 |
| s3 | In contrast to the ordinary Cristoffel's tensor, this tensor is traceless. | contribution 0.83 | 0.65 | 0.04 | asserted 1.00 |
| s4 | Then it has only five independent components and fulfills the depressed cubic characteristic equation. | contribution 0.52 | 0.54 | 0.07 | asserted 1.00 |
| s5 | We show that the conditions for the existence of an acoustic axes in a given propagation direction are determined solely by the reduced acoustic tensor. | result 0.88 | 0.94 | 0.69 | asserted 1.00 |
| s6 | The necessary and sufficient conditions takes a simple compact invariant form. | result 0.84 | 0.90 | 0.39 | asserted 1.00 |
| s7 | We also present the expression for the wave velocities along the acoustic axis. | contribution 0.56 | 0.94 | 0.07 | asserted 0.99 |
| s8 | The conditions for the oblate/prolate types of the axis are derived in terms of reduced acoustic tensor. | result 0.89 | 0.94 | 0.13 | asserted 1.00 |
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The paper
Original abstract
Acoustic axes are spatial directions in media (often crystals) where at least two of the three acoustic waves have the same phase velocity. Identification of such directions for materials with specific elasticity parameters is both theoretically fascinating and practical in acoustic applications. In this paper, we introduce the notion of the reduced acoustic tensor. In contrast to the ordinary Cristoffel's tensor, this tensor is traceless. Then it has only five independent components and fulfills the depressed cubic characteristic equation. We show that the conditions for the existence of an acoustic axes in a given propagation direction are determined solely by the reduced acoustic tensor. The necessary and sufficient conditions takes a simple compact invariant form. We also present the expression for the wave velocities along the acoustic axis. The conditions for the oblate/prolate types of the axis are derived in terms of reduced acoustic tensor.
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How the literature uses it
1 arXiv paper cite this in 5 sentences
Papers citing it · 2 sentences without a confident label
Uses its method or tool 1 paper
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Our objective in this paper is to study the theoretical aspects of the acoustic axes problem using the reduced form of the acoustic tensor, see [ref] .
Acoustic axes conditions revised · Introduction · 0.91
Cites it as background 1 paper
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The theoretical study of the acoustic axes may be addressed in numerous issues: * Formulation of the necessary and sufficient conditions for the existence of acoustic axis in a given direction; see Fedorov [ref] , Norris [ref] , Alshits & Lothe [ref] , [ref] , also [ref] .
Acoustic axes conditions revised · Introduction · 0.94
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