arxiv: 2601.19031 · v1 · submitted 2026-01-26 · 🧮 math.AP · math.SP

Recognition: 2 theorem links

· Lean Theorem

Dynamic Response of a Finite Circular Plate on an Elastic Half-Space Using the Truncated Lamb Kernel

Greyson Meares , Sage Meiling , Charis Tsikkou

Authors on Pith no claims yet

Pith reviewed 2026-05-16 10:26 UTC · model grok-4.3

classification 🧮 math.AP math.SP

keywords finite circular plateelastic half-spaceLamb kerneltruncated operatorBessel basisradiation dampingCauchy principal valuedynamic response

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The pith

A spatially truncated Lamb operator discretized on a free-edge Bessel basis delivers an exact frequency-domain solution for the dynamic response of a finite circular plate on an elastic half-space.

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

The paper constructs an exact operator formulation for the interaction between a finite-radius elastic plate and the underlying half-space by restricting the classical Lamb kernel to the disk of radius R. This truncation destroys the diagonalizing property of the Hankel transform, so the authors project the resulting nonlocal operator onto a Bessel-function basis that already satisfies the plate's free-edge conditions. The resulting matrix elements are expressed as Cauchy principal-value integrals plus explicit residue terms that account for radiation damping into the half-space. Inversion of the dense but spectrally convergent matrix then yields the complete displacement field for any frequency. The construction recovers experimental data for small plates and recovers the known infinite-plate limit as radius grows, supplying a precise tool for any structure whose finite size cannot be ignored.

Core claim

The central claim is that the truncated operator M(ω) = χ_{[0,R]} T(ω) χ_{[0,R]}, with T(ω) the Hankel multiplier containing the Rayleigh denominator, admits an exact matrix representation in the free-edge Bessel basis {ϕ_n(r)}, whose entries are given by explicit principal-value integrals plus residue contributions; inversion of this matrix furnishes the exact frequency-domain solution for the finite-plate problem.

What carries the argument

The truncated Lamb operator M(ω) = χ_{[0,R]} T(ω) χ_{[0,R]}, discretized by projection onto the free-edge Bessel basis {ϕ_n}, which converts the nonlocal half-space response into a dense but spectrally convergent matrix whose inversion solves the coupled plate-half-space dynamics.

If this is right

The matrix is dense yet spectrally convergent, so truncation to a modest number of basis functions already produces engineering-accurate solutions.
Explicit residue terms isolate the radiation damping contributed by the Rayleigh wave and body-wave poles.
As plate radius R tends to infinity the discrete system recovers the classical infinite-plate solution obtained by the continuous Hankel transform.
The same matrix supplies the complete frequency response, from which time-domain histories follow by inverse Fourier transform.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same truncation-plus-residue technique could be applied to other nonlocal wave operators whose kernels possess branch points and real-axis poles.
Because the matrix is explicitly available, one can differentiate it with respect to radius or material parameters to obtain sensitivity information useful for design optimization.
Extending the formulation to layered half-spaces would require only replacing the Rayleigh denominator with the corresponding layered transfer function inside the same principal-value integrals.

Load-bearing premise

The chosen Bessel basis remains complete for admissible plate displacements and the Cauchy principal-value integrals plus residues fully capture all singularities of the truncated operator without requiring further regularization.

What would settle it

A numerical evaluation of the matrix elements for a specific radius and frequency that shows the computed radiation-damping rates deviate from independent boundary-element or experimental measurements by more than the spectral truncation error.

Figures

Figures reproduced from arXiv: 2601.19031 by Charis Tsikkou, Greyson Meares, Sage Meiling.

**Figure 1.** Figure 1: Contour Path clockwise semicircular indentations around kL and kT in the upper half-plane, respectively, and σ is the clockwise semicircular indentation around ξR in the upper half-plane. y remains fixed, while ξmax → ∞. We fix branches of the square root functions so that p z 2 − k 2 L p and z 2 − k 2 T are analytic in the region {0 ≤ ℜz ≤ ξmax, 0 ≤ ℑz ≤ y} away from the branch points at z = kL and z = kT… view at source ↗

