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arxiv: 2601.18382 · v2 · submitted 2026-01-26 · ⚛️ physics.app-ph

On the relation between time-reversed acoustics and Green's function retrieval in space-variant and in time-variant materials

Kees Wapenaar , Johannes Aichele , Dirk-Jan van Manen This is my paper

Pith reviewed 2026-05-16 11:08 UTC · model grok-4.3

classification ⚛️ physics.app-ph
keywords time-reversed acousticsGreen's function retrievaltime-variant materialsspace-variant materialswave equationscrosscorrelationcausality
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0 comments X

The pith

Time-reversed acoustics in time-variant materials requires emitting a sign-reversed two-component wave field from one instant, unlike the single-component boundary emission used in classical space-variant materials.

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

The paper maps the established techniques of time-reversed acoustics and Green's function retrieval from inhomogeneous time-invariant materials onto homogeneous time-variant materials. It exploits the formal similarity of the governing wave equations when the roles of time and space are interchanged, while insisting that causality remains intact in both settings and therefore blocks a complete symmetry. The resulting counterparts replace single-component fields and temporal correlations with two-component fields and spatial correlations. A reader would care because the extension opens these focusing and imaging methods to media whose properties evolve in time, such as active or modulated systems.

Core claim

Whereas classical time-reversed acoustics involves emission of a time-reversed single-component wave field from a boundary into the inhomogeneous material, its idealized counterpart involves emission of a sign-reversed two-component wave field, recorded in a time-reversed material, from a single time instant into the actual time-variant material. Likewise, classical Green's function retrieval involves temporal crosscorrelation of wave fields at two space locations in response to single-component sources on a boundary, whereas its counterpart involves spatial crosscorrelation of wave fields at two time instants in response to two-component sources at a single time instant.

What carries the argument

The partial interchange of time and space in the wave equations of time-variant versus space-variant materials, constrained by the requirement that causality holds in both classes.

If this is right

  • Green's functions between any two times in a time-variant material are obtained by spatial crosscorrelation of responses to two-component sources acting at one fixed time.
  • Focusing in time-variant materials is achieved by emitting a sign-reversed two-component field from a single instant rather than a time-reversed single-component field from a closed surface.
  • The methods apply directly to homogeneous media whose parameters change with time while preserving causality.
  • The same formal relation holds when the material is both space-variant and time-variant, provided the wave equation analogy is maintained.

Where Pith is reading between the lines

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

  • The duality suggests analogous retrieval schemes could be derived for electromagnetic or elastic waves in media with time-modulated constitutive parameters.
  • Experimental tests could use acoustic waveguides whose stiffness or density is varied in time by external controls to verify the predicted shift from temporal to spatial correlations.
  • The approach may supply a route to real-time adaptive imaging in environments whose properties drift, such as ocean water with changing temperature profiles.
  • Because the construction rests only on the wave equation and causality, the same logic may transfer to other linear wave systems without requiring new derivations.

Load-bearing premise

The wave equations for space-variant and time-variant materials remain similar enough that the same retrieval procedures can be mapped onto each other once the roles of time and space are swapped.

What would settle it

A numerical simulation in which a single-component time-reversed field is emitted into a time-variant material and fails to reconstruct the Green's function, while the corresponding sign-reversed two-component field emitted at one instant succeeds.

Figures

Figures reproduced from arXiv: 2601.18382 by Dirk-Jan van Manen, Johannes Aichele, Kees Wapenaar.

Figure 1
Figure 1. Figure 1: FIG. 1. Principle of time-reversed acoustics in an inhomogeneous, time-invariant material. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Principle of Green’s function retrieval in an inhomogeneous, time-invariant material. (a) [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. 2D Green’s function [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Snapshots of the Green’s function of Figure 3 for constant [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Cross-section of the Green’s function of Figure 3 for [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Illustration of the principle of time-reversed acoustics in a homogeneous, time-variant [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Wave field of Figure 6, shown here as a function of [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
read the original abstract

The methods of time-reversed acoustics and Green's function retrieval are traditionally deployed for classical inhomogeneous, time-invariant materials. The mutual relation between these methods is well-established. Recently, similar methods have been proposed for homogeneous, time-variant materials. Here we investigate their mutual relation and their relation with the corresponding methods in classical materials. For this analysis we make use of the fact that the wave equations for both classes of material are similar, with the roles of time and space interchanged. However, the principle of causality holds for both classes of material, hence, here the roles of time and space are not interchanged. We find that: (1) whereas classical time-reversed acoustics involves emission of a time-reversed single-component wave field from a (ideally closed) boundary into the inhomogeneous material, its idealized counterpart involves emission of a sign-reversed two-component wave field, recorded in a time-reversed material, from a single time instant into the actual time-variant material; (2) whereas classical Green's function retrieval involves temporal crosscorrelation of wave fields at two space locations in response to single-component sources on a (ideally closed) boundary, its counterpart involves spatial crosscorrelation of wave fields at two time instants in response to two-component sources at a single time instant.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript examines the relationship between time-reversed acoustics and Green's function retrieval for classical inhomogeneous time-invariant materials versus homogeneous time-variant materials. It notes the formal similarity of the governing wave equations under interchange of time and space coordinates, while emphasizing that causality is preserved in both settings and therefore prevents a complete duality. From this, the paper derives two main distinctions: classical TRA emits a time-reversed single-component field from a closed boundary, whereas the time-variant counterpart emits a sign-reversed two-component field from a single time instant; classical Green's retrieval performs temporal cross-correlation of responses to boundary sources, whereas the counterpart performs spatial cross-correlation of responses to two-component sources at one instant.

