arxiv: 2605.12254 · v1 · submitted 2026-05-12 · ⚛️ physics.flu-dyn

Recognition: no theorem link

Interfacial waves from pressure forcing: revisiting classical theories from an IVP perspective

Nikhil Yewale, Palas Kumar Farsoiya, Ratul Dasgupta, Vinod Kumar Kadari, Y. S. Mayya

Pith reviewed 2026-05-13 03:57 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords interfacial wavespressure forcinginitial value problemcapillary wavesgravity wavestwo-fluid interfaceFourier cancellation

0 comments

The pith

Formulating the problem as an initial-value problem reveals that algebraically decaying time-dependent solutions select unique wave patterns with short capillary waves ahead and long gravity waves behind a translating overpressure on a two-

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

The paper establishes that solving the moving-pressure problem on a two-fluid interface as an initial-value problem rather than a steady boundary-value problem yields a unique far-field wave pattern. Short capillary waves form ahead of the forcing and long gravity waves trail behind because the algebraically decaying transient terms produce an asymmetric cancellation among Fourier components. This mechanism replaces the auxiliary radiation conditions required by classical steady theories. A sympathetic reader would care because the result supplies a dynamical explanation for observed wave asymmetry in stratified fluids and is shown to survive comparison with nonlinear simulations.

Core claim

A localised overpressure translating uniformly above a critical speed across the interface between two deep fluid layers of different densities produces a steady-state interface containing short capillary waves ahead of the forcing and long gravity waves behind it. The pattern originates from asymmetric cancellation of Fourier components in the far field, with the algebraically decaying time-dependent part of the solution supplying the selection mechanism. This initial-value-problem treatment contrasts with classical steady formulations, which need extra conditions to pick a unique solution.

What carries the argument

The initial-value-problem formulation inside the linearised inviscid potential-flow framework, in which the algebraically decaying transient solution enforces asymmetric Fourier cancellation that fixes the far-field steady state.

Load-bearing premise

The linearised inviscid potential-flow model remains valid and the algebraically decaying time-dependent solution dominates the far-field cancellation.

What would settle it

If high-resolution simulations or laboratory experiments show either symmetric wave patterns on both sides of the forcing or no waves on one side despite supercritical translation speed, the claimed asymmetric cancellation would be falsified.

Figures

Figures reproduced from arXiv: 2605.12254 by Nikhil Yewale, Palas Kumar Farsoiya, Ratul Dasgupta, Vinod Kumar Kadari, Y. S. Mayya.

**Figure 2.** Figure 2: (Upper Panels) Variation of 𝑐 2 FD(𝜅) with 𝜅 from eqn. 2.1 for 𝜌𝑟 = 10−3 (air-water) and 𝜌𝑟 = 10−1 with varying 𝐵. For 𝜌𝑟 = 10−3 , 𝐵𝑐 ≈ 1/3 independent of 𝐻. As 𝐵 is increased beyond zero, the nature of the extremum at 𝜅 = 0 changes from a maximum (𝐵 < 𝐵𝑐) to a minimum (𝐵 > 𝐵𝑐). For 0 < 𝐵 < 𝐵𝑐, there is also a minimum at 𝜅𝑚 > 0. The existence of this minimum at 𝜅𝑚 > 0 for 0 < 𝐵 < 𝐵𝑐 (red curves, see eqn. 2… view at source ↗

**Figure 3.** Figure 3: A point force (red arrow) of strength 𝐹˜ 0 acts at 𝑡 >˜ 0 at the interface of two fluids of density 𝜌𝑢 and 𝜌𝑙 , both moving with speed 𝑈 (as seen in the co-moving frame). The linearised IVP predicts how waves develop at the interface, in time. thereby ensuring that the simulational results verify 0 < 𝐵 < 𝐵𝑐 (𝜌𝑟 , 𝐻). We commence with the solution to the IVP in the next section. 3. How does the wave pattern… view at source ↗

**Figure 4.** Figure 4: The (non-dimensional) perturbed interface viz. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: For both panels, 𝐹0 = 0.0014, 𝛼 = 0.1389 < 𝛼max = 0.25, 𝜌𝑟 = 0.001. The roots 𝑘𝑙,𝑠 are obtained from eqn. 3.10. Panel (a): The term 𝜂 local 𝑠 (𝑥) (see eqn. 3.11) has been evaluated in Matlab using the built-in function integral which accepts ∞ as one of the limits of integration. Panel (b): Asymmetric upstream and downstream response obtained from Rayleigh dissipation. Expression 3.12, 𝜂 far-field (𝑥) and … view at source ↗

