Mechanical energy and momentum of wave pulses in a dispersionless, lossless elastic medium, according to the linear theory
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Bibliographic record
Abstract
We re-examine the linear theory of wave propagation through an elastic string under uniform tension or a slender elastic rod from a perspective that focuses on the flow of mechanical energy and mechanical momentum. Continuity equations are established for the flow of energy and momentum, leading to two boundary conditions for the net wave displacement. The important special case of a small amplitude pulse of arbitrary shape traveling through a uniform slender medium joined to another medium with a different linear mass density is examined in detail. The new boundary conditions lead to the correct relative amplitudes for the reflected and transmitted pulses. We obtain the instantaneous mechanical energy and momentum of the incident, reflected, and transmitted pulses and show that the net mechanical energy and momentum are separate constants of motion. The cases of an incoming pulse described by a Lorentzian and a Gaussian distribution are suggested as problems to be solved by the interested reader.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it