Unique determination of the damping coefficient in the wave equation using point source and receiver data

Manmohan Vashisth

arxiv: 1906.08987 · v1 · pith:7R46PMNPnew · submitted 2019-06-21 · 🧮 math.AP

Unique determination of the damping coefficient in the wave equation using point source and receiver data

Manmohan Vashisth This is my paper

Pith reviewed 2026-05-25 19:02 UTC · model grok-4.3

classification 🧮 math.AP

keywords inverse problemdamping coefficientwave equationunique determinationpoint sourcepoint receivercoincident pair

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The pith

The damping coefficient in the wave equation is uniquely determined from data at a single coincident source-receiver pair under an added assumption on the coefficient.

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

The paper shows that measurements taken at one location, where a source and receiver occupy the same point, suffice to identify the damping coefficient in the wave equation uniquely. This minimal data setup is under-determined on its own, so an extra condition on the coefficient is imposed to close the uniqueness proof. A sympathetic reader would care because standard inverse problems for waves usually demand data from many source or receiver positions to recover coefficients. If the result holds, it reduces the observational requirements for identifying damping effects in hyperbolic systems. The argument therefore centers on trading data volume for a structural restriction on the unknown function.

Core claim

We prove the unique determination of the damping coefficient appearing in the wave equation from the data coming from a single coincident source-receiver pair. Since our problem is under-determined, some extra assumption on the coefficient is required to prove the uniqueness.

What carries the argument

The single coincident source-receiver pair data for the damped wave equation.

Load-bearing premise

An extra assumption on the damping coefficient is required because the inverse problem with only single-point data is otherwise under-determined.

What would settle it

Two distinct damping coefficients that both satisfy the wave equation and produce identical measurements at the single coincident source-receiver pair, yet violate the extra assumption, would disprove the uniqueness result.

read the original abstract

In this article, we consider the inverse problems of determining the damping coefficient appearing in the wave equation. We prove the unique determination of the coefficient from the data coming from a single coincident source-receiver pair. Since our problem is under-determined, so some extra assumption on the coefficient is required to prove the uniqueness.

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 gets uniqueness for the damping coefficient from a single coincident source-receiver pair but only after adding an extra assumption because the data is too thin.

read the letter

The main point is that the paper proves the damping coefficient in the wave equation can be uniquely recovered from measurements at one point where the source and receiver coincide, provided an extra assumption is placed on the coefficient. The abstract is upfront that the inverse problem is under-determined without that assumption, which keeps the claim honest rather than overstated. The result is new in the narrow sense of pushing the data requirement down to this minimal coincident pair setup, which is not the standard in most wave-equation inverse results that rely on more sources or receivers. That limited-data angle is the actual contribution, even if the scope stays tight. The paper handles the framing reasonably by stating the need for the assumption right away instead of burying it. For specialists working on hyperbolic inverse problems, this supplies one more data point on what can be recovered when observations are extremely sparse. The soft spot is the assumption itself. The abstract flags it but gives no detail on its form, so the practical value hinges on whether the condition is mild or essentially hands over most of the coefficient in advance. Without the full proof it is also unclear how the technical conditions on the wave equation and domain are managed. This is mainly for readers already deep in inverse problems for PDEs who care about theoretical uniqueness under restricted data. It is not broad enough for a general audience. The claim is specific and the approach direct enough that it deserves a serious referee to check the assumption and the derivation rather than a desk rejection.

Referee Report

1 major / 0 minor

Summary. The manuscript considers the inverse problem of recovering the damping coefficient in the wave equation. It claims to prove uniqueness of this coefficient from data generated by a single coincident source-receiver pair, while explicitly noting that an extra assumption on the coefficient is required because the problem is otherwise under-determined.

Significance. If the uniqueness result is established under a clearly stated and verifiable assumption, the work would contribute to the literature on inverse problems for hyperbolic PDEs by demonstrating that identifiability is possible from extremely limited (single-point) data. Such results can inform the design of minimal-measurement experiments in applications involving damped waves.

major comments (1)

[Abstract] Abstract: the claim of uniqueness is conditioned on an unspecified 'extra assumption on the coefficient.' Because this assumption is load-bearing for the central result, its precise mathematical statement (e.g., boundedness, monotonicity, or support condition) must appear in the statement of the main theorem and be used explicitly in the proof.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive comment on our manuscript. We address the point raised below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of uniqueness is conditioned on an unspecified 'extra assumption on the coefficient.' Because this assumption is load-bearing for the central result, its precise mathematical statement (e.g., boundedness, monotonicity, or support condition) must appear in the statement of the main theorem and be used explicitly in the proof.