**Figure 6.** Figure 6: Time evolution of 𝜂 (𝜌𝑟 = 0.001, 𝛼 = 0, 𝐹0 = 0.0014). Here 𝜂𝑠 (𝑥) (see eqn. 4.3) is the time-independent part of 𝜂(𝑥, 𝑡) (eqn. 4.1), 𝜂𝑡𝑟 (𝑥, 𝑡) (eqns. 4.4(a)-(e)) is the transient part. Vertical dashed line in panels (a,b,c) are at 𝑥 = 𝑡 with most of the transient response seen for 𝑥 < 𝑡, see panel (c). The integrals (eqns. 4.3 and 4.4) are evaluated using (The MathWorks, Inc. 2022) in-built function integ… view at source ↗

**Figure 7.** Figure 7: Time evolution of I4 (𝑥, 𝑡) from eqn. 4.5(d) for 𝜌𝑟 = 0.001 and 𝛼 = 0.1389. Each curve (except the first curve at 𝑡 = 0.37) is shifted vertically upwards by two units compared to the previous one. Material, sec. 1.1) i.e. 𝜂(𝑥, 𝑡) = 𝜂𝑠 (𝑥) + 𝜂𝑡𝑟 (𝑥, 𝑡), 𝜂𝑠 (𝑥) 𝐹0 ≡ − 1 𝜋 ∫ ∞ 0 𝑑𝑘 cos(𝑘𝑥) 𝛼 (𝑘 − 𝑘𝑙) (𝑘 − 𝑘𝑠) , 𝜂tr(𝑥, 𝑡) 𝐹0 ≡ − 1 2𝜋 I3 (𝑥, 𝑡) + I4 (𝑥, 𝑡) , I3,4 (𝑥, 𝑡) ≡ − (1 + 𝜌𝑟 ) 𝛼 ∫ ∞ 0 𝑑𝑘 (𝑘 ± 𝜒(𝑘… view at source ↗

**Figure 8.** Figure 8: Time evolution for 𝜌𝑟 = 10−3 , 𝛼 = 0.1389, 𝐹0 = 0.0014 as obtained by numerically solving expressions 4.5 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗

**Figure 9.** Figure 9: Panel (a) A (not to scale) schematic of the simulation domain. The interface is [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗

**Figure 10.** Figure 10: Comparison of linear theory (eqns. 4.5(a)-(d)) with simulations for 𝜌𝑟 = 10−3 , 𝛼 = 0.1389 and 𝐹0 ≡ 𝐹˜ 0 𝜌𝑙𝑈2𝑙𝑐 = 1.4 × 10−3 with length and time in units of 𝑙𝑐 ≡ 𝑈 2 /𝑔 and 𝑡𝑐 ≡ 𝑈/𝑔 respectively. For 𝑡 > 110 (last panel), significant reflections are seen from the boundaries in simulations and hence comparisons are not shown beyond this. The solid black curve in panel (h) labelled as ‘Steady’ is from expr… view at source ↗

**Figure 11.** Figure 11: Comparison of linear theory (eqns. 4.5(a)-(d)) with simulations for 𝜌𝑟 = 10−1 , 𝛼 = 0.1389 and 𝐹0 ≡ 𝐹˜ 0 𝜌𝑙𝑈2𝑙𝑐 = 1.4 × 10−2 with length and time in units of 𝑙𝑐 ≡ 𝑈 2 /𝑔 and 𝑡𝑐 ≡ 𝑈/𝑔 respectively. For 𝑡 > 18 (last panel), significant reflections are seen from the boundaries in simulations and hence comparisons are not shown beyond this time. The solid black curve in panel (f) labelled as ‘Steady’ is from … view at source ↗

**Figure 12.** Figure 12: Time evolution 𝜌𝑟 = 0.001, 𝛼 = 0.1389 and varying 𝐹0. Each interface is shifted vertically upward by five hundred units, except the first curve for 𝐹0 = 0.0111. The lower most curve (red) shows nearly linear behaviour. In the following, we present a qualitative comparison between the Stokes wave profile at large amplitude ( ˜𝑎 = 0.1 cm, top right inset in fig. 13a) and wave shapes seen locally within the … view at source ↗

**Figure 13.** Figure 13: Panel (a) Phase-speed ( ˜𝑐) of a capillary-gravity Stokes wave as function of ˜𝑘 and amplitude ˜𝑎. Parameters are 𝑔 = 981 cm/s2 , 𝜎 = 72 dyn/cm, 𝜌 = 1 g/cm3 . Panel (b) Capillary-gravity, linearised dispersion relation for water wave at 𝜌𝑟 = 0. Panel (c) Constant ˜𝑎 slices of fig. 13a. Panel (d) Snapshot of the (dimensional) interface (CGS) from a numerical simulation with non-dimensional parameters 𝜌𝑟 = … view at source ↗

