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REVIEW 2 major objections 3 minor 7 references

Tunamis on a deep open sea and on a sloping beach -- a mathematical theory

T0 review · 2 major / 3 minor · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read Shallow water Airy waves suddenly acquire infinite forward speed at crests and infinite backward speed at troughs where surface and bottom slopes coincide on a beach.

desk verdict The infinite-speed claim is an extrapolation of the Airy formula past the point where its own assumptions collapse. read the letter →

arxiv 2409.17269 v7 submitted 2024-09-01 math.AP physics.ao-ph

Tadayoshi Kano This is my paper
classification math.APphysics.ao-ph
keywords tunamisAirywavesshallowwaterslopingbeachpropagationspeedinfinitemathematicaltheorysurface
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that Airy shallow water waves maintain finite cruising speed on deep open sea but lose it near a sloping beach. At the point where the surface tangent matches the bottom tangent, the propagation speed jumps to +∞ just before a crest and -∞ just after a trough. This produces a mathematical crush between the forward-rushing crest and the backward-rushing trough. A sympathetic reader would care because the model supplies a precise mechanism for the sudden amplification and destructive arrival of tunamis on beaches.

What carries the argument

The difference (Γx - bx) between surface and bottom tangents, which sets propagation speed in the Airy shallow-water approximation and drives it to infinity as the difference approaches zero.

What would settle it

A numerical simulation or laboratory measurement of an Airy-type wave on a slope that keeps finite speed when the surface and bottom tangents coincide.

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Extended reading notes

Core claim

Approaching a sloping beach, shallow water surface waves of Airy get suddenly +∞ or -∞ propagation speed at the point of surface x = x0 where the tangent Γx of the surface y = Γ coincide with that bx of the water-bottom y = b(x), losing the cruising sound speed of propagation so high on a deep open sea. That is, the tunamis gain instantaneously a +∞ propagation speed just before the crest as (Γx - bx)(x) → +0, x → x0−0, and a -∞ propagation speed just after the trough as (Γx - bx)(x) → -0, x → x0−0. We would have thus a big crush between the crest rushing forward and the trough rushing backward.

Load-bearing premise

Propagation speed is fixed by the Airy approximation as a direct function of the tangent difference (Γx - bx) approaching zero.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The manuscript presents a mathematical theory for Airy shallow-water waves (termed 'tunamis') on a deep open sea and a sloping beach. It asserts that as the surface tangent Γ_x approaches the bottom tangent b_x at a point x0 on the beach, the propagation speed diverges instantaneously to +∞ just before the crest and to -∞ just after the trough (as (Γ_x - b_x) → ±0), producing a 'big crush' between crest and trough, in contrast to finite 'cruising sound speed' on deep water.

Significance. If the central divergence result could be shown to hold rigorously inside a well-defined model, it would constitute a striking formal prediction about limiting wave behavior on beaches. The manuscript contains no machine-checked proofs, reproducible code, or parameter-free derivations, and the result rests entirely on an unverified extrapolation of the Airy speed formula.

major comments (2)
  1. [Abstract] Abstract: the claim that propagation speed diverges to ±∞ as (Γ_x - b_x)(x) → 0 is presented as a direct consequence of the Airy model, yet no derivation steps, dispersion relation, or characteristic-speed calculation is supplied to show how the factor 1/(Γ_x - b_x) arises or remains valid in the limit.
  2. [Abstract] Abstract: the Airy linearised shallow-water approximation is invoked, but the stated limit violates its explicit hypotheses of small surface slopes and scale separation between local depth and wavelength; the divergence is therefore an extrapolation outside the regime in which the speed formula was derived, rendering the 'big crush' conclusion unsupported.
minor comments (3)
  1. The spelling 'tunamis' is non-standard and should be replaced by the conventional term 'tsunamis' throughout.
  2. Notation Γ_x and b_x for derivatives is acceptable but should be defined explicitly on first use; the variant form varGamma is unnecessary.
  3. The phrase 'cruising sound speed of propagation' is unclear; replace with a precise term such as 'phase speed' or 'characteristic speed' and cite the relevant dispersion relation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that propagation speed diverges to ±∞ as (Γ_x - b_x)(x) → 0 is presented as a direct consequence of the Airy model, yet no derivation steps, dispersion relation, or characteristic-speed calculation is supplied to show how the factor 1/(Γ_x - b_x) arises or remains valid in the limit.

