Elastodynamic Equations: Characteristics, Wavefronts and Rays
Bibliographic record
Abstract
We derive the characteristic equations, and the so-called Christoffel equation, for the vector elastodynamic equations in terms of both hypersurfaces of nonuniqueness and as wavefronts based on a physical definition. We follow Courant and Hilbert in defining a wavefront as a surface for which a solution may be zero on one side but nonzero on the other. We show that wavefronts defined in this way must be characteristics. Moreover, we give a partial converse showing that in the case of analytic coefficients, characteristics must (locally) be wavefronts. Furthermore, we show how the equations defining wavefronts, which are characteristics, can be obtained by letting frequency tend to infinity in a certain trial solution expressed in the frequency domain. The last approach might suggest that ray theory, which results from the Christoffel equation, is an asymptotic theory. Taking the limit, however, is just a technicality and is unnecessary to obtain the Christoffel equation, which is contained in the elastodynamic equations and their characteristic equations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".