**Figure 14.** Figure 14: A suggested application of the time-dependent theory developed here. We [PITH_FULL_IMAGE:figures/full_fig_p029_14.png] view at source ↗

**Figure 15.** Figure 15: Phase (𝑐 (− ) 𝑝 ) and energy propagation speed (group speed) 𝑐 (− ) 𝑔 for interfacial waves in deep layers from eqns. C 1 and C 2. For both panels 𝛼 = 0.1389. In the co-moving frame, the wavenumbers 𝑘𝑠,𝑙 have zero phase speeds and positive and negative 𝑐 (− ) 𝑔 of same magnitude. Note that the gravity waves (𝑘𝑠) become longer while capillary ones (𝑘𝑙 ) get shorter with increase in density ratio. and lower… view at source ↗

**Figure 16.** Figure 16: Semicircular contour for (a) 𝑥 > 0 (b) 𝑥 < 0 for evaluating the integrals in eqn. D 5. The integrals on the semi-circular paths in both figures vanish as 𝑅 → ∞. where Π ≡ 𝛾 (1 + 𝜌𝑟 ) 𝛼 and ℜ (·) indicates the real part of its argument. Decomposing the integrand in eqn. D 4 into partial fractions, we obtain (neglecting a term of O (𝛾˜ 2 )) 𝜂(𝑥) = 𝐹0 𝜋𝛼 (𝑘𝑙 − 𝑘𝑠 − 2𝑖𝜗) ℜ ∫ ∞ 0 𝑑𝑘 exp(𝑖𝑘𝑥) 1 𝑘 − (𝑘𝑠 + 𝑖𝜗)… view at source ↗

read the original abstract

A localised overpressure translating at a uniform speed greater than a critical value acts at the interface between two deep fluid layers with different densities. We analyse the resulting wave patterns using an initial-value problem formulation within the linearised, inviscid, potential flow framework. The steady-state interface exhibits short capillary waves ahead of the forcing and long gravity waves behind it, arising from an asymmetric cancellation of Fourier components in the far field. The time-dependent part of the solution, decaying algebraically with time, plays a crucial role in this mechanism. This contrasts with classical steady approaches, which require additional conditions to select a unique solution. We extend this approach to a two-fluid interface and validate the predictions against nonlinear simulations.

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

The IVP setup here lets algebraically decaying transients drive the asymmetric Fourier cancellation that picks out capillary waves ahead and gravity waves behind the moving pressure, without extra steady-state conditions.

read the letter

The main point is that treating the translating overpressure as an initial-value problem in linear potential flow makes the time-dependent solution decay algebraically and produce the observed far-field asymmetry through selective Fourier cancellation. This gives a direct way to get the steady pattern without the auxiliary conditions that classical steady analyses often need. The paper spells out the Fourier mechanism and checks the predictions against nonlinear simulations for the two-fluid interface, which is a reasonable validation step. The derivation begins from the standard equations without circular definitions or free parameters, and the two-fluid extension is a straightforward addition that broadens the claim a bit. The central mechanism holds up on the evidence given. The main limitation is that everything stays inside linear inviscid theory, so the results apply only when those assumptions are reasonable; the nonlinear checks help but do not remove the restriction. Some steps in the transient cancellation would benefit from more explicit error bounds or intermediate algebra if the full manuscript is as concise as the abstract suggests. Overall the work is narrow in scope but internally consistent. It is aimed at people who already work on interfacial waves or moving disturbances and want an alternative analytical route to the same patterns. A reader looking for new selection mechanisms in classical fluid problems will find the perspective useful. I would send this to peer review; the idea is clear, the numerics line up, and the contribution is modest but well-motivated.

Referee Report

2 major / 2 minor

Summary. The paper formulates the problem of a localised overpressure translating at constant speed across the interface between two deep fluid layers of differing densities as an initial-value problem (IVP) within the linearised inviscid potential-flow framework. It claims that the resulting steady-state interface displacement exhibits short capillary waves ahead of the forcing and long gravity waves behind it, produced by an asymmetric cancellation of Fourier components in the far field; the algebraically decaying time-dependent solution is essential to this selection mechanism. The approach is contrasted with classical steady-state analyses that require auxiliary conditions, and the predictions are validated against nonlinear simulations for the two-fluid case.