    Authors: The factor 1/(Γ_x - b_x) follows from the characteristic speeds of the linearized Airy shallow-water system on a variable bottom when the surface and bottom tangents coincide. We agree that the abstract omits the intermediate steps and will add a concise derivation outline, including the relevant dispersion relation and limiting process, in the revised version. revision: yes

  2. Referee: [Abstract] Abstract: the Airy linearised shallow-water approximation is invoked, but the stated limit violates its explicit hypotheses of small surface slopes and scale separation between local depth and wavelength; the divergence is therefore an extrapolation outside the regime in which the speed formula was derived, rendering the 'big crush' conclusion unsupported.

    Authors: The manuscript presents the divergence as a formal mathematical consequence obtained by applying the Airy characteristic-speed formula in the indicated limit on a sloping beach. While this limit lies at the edge of the classical regime, the paper's purpose is precisely to examine the model's behavior under that extrapolation. We therefore maintain that the 'big crush' statement is supported within the stated mathematical theory and do not intend to retract the claim. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: infinite-speed claim is direct substitution into standard Airy formula, not a self-referential reduction.

full rationale

The paper applies the known Airy shallow-water propagation speed to the geometric condition (Γ_x - b_x) → 0 and obtains divergence by algebra. No equations are shown to be fitted to the target result, no self-citation chain is invoked to justify the speed formula itself, and the derivation does not rename or smuggle an ansatz that presupposes the claimed infinity. The noted breakdown of Airy assumptions in the limit is a question of domain of validity rather than a circularity in the formal steps.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided text.

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Cite this review

Pith. "Pith review of Tunamis on a deep open sea and on a sloping beach -- a mathematical theory." pith.science (2026). https://pith.science/paper/2409.17269

@misc{pith2026240917269,
  author       = {Pith},
  title        = {Pith review of: Tunamis on a deep open sea and on a sloping beach -- a mathematical theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2409.17269}},
  note         = {Machine review of arXiv:2409.17269}
}
abstract

Approaching a sloping beach, shallow water surface waves of Airy get suddenly $ +\infty$ or $ -\infty$ propagation speed at the point of surface $x = x_0$, say, where the tangent $\varGamma_x$ of the surface $y = \varGamma$ "coincide" with that $b_x$ of the water-bottom $y = b(x)$, losing the cruising sound speed of propagation so high on a deep open sea. That is, the tunamis gain instantaneously a $ +\infty$ propagation speed just before the crest as $(\varGamma_x - b_x)(x) \to +0$, $x \to x_0\!-\!0$ , and a $ -\infty$ propagation speed just after the trough as $(\varGamma_x - b_x)(x) \to -0$, $x \to x_0\!-\!0$. We would have thus a big crush between the crest rushing forward and the trough rushing backward. This is a mathematical structure of tunamis "on" a sloping beach, in particular.

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Lean theorems connected to this paper

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

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

Works this paper leans on

7 extracted references · 7 canonical work pages

  1. [1]

    The formation of breakers and bores

    K. - O. Friedrichs: On the derivation of the shallow water theory, Appendix to: “The formation of breakers and bores” by J.J. Stoker in CPAM 1, 1 – 87 (1948)

    work page 1948

  2. [2]

    Nishida: Sur les ondes de surface de l’eau avec une justification math´ ematique des ´ equations des ondes en eau peu profonde, J

    T.Kano -T. Nishida: Sur les ondes de surface de l’eau avec une justification math´ ematique des ´ equations des ondes en eau peu profonde, J. Math.Kyoto Univ., 19, 335 – 370 (1979)

    work page 1979

  3. [3]

    Nishida: Water waves and Friedrichs expansion, Lect

    T.Kano -T. Nishida: Water waves and Friedrichs expansion, Lect. Note. Num.Appl. Anal., Kinokuniya-North Holland, 6, 39-57 (1983)

    work page 1983

  4. [4]

    Annual Report 2017, ESI

    T.Kano, Tunamis on a deep open sea and on a gentle sloping beach, in “Annual Report 2017, ESI”, Vienna (2017)

    work page 2017

  5. [5]

    D.Lannes, The water waves problem: mathematical analysis and asymptotics, Mathematical surveys and monograph, AMS, 2013

    work page 2013

  6. [6]

    L.V.Ovsjannikov: A nonlinear Cauchy problem in a scale of Banach spaces (Russian), Dokl. Akad. Nauk URSS, 200, 789-792 (1971)

    work page 1971

  7. [7]

    Note in Math., Springer-Verlag, 503, 426-437 (1976)

    L.V.Ovsjannikov: Cauchy problem in a scale of Banach spaces and its application to the shallow water theory justifi- cation, Lect. Note in Math., Springer-Verlag, 503, 426-437 (1976). Institut Vercors, Ushinomiyatyo 13-4, Sakyoku-Yoshida, 6068302 KYOTO, Japan Email address : institut vercors@cpost.plala.or.jp

    work page 1976

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Reviewed May 23, 2026 · model on record in the stance chip above.