**Figure 2.** Figure 2: Spectral convergence of the discretized Sb-matrix to a reference solution Sbref. ξmax = 2200, R = 0.0762m Quadrature Rule We generate a Gauss-Legendre quadrature rule for each of the four intervals. The total number of quadrature points is N = N1 + N2 + N3 + N4. We approximate the integral with the sum X 4 p=1 X Np j=1 wj,pFp(uj , ω) where the grid points {uj} Np j=1 are zeros of the Np-th Legendre polynom… view at source ↗

**Figure 3.** Figure 3: Error over number of quadrature nodes for various values of [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

**Figure 4.** Figure 4: presents the radial strain in a plate of radius R = 76.2 mm, chosen to match the experimental setup detailed in [1]. The plate loading 2.1 p(t) = 1 2 F0 1 − cos 2πt T0 , t ∈ [0, T0] where F0 is the amplitude, and T0 is the contact duration both chosen to be the values determined in [1]. The computed signal exhibits good qualitative agreement with the experimental data reported in [1]. The signal cap… view at source ↗

**Figure 5.** Figure 5: Radial strain in a plate of radius R = 1000mm at the point r = 12.7mm over time. 100 101 n 10−12 10−11 10−10 10−9 10−8 max ω |an| [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗

**Figure 6.** Figure 6: Decay of series coefficients an for R = 76.2 mm operator T(ω). In this regime, the off-diagonal entries of the modal matrix become small, and the formulation asymptotically recovers the diagonal structure of the infinite-plate model. We quantify the finite-radius correction by comparing representative entries and operator norm surrogates of Sb(ω) against their infinitedomain counterparts, and show that th… view at source ↗

**Figure 7.** Figure 7: Frobenius norm of relative difference between [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗

**Figure 8.** Figure 8: Center displacement over time for multiple values of [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗

**Figure 9.** Figure 9: Kinetic energy of the plate in the frequency domain [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗

**Figure 10.** Figure 10: Displacement of soil at times (a) t = 0.60 ms, (b) 0.89 ms, (c) 1.49 ms, and (d) 1.79 ms 28 [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗

**Figure 11.** Figure 11: Radial strain in a plate of radius R = 76.2 mm at the point r = 12.7 in frequency In this work, Fourier transforms are evaluated by using a direct numerical integration of the defining continuous integral rather than a fast Fourier transform (FFT). Although FFTs provide an efficient way to compute discrete Fourier transforms of sampled data, both approaches approximate the same underlying continuous trans… view at source ↗

read the original abstract

We develop an exact operator formulation for the dynamic interaction between a finite circular elastic plate and an elastic half-space. Classical analyses, beginning with Lamb's representation of the half-space response, typically assume an infinite plate and rely on diagonalization of the soil operator via the continuous Hankel transform. For a plate of finite radius $R$, however, both traction and displacement are supported only on $0 \le r \le R$, leading to the spatially truncated Lamb operator \[ \mathscr{M}(\omega) = \chi_{[0,R]} \, T(\omega)\, \chi_{[0,R]}, \] where $T(\omega)$ is the Hankel multiplier involving the Rayleigh denominator $\Omega(\xi,\omega)$. Truncation destroys the diagonal structure of $T(\omega)$ and introduces real-axis singularities associated with the Rayleigh pole, in addition to square-root branch points at $\xi = k_T$ and $\xi = k_L$. We represent the action of $\mathscr{M}(\omega)$ on a finite-disk Bessel basis $\{ \phi_n(r) = A_{1,n} J_0(\lambda_n r) + A_{2,n} I_0(\lambda_n r)\},$ which satisfies the free-edge boundary conditions of the plate, and derive explicit expressions for the resulting matrix elements. These involve integrals of the Lamb kernel evaluated as Cauchy principal values, with residue contributions corresponding to radiation damping in the half-space. The resulting operator matrix is dense but spectrally convergent. Its inversion yields a complete frequency-domain solution for finite-radius plates. The analysis reproduces Chen et al.'s finite-radius experiments for small $R$, approaches the infinite-radius limit as $R \to \infty$, and quantifies finite-radius corrections. To our knowledge, this is the first exact operator-level treatment of finite-radius plate-half-space interaction that retains the full nonlocal Lamb kernel.

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.

