Analysis of Damped Bloch Waves by the Rayleigh Perturbation Method
Bibliographic record
Abstract
Bloch waves in viscously damped periodic material and structural systems are analyzed using a perturbation method originally developed by Rayleigh for vibration analysis of finite structures. The extended method, called the Bloch–Rayleigh perturbation method here, utilizes the Bloch waves of an undamped unit cell as basis functions to provide approximate closed-form expressions for the complex eigenvalues and eigenvectors of the damped unit cell. In doing so, we circumvent the solution of a quadratic Bloch eigenvalue problem and subsequent computationally intensive transformation to first order/state-space form. Dispersion curves of a one-dimensional damped spring-mass chain and a two-dimensional phononic crystal with square inclusions are calculated using the state-space method and the proposed method. They are compared and found to be in excellent quantitative agreement for both proportional and nonproportional viscous damping models. The perturbation method is able to capture anomalous dispersion phenomena—branch overtaking, branch cut-on/cut-off, and frequency contour transformation—in parametric ranges where state-space formulations encounter numerical issues. Generalization to other linear nonviscous damping models is permissible.
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How this classification was reachedexpand
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".