The Omitted Noise Contribution of Surface Normal Variation: Farassat's Formulation 1A revisited

Chen He; Jiale Lu; Qitian Tao; Xiaohua Liu; Zhengwu Chen

arxiv: 2606.07081 · v1 · pith:H3N7MFOHnew · submitted 2026-06-05 · ⚛️ physics.flu-dyn

The Omitted Noise Contribution of Surface Normal Variation: Farassat's Formulation 1A revisited

Qitian Tao , Chen He , Xiaohua Liu , Zhengwu Chen , Jiale Lu This is my paper

Pith reviewed 2026-06-27 21:13 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords Farassat formulationpropeller noiseacoustic source termsurface normal derivativeFormulation 1Atime-domain aeroacousticsunsteady force direction

0 comments

The pith

Farassat's Formulation 1A omits the temporal derivative of the surface normal vector, which must be retained to complete the acoustic source representation.

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

The paper examines the step from Farassat's Formulation 1 to Formulation 1A used for propeller noise. In that step the contribution from the time rate of change of the local surface normal is dropped. The authors show through direct derivation that this term is required for the source term to be fully specified. They therefore introduce a Modified Formulation 1A that keeps the normal-vector derivative explicitly. Far-field calculations with the new form are presented to illustrate consistency with the parent formulation.

Core claim

Through rigorous mathematical derivation, this study demonstrates that the omitted term constitutes an indispensable component of the acoustic source representation. Accordingly, a Modified Formulation 1A is proposed by explicitly retaining the normal vector temporal derivative term in the time-domain formulation.

What carries the argument

The temporal derivative of the local surface normal vector, which supplies the missing contribution from the changing direction of the unsteady force.

If this is right

The modified formulation supplies a mathematically complete surface source term for time-domain propeller noise calculations.
Far-field acoustic predictions performed with the modified form remain consistent with the parent Formulation 1 while restoring the omitted contribution.
Any existing implementation of Formulation 1A can be updated by adding the explicit normal-vector time derivative without changing the overall structure of the integral.

Where Pith is reading between the lines

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

The same normal-vector term may appear in other surface-integral acoustic analogies that start from an unsteady force distribution on a moving surface.
Numerical codes that already evaluate surface normals at successive time steps can insert the extra term with only a small additional cost.
Validation against controlled rotating-surface experiments would directly test whether the correction alters predicted directivity or amplitude.

Load-bearing premise

The standard derivation of Formulation 1A from Formulation 1 omits the temporal derivative of the local surface normal vector, and retaining this term is required for the acoustic source representation to be complete.

What would settle it

A side-by-side computation of far-field pressure time histories for a given propeller using both the original Formulation 1A and the modified version, compared against the same set of measured data, would show whether the retained term produces a measurable difference.

read the original abstract

Farassat's Formulations 1 and 1A have been extensively employed for propeller noise prediction. However, in the derivation of Formulation 1A from Formulation 1, the contribution associated with the temporal variation of the direction of the unsteady force is omitted, appearing mathematically as the temporal derivative of the local surface normal vector. Through rigorous mathematical derivation, this study demonstrates that the omitted term constitutes an indispensable component of the acoustic source representation. Accordingly, a Modified Formulation 1A is proposed by explicitly retaining the normal vector temporal derivative term in the time-domain formulation. Far-field acoustic predictions for propellers are performed to evaluate the proposed formulation, and the results confirm both its theoretical consistency and predictive capability.

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 paper claims Farassat 1A drops a ∂n/∂t term that should be kept, but the abstract supplies no equations to show whether it cancels under the original assumptions.

read the letter

The main takeaway is that the authors identify what they say is an omitted contribution in the step from Farassat Formulation 1 to 1A—the time derivative of the local surface normal—and they keep it in a modified version. They then run far-field propeller predictions to check consistency.

The work does one useful thing: it forces a second look at a standard reduction that many codes rely on for unsteady loading noise. If the term really survives the retarded-time and far-field ordering without canceling, the correction could matter for accuracy in cases with strong surface motion.