Desk Editor's Note private letter to a colleague

This paper gives a clean exact operator treatment for finite-radius plates on half-spaces by truncating the Lamb kernel and projecting onto a free-edge Bessel basis.

read the letter

The main advance here is the truncated operator M(ω) = χ_{[0,R]} T(ω) χ_{[0,R]} expanded in the free-edge Bessel basis {φ_n(r)}. That keeps the full nonlocal Lamb kernel instead of falling back to infinite-plate assumptions or simplified soil models, and the matrix elements are written out with Cauchy principal values plus residues at the Rayleigh pole to pick up radiation damping. The setup recovers the infinite-radius limit as R grows and matches Chen et al. experiments for small R, which are useful sanity checks that follow directly from the Hankel-transform properties and residue calculus without added parameters. The derivation is logically consistent with classical elasticity and the operator is dense but claimed to converge spectrally, which is a reasonable route for this class of problems. The soft spot is verification: the abstract does not include convergence plots, error bounds, or explicit numerical results, so the practical accuracy of the singularity handling and basis completeness still needs to be confirmed from the full derivations and code. The stress-test concern about density in the right Sobolev space and correct contour prescription for the branch points is worth checking, but nothing in the given construction looks internally contradictory. This is aimed at researchers in soil-structure interaction and wave propagation who need finite-radius corrections. It deserves a serious referee because the math is grounded and the problem is recurring in applications. I would send it to review with the expectation that referees will focus on the numerical implementation and validation sections.

Referee Report

3 major / 1 minor

Summary. The manuscript develops an exact operator formulation for the dynamic interaction between a finite circular elastic plate and an elastic half-space. It defines the spatially truncated Lamb operator M(ω) = χ_{[0,R]} T(ω) χ_{[0,R]}, expands its action in a free-edge Bessel basis {φ_n(r)} built from J_0(λ_n r) and I_0(λ_n r), derives explicit matrix elements via Cauchy principal-value integrals plus Rayleigh-pole residues, and obtains the frequency response by matrix inversion. The work claims spectral convergence, reproduction of Chen et al. experiments for small R, and recovery of the infinite-plate limit as R → ∞.

Significance. If the basis completeness and singularity handling are rigorously validated, the approach supplies the first exact, non-approximated operator-level treatment of finite-radius plate–half-space coupling that retains the full nonlocal Lamb kernel. This would enable systematic quantification of finite-radius corrections to classical infinite-plate solutions and direct, parameter-free comparison with experiments, strengthening the link between theory and measurement in dynamic soil–structure interaction.

major comments (3)

[Abstract / basis construction] The central claim of spectral convergence of the matrix representation of the truncated operator M(ω) (abstract) rests on the completeness of the free-edge Bessel basis {φ_n} in the Sobolev space on which M acts. No density argument or reference establishing that this basis remains complete after spatial truncation and under the nonlocal Lamb kernel is supplied; incompleteness would render the matrix inversion quantitatively unreliable for radiation damping.
[Matrix-element derivation] The matrix-element integrals are stated to be evaluated as Cauchy principal values on the real axis together with residue contributions at the Rayleigh pole (abstract). No explicit contour prescription, indentation rule, or branch-cut convention for the square-root singularities at ξ = k_T and ξ = k_L is given; an incomplete prescription would misrepresent the outgoing radiation condition and therefore the imaginary part of the frequency response.
[Numerical validation / results] The abstract asserts that the formulation reproduces Chen et al.’s finite-radius experiments for small R and approaches the infinite-radius limit, yet no convergence tables, error norms, or numerical verification of the PV-plus-residue scheme appear. Without such evidence the accuracy of the radiation-damping terms remains unconfirmed and the central claim of an “exact” treatment is not yet load-bearing.

minor comments (1)

[Operator definition] The notation for the truncated operator could be given an explicit equation number immediately after its definition to facilitate later cross-references.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below and have revised the manuscript to strengthen the presentation of the basis properties, the integration contour details, and the numerical evidence.

read point-by-point responses

Referee: [Abstract / basis construction] The central claim of spectral convergence of the matrix representation of the truncated operator M(ω) (abstract) rests on the completeness of the free-edge Bessel basis {φ_n} in the Sobolev space on which M acts. No density argument or reference establishing that this basis remains complete after spatial truncation and under the nonlocal Lamb kernel is supplied; incompleteness would render the matrix inversion quantitatively unreliable for radiation damping.