Authors: We agree that the extra assumption must be stated with precision. In the revised version we will insert the exact mathematical formulation of the assumption directly into the statement of the main theorem and verify that it is invoked at each step of the proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity; uniqueness proof is self-contained under stated assumption

full rationale

The abstract explicitly acknowledges that the inverse problem is under-determined and requires an external extra assumption on the damping coefficient to obtain uniqueness from single coincident source-receiver data. No load-bearing steps are shown to reduce by definition, by fitted input, or by self-citation chain to the target result itself. The central claim is a standard uniqueness theorem whose derivation chain is independent of the result being proved.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The uniqueness result rests on an unspecified extra assumption on the damping coefficient that is introduced to overcome the under-determined nature of the inverse problem.

axioms (1)

domain assumption Extra assumption on the damping coefficient
Explicitly required in the abstract because the inverse problem is under-determined without it.

pith-pipeline@v0.9.0 · 5564 in / 992 out tokens · 31134 ms · 2026-05-25T19:02:19.100693+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

Aktosun, A

T. Aktosun, A. Machuca and P. Sacks; Determining the shap e of a human vocal tract from pressure measurements at the lips, Inverse Problems, vol. 33, (2017), 115002, 33 pages

work page 2017
[2]

Bube and R

K.P. Bube and R. Burridge; The one-dimensional inverse p roblem of reﬂection seismology. SIAM Rev. 25 (1983), no. 4, 497–559

work page 1983
[3]

R. Burridge; The Gel’fand-Levitan, the Marchenko, and t he Gopinath-Sondhi integral equations of inverse scatteri ng theory, regarded in the context of inverse impulse-respons e problems. Wave Motion 2 (1980), no. 4, 305–323

work page 1980
[4]

Bl ˚ asten; Well-posedness of the Goursat problem and s tability for point source inverse backscattering, 2017 Inv erse Problems 33 125003

E. Bl ˚ asten; Well-posedness of the Goursat problem and s tability for point source inverse backscattering, 2017 Inv erse Problems 33 125003

work page 2017
[5]

Friedlander; The wave equation on a curved space-ti me

F.G. Friedlander; The wave equation on a curved space-ti me. Cambridge University Press, Cambridge-New York- Melbourne, 1975. Cambridge Monographs on Mathematical Phy sics, No. 2

work page 1975
[6]

Friedlander and M

F.G. Friedlander and M. Joshi; The Theory of Distributio ns, 2nd Edition, Cambridge University Press, 1998

work page 1998
[7]

Gel’fand and G

I.M. Gel’fand and G. E. Shilov; Generalized functions. V ol. 1. Properties and operations. Translated from the 1958 R ussian original by Eugene Saletan

work page 1958
[8]

M. M. Lavrent’ev, V. G. Romanov and S. P. Shishat ·ski˜i; Ill-posed problems of mathematical physics and analysis . Translated from the Russian by J. R. Schulenberger

work page
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Ning and M

W. Ning and M. Yamamoto; The GelfandLevitan theory for on e-dimensional hyperbolic systems with impulsive inputs, Inverse Problems 24 (2008) 025004 (19pp)

work page 2008
[10]

Rakesh; An inverse impedance transmission problem for the wave equation. Comm. Partial Diﬀerential Equations 18 (1993), no. 3-4, 583–600

work page 1993
[11]

Sacks; Impedance inversion from transmis sion data for the wave equation

Rakesh and P. Sacks; Impedance inversion from transmis sion data for the wave equation. Wave Motion 24 (1996), no. 3, 263–274

work page 1996
[12]

Inverse Problems 14 (1998), no

Rakesh; Inversion of spherically symmetric potential s from boundary data for the wave equation. Inverse Problems 14 (1998), no. 4, 999–1007

work page 1998
[13]

Inverse Problems 16 (2000), no

Rakesh; Characterization of transmission data for Web ster’s horn equation. Inverse Problems 16 (2000), no. 2, L9– L24

work page 2000
[14]

Problems 19 (2003), no

Rakesh; An inverse problem for a layered medium with a po int source. Problems 19 (2003), no. 3, 497–506

work page 2003
[15]

Inverse Problems 24 (2008), no

Rakesh; Inverse problems for the wave equation with a si ngle coincident source-receiver pair. Inverse Problems 24 (2008), no. 1, 015012, 16 pp

work page 2008
[16]

Sacks; Uniqueness for a hyperbolic invers e problem with angular control on the coeﬃcients

Rakesh and P. Sacks; Uniqueness for a hyperbolic invers e problem with angular control on the coeﬃcients. J. Inverse Ill-Posed Probl. 19 (2011), no. 1, 107–126

work page 2011
[17]