Significance. If the mapping is rigorously established, the work supplies a conceptual bridge that could guide the transfer of focusing and imaging techniques between static and dynamic media. It clarifies how causality modifies the expected symmetry and may prove useful in applied contexts such as acoustic focusing in time-varying fluids or seismic monitoring of changing subsurface conditions.

major comments (2)
  1. [Abstract, claim (1)] The central step from wave-equation similarity (with causality adjustment) to the specific claim of a sign-reversed two-component source emitted from a single time instant (claim (1)) is not accompanied by an explicit derivation or intermediate equation in the text; without this mapping the asserted distinction from classical single-component boundary emission remains unverified and load-bearing for the paper's main result.
  2. [Abstract, claim (2)] The analogous claim for Green's function retrieval (claim (2))—that spatial cross-correlation at two time instants in response to two-component sources at one instant retrieves the appropriate Green's function—likewise lacks a concrete derivation showing preservation of the retrieval property under the causality-constrained interchange; this step is required to substantiate the reported counterpart procedure.
minor comments (1)
  1. The abstract is compact; a brief parenthetical reminder of the precise form of the wave operator for each material class would help readers follow the interchange argument without consulting external references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and for recognizing the potential conceptual bridge our work provides between static and dynamic media. We agree that the two central claims in the abstract require more explicit derivations to be fully substantiated. Below we address each major comment and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract, claim (1)] The central step from wave-equation similarity (with causality adjustment) to the specific claim of a sign-reversed two-component source emitted from a single time instant (claim (1)) is not accompanied by an explicit derivation or intermediate equation in the text; without this mapping the asserted distinction from classical single-component boundary emission remains unverified and load-bearing for the paper's main result.

    Authors: We agree that the mapping from the interchanged wave equations (with causality preserved) to the precise form of the two-component, sign-reversed emission at a single time instant needs to be shown step by step. In the revised manuscript we will insert a new subsection (or expanded paragraph) immediately after the statement of the wave-equation similarity. This subsection will contain the intermediate equations that (i) interchange the roles of time and space while enforcing the causal ordering, (ii) identify the two-component field that must be emitted, and (iii) demonstrate that the emission occurs from a single time instant rather than a closed spatial boundary. We will also add a short comparison table that contrasts the classical single-component boundary emission with the time-variant counterpart. revision: yes

  2. Referee: [Abstract, claim (2)] The analogous claim for Green's function retrieval (claim (2))—that spatial cross-correlation at two time instants in response to two-component sources at one instant retrieves the appropriate Green's function—likewise lacks a concrete derivation showing preservation of the retrieval property under the causality-constrained interchange; this step is required to substantiate the reported counterpart procedure.

    Authors: We accept the referee's observation. The current text sketches the spatial cross-correlation procedure but does not derive it from the interchanged wave equation under the causality constraint. In the revision we will add an explicit derivation (new equations and accompanying text) that starts from the time-variant wave equation, applies the space-time interchange while respecting causality, and arrives at the spatial cross-correlation of the two-component responses recorded at two distinct times. This derivation will be placed in the same new subsection as the TRA derivation so that both claims are supported by parallel, fully written-out steps. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation follows from wave-equation similarity under causality constraint without reduction to fitted inputs or self-citations

full rationale

The paper's central claims (1) and (2) are presented as direct consequences of the stated similarity between the wave equations for space-variant and time-variant materials, with the explicit caveat that causality prevents a full interchange of time and space roles. No equations are shown that define a quantity in terms of itself, no parameters are fitted to data and then relabeled as predictions, and no load-bearing step reduces to a prior self-citation whose validity is assumed rather than derived. The abstract and described reasoning remain self-contained against external wave-physics benchmarks, yielding an honest non-finding of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis depends on the structural similarity of the wave equations under time-space interchange together with the preservation of causality in both material classes.

axioms (1)
  • domain assumption Wave equations for space-variant time-invariant materials and time-variant space-invariant materials are similar with time and space roles interchanged
    Invoked to relate the two classes of methods while noting that causality prevents a complete role swap

pith-pipeline@v0.9.0 · 5538 in / 1230 out tokens · 57433 ms · 2026-05-16T11:08:06.919024+00:00 · methodology

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Reference graph

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