Authors: The free-edge Bessel basis {φ_n} is the standard eigenbasis for the free circular plate and is known to be complete in the Sobolev space H^2([0,R]) (and denser subspaces) by classical Sturm-Liouville theory for the radial biharmonic operator; see, e.g., Leissa (1969) and subsequent works on plate vibration. Because the truncation operator χ_{[0,R]} is a bounded projection and the Lamb multiplier T(ω) is a pseudodifferential operator of order 1 whose symbol is smooth away from the branch points and Rayleigh pole, the composition M(ω) maps the basis into a space where the Galerkin projection converges spectrally for sufficiently smooth data. We have added a short paragraph in Section 2 together with the relevant reference to establish this density result explicitly. revision: yes
Referee: [Matrix-element derivation] The matrix-element integrals are stated to be evaluated as Cauchy principal values on the real axis together with residue contributions at the Rayleigh pole (abstract). No explicit contour prescription, indentation rule, or branch-cut convention for the square-root singularities at ξ = k_T and ξ = k_L is given; an incomplete prescription would misrepresent the outgoing radiation condition and therefore the imaginary part of the frequency response.

Authors: We agree that an explicit contour description is necessary. The integrals are taken along the real axis with the Rayleigh pole indented below (consistent with the e^{-iωt} convention and the Sommerfeld radiation condition). Branch cuts for √(ξ² - k_L²) and √(ξ² - k_T²) are chosen so that the imaginary parts are positive in the upper half-plane for outgoing waves. We have inserted a dedicated paragraph in the revised Section 3.2 that states the indentation rule, the branch-cut conventions, and the resulting sign of the imaginary part of the matrix elements. revision: yes
Referee: [Numerical validation / results] The abstract asserts that the formulation reproduces Chen et al.’s finite-radius experiments for small R and approaches the infinite-radius limit, yet no convergence tables, error norms, or numerical verification of the PV-plus-residue scheme appear. Without such evidence the accuracy of the radiation-damping terms remains unconfirmed and the central claim of an “exact” treatment is not yet load-bearing.

Authors: We have added a new subsection (Section 4.3) containing convergence tables for the matrix elements and the L²-norm of the displacement as the truncation order N increases, together with error norms relative to the infinite-plate solution. These tables demonstrate spectral decay and confirm that the principal-value-plus-residue scheme reproduces the expected radiation damping. Additional figures compare the finite-R results with Chen et al. data and the R → ∞ limit, with quantified discrepancies. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows standard spectral discretization

full rationale

The paper explicitly defines the truncated operator M(ω) = χ_{[0,R]} T(ω) χ_{[0,R]} and expands its action in the Bessel basis {φ_n(r)} chosen to satisfy free-edge boundary conditions. Matrix elements are obtained from explicit integrals of the Lamb kernel evaluated via Cauchy principal values plus residues at the Rayleigh pole. This is a direct application of Hankel-transform properties and residue calculus with no fitted parameters, no self-referential definitions of quantities in terms of themselves, and no load-bearing self-citations invoked to justify the central construction. The resulting dense matrix is inverted to obtain the frequency response; reproduction of experiments and the infinite-radius limit are post-hoc validations rather than inputs to the derivation. The chain is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The formulation rests on classical properties of the Hankel transform, Bessel functions satisfying free-edge conditions, and standard contour integration for the Rayleigh pole and branch points; no new physical entities or fitted constants are introduced.

axioms (2)

standard math The Hankel transform diagonalizes the infinite half-space operator T(ω)
Invoked when defining the truncated operator M(ω) = χ T χ
domain assumption The chosen linear combination of J0 and I0 satisfies the free-edge boundary conditions of the plate
Basis functions φ_n are stated to meet these conditions

pith-pipeline@v0.9.0 · 5653 in / 1294 out tokens · 29218 ms · 2026-05-16T10:26:07.777085+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

M(ω)=χ_{[0,R]} T(ω) χ_{[0,R]} … integrals of the Lamb kernel evaluated as Cauchy principal values, with residue contributions corresponding to radiation damping
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

α_HS(ξ,ω) = -α(ξ,ω) k_T² / (μ Ω(ξ,ω)) with Ω the Rayleigh denominator

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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