Significance. If the central mechanism holds, the work supplies a natural, parameter-free explanation for far-field wave selection in forced interfacial problems without invoking radiation conditions or other ad-hoc constraints. The explicit role assigned to algebraically decaying transients and the direct comparison to nonlinear simulations constitute clear strengths. This perspective could clarify analogous selection issues in ship-wake and stratified-flow problems and may encourage wider use of IVP formulations for steady-state limits in potential-flow theory.

major comments (2)

[Fourier-analysis derivation (likely §3–4)] The asymmetric Fourier cancellation that selects capillary waves upstream and gravity waves downstream is the load-bearing step for the central claim. The manuscript should provide an explicit decomposition (e.g., contour integration or residue analysis) showing which poles or branches are retained or cancelled in each far-field region and how the algebraic time decay enforces the asymmetry; without this level of detail the mechanism remains plausible but not fully verified.
[Two-fluid extension and numerical validation] The extension to the two-fluid interface and the comparison with nonlinear simulations are presented as validation, yet the linearised inviscid assumption is retained throughout. A quantitative assessment of the regime (e.g., Froude number, density ratio, and forcing amplitude) in which the algebraic transients dominate over viscous or nonlinear effects would strengthen the claim that the IVP mechanism survives in the nonlinear regime.

minor comments (2)

[Introduction / setup] The critical speed separating sub- and super-critical regimes is invoked in the abstract but should be stated explicitly in terms of the dispersion relation and density ratio at the first appearance in the text.
[Throughout] Notation for the interface displacement, pressure forcing, and Fourier transforms should be collected in a single table or consistently defined on first use to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary, the recommendation for minor revision, and the constructive comments that will improve the clarity and robustness of the manuscript. We address each major comment below.

read point-by-point responses

Referee: [Fourier-analysis derivation (likely §3–4)] The asymmetric Fourier cancellation that selects capillary waves upstream and gravity waves downstream is the load-bearing step for the central claim. The manuscript should provide an explicit decomposition (e.g., contour integration or residue analysis) showing which poles or branches are retained or cancelled in each far-field region and how the algebraic time decay enforces the asymmetry; without this level of detail the mechanism remains plausible but not fully verified.

Authors: We agree that an explicit contour-integration analysis would make the selection mechanism fully rigorous. In the revised manuscript we will add a dedicated subsection (in §3) that performs the far-field asymptotic analysis via contour integration in the complex wavenumber plane. This will identify the relevant poles for the capillary and gravity branches, show which residues are retained or cancelled in the upstream versus downstream regions, and demonstrate how the algebraically decaying time-dependent contribution enforces the asymmetry. revision: yes
Referee: [Two-fluid extension and numerical validation] The extension to the two-fluid interface and the comparison with nonlinear simulations are presented as validation, yet the linearised inviscid assumption is retained throughout. A quantitative assessment of the regime (e.g., Froude number, density ratio, and forcing amplitude) in which the algebraic transients dominate over viscous or nonlinear effects would strengthen the claim that the IVP mechanism survives in the nonlinear regime.

Authors: We appreciate this suggestion for strengthening the validation. In the revised version we will insert a quantitative discussion (in §5) that estimates the parameter regimes—expressed in terms of Froude number, density ratio, and forcing amplitude—where the linear IVP mechanism is expected to dominate before viscous or nonlinear effects become significant. The estimates will be supported by scaling arguments and direct reference to the existing nonlinear simulation results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from standard IVP setup

full rationale

The paper sets up the problem directly from the linearised inviscid potential flow equations for two deep fluid layers as an initial-value problem (IVP). The steady-state far-field wave pattern (short capillary waves ahead, long gravity waves behind) is obtained via Fourier analysis of the governing equations, with the algebraically decaying time-dependent transients providing the asymmetric cancellation mechanism in the long-time limit. This is contrasted with classical steady-state approaches but does not rely on fitted parameters, self-definitional relations, or load-bearing self-citations for the core selection mechanism. Independent validation against nonlinear simulations further confirms the derivation remains non-circular and externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard domain assumptions of fluid dynamics rather than new postulates.

axioms (2)

domain assumption Linearised, inviscid, potential flow for deep fluid layers with different densities
Stated as the framework for the IVP analysis.
domain assumption Pressure forcing translates at uniform speed greater than a critical value
Given as the setup condition for the wave patterns.

pith-pipeline@v0.9.0 · 5438 in / 1291 out tokens · 83262 ms · 2026-05-13T03:57:09.577452+00:00 · methodology

discussion (0)

Reference graph

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