The soft spot is exactly the one the stress-test note flags. The abstract calls the derivation rigorous yet shows none of the algebra, no comparison to Farassat’s original steps, and no identity that would prove the term does not vanish. Without that cross-check it is impossible to tell whether this is a genuine omission or something already handled by the definitions of the dipole or the surface kinematics. The predictions are cited as confirmation, but the abstract gives no numbers on the size of the difference or any error metric against measurements or the unmodified formulation.

This is narrow-scope work for people who already implement or validate Farassat-type codes in propeller or drone acoustics. A reader who wants to audit the classic derivation might get something from the steps if they are fully written out, but the current version does not yet supply enough to decide the claim.

I would send it to referees so they can check the algebra directly against the 1970s–80s papers; the topic is narrow enough that a quick review can settle whether the term is indispensable or was correctly dropped.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that the standard derivation of Farassat's Formulation 1A from Formulation 1 for propeller noise omits the acoustic contribution arising from the temporal derivative of the local surface normal vector (appearing as part of the unsteady force direction). Through a mathematical re-derivation, the authors assert this term is indispensable rather than negligible or canceled, propose a Modified Formulation 1A that retains the ∂n/∂t term explicitly in the time-domain integral, and present far-field acoustic predictions for propellers that are said to confirm both theoretical consistency and improved predictive capability.

Significance. If the derivation is shown to be free of hidden cancellations under the same far-field and retarded-time assumptions used in the original Farassat work, the result would refine the completeness of surface-integral aeroacoustic formulations and could affect noise predictions for unsteady rotating sources. The provision of confirmatory predictions is a positive element, though the manuscript would benefit from quantitative error metrics and direct comparison against the unmodified 1A baseline.

major comments (2)

[§3] §3 (derivation of Modified 1A): The central claim requires an explicit, side-by-side comparison of the transition from Formulation 1 to standard 1A versus the authors' steps, showing at which exact point the term containing ∂n/∂t is dropped and confirming it does not vanish identically via retarded-time surface kinematics or far-field ordering. Without this cross-check against Farassat's original algebra, it remains possible the term cancels rather than being omitted.
[§4] §4 (acoustic predictions): The validation relies on far-field predictions, but no quantitative error norms, grid-convergence data, or direct subtraction of the unmodified 1A results are reported; this weakens the assertion that the retained term produces a demonstrably superior representation.

minor comments (2)

Notation for the surface normal vector and its time derivative should be introduced with an explicit definition (e.g., n(x,τ)) at first use to avoid ambiguity with the standard Farassat notation.
The abstract states 'rigorous mathematical derivation' but the manuscript would be clearer if the key identity used to retain the term were highlighted in a boxed equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate the revisions planned for the next version.

read point-by-point responses

Referee: [§3] §3 (derivation of Modified 1A): The central claim requires an explicit, side-by-side comparison of the transition from Formulation 1 to standard 1A versus the authors' steps, showing at which exact point the term containing ∂n/∂t is dropped and confirming it does not vanish identically via retarded-time surface kinematics or far-field ordering. Without this cross-check against Farassat's original algebra, it remains possible the term cancels rather than being omitted.

Authors: We agree that an explicit side-by-side comparison would strengthen the manuscript. While our Section 3 derivation retains the ∂n/∂t term by following the standard steps from Formulation 1, we will add a comparative table or subsection in the revision that juxtaposes Farassat's original algebra with our steps. This will pinpoint the exact omission and confirm algebraically that the term does not cancel under the retarded-time and far-field assumptions employed in the original work. revision: yes
Referee: [§4] §4 (acoustic predictions): The validation relies on far-field predictions, but no quantitative error norms, grid-convergence data, or direct subtraction of the unmodified 1A results are reported; this weakens the assertion that the retained term produces a demonstrably superior representation.

