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arxiv: 2604.23253 · v1 · submitted 2026-04-25 · 🧮 math-ph · math.MP

Matched asymptotics of Rayleigh-wave fields near cuspidal ridges and gorges

Oleg Kiselev This is my paper

Pith reviewed 2026-05-08 07:05 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords cuspidal ridgecuspidal gorgeRayleigh wavesmatched asymptoticselastic hornLamé equationsstress singularitytraction-free surface
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The pith

Cuspidal ridges behave like vanishing-width horns and support bounded-stress Rayleigh-wave fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a matched-asymptotic description of time-harmonic elastic fields produced by Rayleigh waves near cusps in a traction-free surface. For a cuspidal gorge, modeled as a zero-opening re-entrant notch, the leading field follows the Williams crack-tip singularity with stresses varying as r to the power -1/2. In contrast, a cuspidal ridge is treated as an elastic horn of vanishing width, where the leading free-tip field remains asymptotically rigid with bounded stresses. This differs from cases with a finite tip truncation, which admit a high-energy solution branch featuring stresses growing as rho to the power -m. Finite-element computations for the associated static Lamé problems confirm these asymptotic predictions for both ridges and gorges.

Core claim

The central claim is that near a cuspidal ridge the elastic field generated by an incident Rayleigh wave can be described by matching to a local solution in a vanishing-width horn geometry, yielding an asymptotically rigid leading term with bounded stress at the free tip; this is distinct from the singular high-energy field that appears when the cusp is replaced by a truncated tip with finite opening.

What carries the argument

Matched asymptotics linking the far-field Rayleigh-wave solution to local solutions of the Lamé equations in cusp or horn geometries defined by a graph exponent 0 < α < 1.

Load-bearing premise

The traction-free surface can be locally approximated by a cusp-shaped graph whose width vanishes as a power law with exponent greater than one, allowing standard matched asymptotics to connect the Rayleigh far field to the near-tip Lamé solution.

What would settle it

A direct numerical solution or measurement of the stress field at the tip of a free cuspidal ridge showing divergence like ρ to the power -m rather than remaining bounded.

Figures

Figures reproduced from arXiv: 2604.23253 by Oleg Kiselev.

Figure 1
Figure 1. Figure 1: Local cusp geometries. Left: a cuspidal ridge is a vanishing-width elastic horn. view at source ↗
Figure 2
Figure 2. Figure 2: Model local laws. Left: in an elastic horn, a non-zero tip-load branch requires a view at source ↗
Figure 3
Figure 3. Figure 3: Matching structure. The outer wave region is governed by the Rayleigh-scale view at source ↗
Figure 4
Figure 4. Figure 4: Expected rounding law. A cuspidal gorge inherits the crack cut-off view at source ↗
Figure 5
Figure 5. Figure 5: Numerical illustration of the first-correction matching. Left: the curvature view at source ↗
Figure 6
Figure 6. Figure 6: Reduced numerical checks of first corrections. Top: the regular free-tip horn view at source ↗
Figure 7
Figure 7. Figure 7: Local FEM stress fields for the cuspidal ridge and gorge. The material region is view at source ↗
Figure 8
Figure 8. Figure 8: Numerical scaling check. The non-destructive free-tip ridge test does not show view at source ↗
Figure 9
Figure 9. Figure 9: Numerical scaling check for the general high-energy ridge branch. A fixed non view at source ↗
read the original abstract

We construct a local matched-asymptotic description of time-harmonic elastic fields generated by Rayleigh waves near cuspidal elements of a traction-free surface. The free surface is represented locally by a cusp graph with exponent $0<\alpha<1$, or equivalently by a vanishing-width horn $b(s)=B s^m$, $m=1/\alpha>1$. A cuspidal gorge is a zero-opening re-entrant notch: its leading field is the Williams crack-tip field, and the stresses behave as $r^{-1/2}$. The cusp exponent affects the gorge through lower-order corrections and through the stress cut-off produced by rounding the bottom. In contrast, a cuspidal ridge behaves as an elastic horn with vanishing width. The leading admissible free-tip field is asymptotically rigid (bounded stress), distinct from the high-energy branch supported by a finite tip truncation, where stresses grow as $\sigma \sim \rho^{-m}$. Finite-element calculations for the local static Lam\'e problems support these predictions: the free-tip ridge test confirms the absence of crack-like growth, the truncated ridge recovers the high-energy law, and the gorge stress slope is found to be close to $-1/2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper develops a matched-asymptotic description of time-harmonic Rayleigh-wave fields near cuspidal ridges and gorges on a traction-free elastic surface. The local geometry is represented as a cusp graph with exponent 0<α<1 (equivalently a vanishing-width horn b(s)=B s^m, m>1). For gorges the leading admissible field is the Williams crack-tip singularity with stresses behaving as r^{-1/2}; for free-tip ridges the leading field is asymptotically rigid (bounded stress), while a finite tip truncation supports a high-energy branch with stresses growing as σ∼ρ^{-m}. The constructions are illustrated by explicit local Lamé solutions and are stated to be supported by finite-element computations of the static inner problems.

Significance. If the central constructions hold, the work supplies explicit local fields that clarify how geometric singularities of different types (ridge versus gorge, free versus truncated) control stress concentration and energy distribution for surface waves. This distinction is useful for asymptotic modeling in elastodynamics and could inform numerical schemes or applications in seismology and materials with complex surface topography. The combination of matched asymptotics with numerical checks of the inner static problems is a positive feature.

minor comments (2)
  1. [Abstract and numerical validation] The abstract states that finite-element calculations support the predictions (free-tip ridge shows no crack-like growth, truncated ridge recovers the high-energy law, gorge stress slope is close to -1/2), yet no quantitative error measures, mesh-refinement studies, convergence rates, or comparison plots are supplied. Adding these details would allow readers to assess the numerical confirmation of the asymptotic claims.
  2. A short paragraph clarifying the range of validity of the scale-separation assumption (cusp size ≪ wavelength) for the time-harmonic case would help readers understand the limits of the static-inner-problem approximation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on matched asymptotics for Rayleigh-wave fields near cuspidal ridges and gorges. The recommendation for minor revision is noted, and we will incorporate any editorial or minor clarifications in the revised version. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds by standard matched asymptotics: the surface is represented locally as a cusp graph (0<α<1) or equivalent horn b(s)=B s^m (m>1), the outer Rayleigh field is matched to inner solutions of the static Lamé system on the respective domains, and admissible free-tip fields are identified by bounded-stress conditions. These steps rely on the scale separation (cusp size ≪ wavelength) and explicit local constructions rather than any fitted parameters or self-referential definitions. Finite-element solutions of the inner static problems are presented as independent numerical checks that confirm the analytic stress exponents and boundedness claims; they do not supply the exponents themselves. No load-bearing self-citations, uniqueness theorems imported from prior work by the same author, or ansätze smuggled via citation appear in the argument chain. The central distinction between the asymptotically rigid free-tip ridge field and the high-energy truncated branch is obtained directly from the boundary-value problems and is not equivalent to the input data by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard linear isotropic elasticity and traction-free boundary conditions; the cusp/horn geometry is introduced as a modeling choice rather than derived.

free parameters (1)
  • cusp exponent α (or m=1/α)
    Shape parameter defining local surface geometry; treated as given input in (0,1) range.
axioms (2)
  • standard math Linear isotropic elasticity governed by Lamé equations
    Invoked for the time-harmonic elastic fields throughout the matched regions.
  • domain assumption Traction-free boundary conditions on the cuspidal surface
    Stated as the physical setup for the free surface.

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