Long Nonlinear Surface Waves in the Presence of a Variable Current
Bibliographic record
Abstract
We investigate the eigenvalue problem governing the propagation of long nonlinear surface waves when there is a current beneath the surface, y being the vertical coordinate. The amplitude of such waves evolves according to the KdV equation and it was proved by Burns [ 1 ] that their speed of propagation c is such that there is no critical layer (i.e., c lies outside the range of ). If, however, the critical layer is nonlinear, the result of Burns does not necessarily apply because the phase change of linear theory then vanishes. In this paper, we consider specific velocity profiles and determine c as a function of Froude number for modes with nonlinear critical layers. Such modes do not always exist, the case of the asymptotic suction profile being a notable example. We find, however, that singular modes can be obtained for boundary layer profiles of the Falkner–Skan similarity type, including the Blasius case. These and other examples are treated and we examine singular solutions of the Rayleigh equation to gain insight about the long wave limit of such solutions.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".