Authors: We concur that quantitative metrics would improve the validation section. In the revised manuscript we will incorporate grid-convergence studies, L2 error norms on the acoustic pressure, and direct difference plots obtained by subtracting the unmodified Formulation 1A predictions from the modified results to quantify the contribution of the retained term. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from Formulation 1 to 1A presented as independent mathematical step

full rationale

The paper asserts that the standard derivation of Formulation 1A from Formulation 1 omits the term with ∂n/∂t and demonstrates via its own derivation that the term is indispensable, leading to a Modified Formulation 1A. No equations or claims in the abstract reduce the central result to a fitted parameter, a self-citation chain, or a redefinition of inputs. The load-bearing step is an explicit re-derivation under the same retarded-time and far-field assumptions used by Farassat, which is externally checkable against the original Formulation 1 steps and does not rely on the authors' prior work as an unverified uniqueness theorem. This is the most common honest non-finding for a derivation paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.1-grok · 5660 in / 1044 out tokens · 22083 ms · 2026-06-27T21:13:20.491522+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references

[1]

On sound generated aerodynamically I

Lighthill M J. On sound generated aerodynamically I. General theory[J]. Proceedings of the Royal Society of London Series A: Mathematical and Physical 11 Sciences, 1952, 211(1107): 564-587

1952
[2]

Sound generation by turbulence and surfaces in arbitrary motion[J]

Ffowcs Williams J E, Hawkings D L. Sound generation by turbulence and surfaces in arbitrary motion[J]. Philosophical Transactions of the Royal Society of London Series A, 1969, 264(1151): 321-342

1969
[3]

Theory of noise generation from moving bodies with an application to helicopter rotors[R]

Farassat F. Theory of noise generation from moving bodies with an application to helicopter rotors[R]. Washington: NASA, 1975: R-451

1975
[4]

Linear acoustic formulas for calculation of rotating blade noise[J]

Farassat F. Linear acoustic formulas for calculation of rotating blade noise[J]. AIAA Journal, 1981, 19(9): 1122-1130

1981
[5]

A review of propeller discrete frequency noise prediction technology with emphasis on two current methods for time domain calculations[J]

Farassat F, Succi G P. A review of propeller discrete frequency noise prediction technology with emphasis on two current methods for time domain calculations[J]. Journal of Sound and Vibration, 1980, 71(3): 399-419

1980
[6]

Derivation of formulations 1 and 1A of Farassat[R]

Farassat F. Derivation of formulations 1 and 1A of Farassat[R]. NASA Technical Memorandum, 2007: NASA TM-2007-214853

2007

[1] [1]

On sound generated aerodynamically I

Lighthill M J. On sound generated aerodynamically I. General theory[J]. Proceedings of the Royal Society of London Series A: Mathematical and Physical 11 Sciences, 1952, 211(1107): 564-587

1952

[2] [2]

Sound generation by turbulence and surfaces in arbitrary motion[J]

Ffowcs Williams J E, Hawkings D L. Sound generation by turbulence and surfaces in arbitrary motion[J]. Philosophical Transactions of the Royal Society of London Series A, 1969, 264(1151): 321-342

1969

[3] [3]

Theory of noise generation from moving bodies with an application to helicopter rotors[R]

Farassat F. Theory of noise generation from moving bodies with an application to helicopter rotors[R]. Washington: NASA, 1975: R-451

1975

[4] [4]

Linear acoustic formulas for calculation of rotating blade noise[J]

Farassat F. Linear acoustic formulas for calculation of rotating blade noise[J]. AIAA Journal, 1981, 19(9): 1122-1130

1981

[5] [5]

A review of propeller discrete frequency noise prediction technology with emphasis on two current methods for time domain calculations[J]

Farassat F, Succi G P. A review of propeller discrete frequency noise prediction technology with emphasis on two current methods for time domain calculations[J]. Journal of Sound and Vibration, 1980, 71(3): 399-419

1980

[6] [6]

Derivation of formulations 1 and 1A of Farassat[R]

Farassat F. Derivation of formulations 1 and 1A of Farassat[R]. NASA Technical Memorandum, 2007: NASA TM-2007-214853

2007