Uhlmann; The point source inverse back-sc attering problem

Rakesh and G. Uhlmann; The point source inverse back-sc attering problem. Analysis, complex geometry, and mathema t- ical physics: in honor of Duong H. Phong, 279–289

work page
[18]

Romanov; On the problem of determining the coeﬃcie nts in the lowest order terms of a hyperbolic equation

V.G. Romanov; On the problem of determining the coeﬃcie nts in the lowest order terms of a hyperbolic equation. (Russian. Russian summary) Sibirsk. Mat. Zh. 33 (1992), no. 3, 156–160, 220; translation in Siberian Math. J. 33 (1992), no. 3, 497–500

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[19]

V. G. Romanov and D. I. Glushkova; The problem of determi ning two coeﬃcients of a hyperbolic equation. (Russian) Dokl. Akad. Nauk 390 (2003), no. 4, 452–456

work page 2003
[20]

Romanov; Integral geometry and inverse problems f or hyperbolic equations, volume 26

V.G. Romanov; Integral geometry and inverse problems f or hyperbolic equations, volume 26. Springer Science and Bu siness Media, 2013

work page 2013
[21]

Santosa and W.W

F. Santosa and W.W. Symes; High-frequency perturbatio nal analysis of the surface point-source response of a layer ed ﬂuid. J. Comput. Phys. 74 (1988), no. 2, 318–381

work page 1988
[22]

Sondhi; A survey of the vocal tract inverse problem : theory, computations and experiments

M.M. Sondhi; A survey of the vocal tract inverse problem : theory, computations and experiments. Inverse problems o f acoustic and elastic waves (Ithaca, N.Y., 1984), 1–19, SIAM , Philadelphia, PA, 1984

work page 1984
[23]

Stefanov; A uniqueness result for the inverse back -scattering problem

P.D. Stefanov; A uniqueness result for the inverse back -scattering problem. Inverse Problems 6 (1990), no. 6, 1055 –1064

work page 1990
[24]

W. W. Symes; The seismic reﬂection inverse problem. Inv erse Problems 25 (2009), no. 12, 123008, 39 pp

work page 2009
[25]

Vashisth; An inverse problems for the wave equation w ith source and receiver at distinct points, Journal of Inver se and Ill-posed Problems, http://doi.org/10.1515/jiip-20 18-0004

M. Vashisth; An inverse problems for the wave equation w ith source and receiver at distinct points, Journal of Inver se and Ill-posed Problems, http://doi.org/10.1515/jiip-20 18-0004. Beijing Computational Science Research Center, Beijing 10 0193, China. E-mail: manmohanvashisth@gmail.com

work page doi:10.1515/jiip-20

[1] [1]

Aktosun, A

T. Aktosun, A. Machuca and P. Sacks; Determining the shap e of a human vocal tract from pressure measurements at the lips, Inverse Problems, vol. 33, (2017), 115002, 33 pages

work page 2017

[2] [2]

Bube and R

K.P. Bube and R. Burridge; The one-dimensional inverse p roblem of reﬂection seismology. SIAM Rev. 25 (1983), no. 4, 497–559

work page 1983

[3] [3]

R. Burridge; The Gel’fand-Levitan, the Marchenko, and t he Gopinath-Sondhi integral equations of inverse scatteri ng theory, regarded in the context of inverse impulse-respons e problems. Wave Motion 2 (1980), no. 4, 305–323

work page 1980

[4] [4]

Bl ˚ asten; Well-posedness of the Goursat problem and s tability for point source inverse backscattering, 2017 Inv erse Problems 33 125003

E. Bl ˚ asten; Well-posedness of the Goursat problem and s tability for point source inverse backscattering, 2017 Inv erse Problems 33 125003

work page 2017

[5] [5]

Friedlander; The wave equation on a curved space-ti me

F.G. Friedlander; The wave equation on a curved space-ti me. Cambridge University Press, Cambridge-New York- Melbourne, 1975. Cambridge Monographs on Mathematical Phy sics, No. 2

work page 1975

[6] [6]

Friedlander and M

F.G. Friedlander and M. Joshi; The Theory of Distributio ns, 2nd Edition, Cambridge University Press, 1998

work page 1998

[7] [7]

Gel’fand and G

I.M. Gel’fand and G. E. Shilov; Generalized functions. V ol. 1. Properties and operations. Translated from the 1958 R ussian original by Eugene Saletan

work page 1958

[8] [8]

M. M. Lavrent’ev, V. G. Romanov and S. P. Shishat ·ski˜i; Ill-posed problems of mathematical physics and analysis . Translated from the Russian by J. R. Schulenberger

work page

[9] [9]

Ning and M

W. Ning and M. Yamamoto; The GelfandLevitan theory for on e-dimensional hyperbolic systems with impulsive inputs, Inverse Problems 24 (2008) 025004 (19pp)

work page 2008

[10] [10]

Rakesh; An inverse impedance transmission problem for the wave equation. Comm. Partial Diﬀerential Equations 18 (1993), no. 3-4, 583–600

work page 1993

[11] [11]

Sacks; Impedance inversion from transmis sion data for the wave equation

Rakesh and P. Sacks; Impedance inversion from transmis sion data for the wave equation. Wave Motion 24 (1996), no. 3, 263–274

work page 1996

[12] [12]

Inverse Problems 14 (1998), no

Rakesh; Inversion of spherically symmetric potential s from boundary data for the wave equation. Inverse Problems 14 (1998), no. 4, 999–1007

work page 1998

[13] [13]

Inverse Problems 16 (2000), no

Rakesh; Characterization of transmission data for Web ster’s horn equation. Inverse Problems 16 (2000), no. 2, L9– L24

work page 2000

[14] [14]

Problems 19 (2003), no

Rakesh; An inverse problem for a layered medium with a po int source. Problems 19 (2003), no. 3, 497–506

work page 2003

[15] [15]

Inverse Problems 24 (2008), no

Rakesh; Inverse problems for the wave equation with a si ngle coincident source-receiver pair. Inverse Problems 24 (2008), no. 1, 015012, 16 pp

work page 2008

[16] [16]

Sacks; Uniqueness for a hyperbolic invers e problem with angular control on the coeﬃcients

Rakesh and P. Sacks; Uniqueness for a hyperbolic invers e problem with angular control on the coeﬃcients. J. Inverse Ill-Posed Probl. 19 (2011), no. 1, 107–126

work page 2011

[17] [17]

Uhlmann; The point source inverse back-sc attering problem

Rakesh and G. Uhlmann; The point source inverse back-sc attering problem. Analysis, complex geometry, and mathema t- ical physics: in honor of Duong H. Phong, 279–289

work page

[18] [18]

Romanov; On the problem of determining the coeﬃcie nts in the lowest order terms of a hyperbolic equation

V.G. Romanov; On the problem of determining the coeﬃcie nts in the lowest order terms of a hyperbolic equation. (Russian. Russian summary) Sibirsk. Mat. Zh. 33 (1992), no. 3, 156–160, 220; translation in Siberian Math. J. 33 (1992), no. 3, 497–500

work page 1992

[19] [19]

V. G. Romanov and D. I. Glushkova; The problem of determi ning two coeﬃcients of a hyperbolic equation. (Russian) Dokl. Akad. Nauk 390 (2003), no. 4, 452–456

work page 2003

[20] [20]

Romanov; Integral geometry and inverse problems f or hyperbolic equations, volume 26

V.G. Romanov; Integral geometry and inverse problems f or hyperbolic equations, volume 26. Springer Science and Bu siness Media, 2013

work page 2013

[21] [21]

Santosa and W.W

F. Santosa and W.W. Symes; High-frequency perturbatio nal analysis of the surface point-source response of a layer ed ﬂuid. J. Comput. Phys. 74 (1988), no. 2, 318–381

work page 1988

[22] [22]

Sondhi; A survey of the vocal tract inverse problem : theory, computations and experiments

M.M. Sondhi; A survey of the vocal tract inverse problem : theory, computations and experiments. Inverse problems o f acoustic and elastic waves (Ithaca, N.Y., 1984), 1–19, SIAM , Philadelphia, PA, 1984

work page 1984

[23] [23]

Stefanov; A uniqueness result for the inverse back -scattering problem

P.D. Stefanov; A uniqueness result for the inverse back -scattering problem. Inverse Problems 6 (1990), no. 6, 1055 –1064

work page 1990

[24] [24]

W. W. Symes; The seismic reﬂection inverse problem. Inv erse Problems 25 (2009), no. 12, 123008, 39 pp

work page 2009

[25] [25]

Vashisth; An inverse problems for the wave equation w ith source and receiver at distinct points, Journal of Inver se and Ill-posed Problems, http://doi.org/10.1515/jiip-20 18-0004

M. Vashisth; An inverse problems for the wave equation w ith source and receiver at distinct points, Journal of Inver se and Ill-posed Problems, http://doi.org/10.1515/jiip-20 18-0004. Beijing Computational Science Research Center, Beijing 10 0193, China. E-mail: manmohanvashisth@gmail.com

work page doi:10.1515/jiip-20