🔬 materials science

On the Doppler Effect in Static Materials: Vortices and Black, White and Worm-Hole Like Wave Trajectories

This paper demonstrates that the Doppler effect can be simulated in static materials by employing a network of direction-dependent waveguides to create field patterns equivalent to moving sources, thereby enabling the design of static analogues for vortices, black holes, and wormholes.

Original authors: Tristan Lawrie, Simon Horsley

Published 2026-10-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tristan Lawrie, Simon Horsley

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Sound and light travel as waves, and when the source of those waves moves, the waves change. If a car honks its horn while driving past you, the pitch sounds higher as it approaches and lower as it recedes. This is the Doppler effect, a phenomenon caused by the motion of the source compressing the waves in front of it and stretching them out behind it. Usually, to create this effect in a laboratory, scientists must either move the source physically or make the medium through which the waves travel flow, like wind blowing through a pipe. However, moving parts are often difficult to control, and creating a steady flow of air or water at the speed of sound is rarely practical. For decades, researchers have sought a way to mimic the behavior of a moving medium without any actual movement, hoping to study complex wave behaviors in a stationary, stable environment.

In a recent study, researchers at the University of Exeter have demonstrated exactly this: a way to create the Doppler effect in a completely static material. They did not move the source, nor did they make the air or the material flow. Instead, they built a network of thin tubes, or waveguides, and carefully adjusted the length of each tube depending on which direction a wave was traveling. By making the path longer for waves moving in one direction and shorter for waves moving in the opposite direction, they tricked the waves into behaving as if they were traveling through a moving medium. The result is a stationary structure that produces the exact same wave patterns as a source speeding through space, all without a single moving part.

The researchers modeled their system as a grid of junctions connected by these tubes. In a normal, stationary grid, a wave traveling from one junction to the next takes the same amount of time regardless of direction. To simulate motion, the team split each connection into two separate paths. One path was designed to be slightly longer, and the other slightly shorter. This difference in length meant that a wave traveling "with the flow" would take less time to cross a junction, while a wave traveling "against the flow" would take more time. This time difference is mathematically identical to what happens when a medium is actually moving. The team used a computer to design these networks, calculating the precise lengths needed to mimic specific speeds of motion. They verified that the waves traveling through this static grid accumulated the correct phases, creating the compressed and stretched patterns characteristic of the Doppler effect.

To prove the concept worked, the researchers designed three distinct scenarios that are notoriously difficult to create in real life. First, they created a vortex. In a real fluid, a vortex is a swirling motion where the speed of the flow changes as you get closer to the center. In their static network, they arranged the tube lengths so that waves spiraled around a central point, mimicking the swirling motion of a fluid without any actual rotation. The waves wound around the center, creating a spiral pattern that looked exactly like a source moving in a circle, even though the entire structure was perfectly still.

Next, they built an analogue black hole. In astrophysics, a black hole is a region where gravity is so strong that nothing, not even light, can escape once it crosses a certain boundary called the horizon. In an acoustic version, this happens when a fluid flows inward faster than the speed of sound, dragging any sound waves with it. The researchers simulated this by making the tubes leading away from a central point infinitely long in their mathematical model. In practice, they replaced these infinite tubes with special absorbers that swallowed the waves. They found that waves could easily fall into this central region, but once they entered, they could not get out. The structure acted as a perfect trap for the waves, bending their paths inward just as a real black hole bends light, all while the material itself remained motionless.

Finally, they connected this black hole to a "white hole," which is the theoretical time-reverse of a black hole, a place where things can only come out and nothing can go in. By linking the two structures with a tunnel, they created a wormhole. In their simulation, a beam of waves entered the black hole, traveled through the connecting tunnel, and emerged from the white hole on the other side. The waves passed through the structure as if they had traveled through a shortcut in space, emerging from a different location than where they entered. This demonstrated that the network could not only mimic the trapping of waves but also the complex geometry of connecting two distant points in space.

The researchers were careful to note what their model could and could not do. While the network successfully reproduced the movement of waves and the trapping of energy, it did not produce Hawking radiation, the faint glow of particles that theoretical physics suggests black holes should emit. This is because the specific conditions required to generate that radiation would need parts of the network to have "negative lengths," a physical impossibility in their static setup. The study confirms that the Doppler effect and the behavior of waves in moving media can be fully replicated by simply changing the geometry of a stationary network. This approach opens the door for scientists to study complex wave phenomena, such as those found in flowing fluids or curved spacetime, using simple, static structures made of sound, light, or radio waves, without the need for complex moving machinery or time-varying materials.

Technical Summary: The Doppler Effect in Static Materials via Direction-Dependent Waveguide Lengths

Problem Statement
The Doppler effect, characterized by the compression and stretching of waves ahead of and behind a moving source, typically arises from the motion of a medium or the source itself. In classical physics, reproducing this effect requires bulk motion at appreciable fractions of the wave speed, which is often impractical. While active communities have simulated moving frames using materials with properties that vary in space and time (e.g., non-reciprocal isolation, synthetic Fresnel drag), these approaches often require temporal modulation or complex active components. The central problem addressed is how to reproduce the propagation laws of a moving frame—including the Doppler shift and associated wavefront distortions—within a completely static material, without modulating any parameters in time.

Methodology
The authors propose a static medium modeled as a network of thin waveguides (a quantum graph), where vertices act as junctions and edges act as waveguides. The core theoretical insight is that the phase accumulation of a wave in a moving frame can be replicated in a static frame by manipulating the physical length of the waveguides based on the direction of travel.

Theoretical Basis: Starting from the 1D wave equation for a source moving with velocity $v$ , the authors derive that the wavenumbers for waves traveling with ( $+$ ) and against ($-$) the flow are $\kappa_\pm = \pm k / (1 \mp M)$ , where $M = v/c$ .
Length-Velocity Equivalence: Instead of altering the wave speed or frequency, the authors demonstrate that the phase shift $\kappa_\pm \ell$ can be achieved in a static medium by assigning different lengths to the two directions of travel along an edge. The effective lengths are defined as $L_\pm = \ell / (1 \mp M)$ .
Implementation: To realize direction-dependent lengths, the network breaks time-reversal symmetry on every edge. This is achieved by routing the two directions of travel onto separate paths (e.g., using circulators, magnetically biased materials, or space-time modulated elements). The network itself remains static; only the path lengths differ.
Simulation: The authors model the system using the scattering language of quantum graph theory. They solve for the Green's function of the graph to determine the field patterns generated by point sources.

Key Contributions and Results
The paper demonstrates that a static network of waveguides can emulate complex flow dynamics and the Doppler effect through four specific examples:

Doppler Effect on a Periodic Lattice:
- By applying the length rule $L_\pm$ to a 2D square lattice, the authors reproduce the dispersion relation of a moving source.
- Result: A point source on the static lattice generates wavefronts that bunch ahead of the source and spread out behind it, exactly mimicking a source moving with Mach number $M_x = 0.6$ . The isofrequency contours are displaced, and the field pattern matches that of a moving source in a stationary medium.
Vortex Generation:
- The authors design a network where the path lengths circulate around a central core. The length difference is proportional to the angle subtended at the core, creating a phase winding.
- Result: The network generates a spiral wave pattern with $|q|$ arms, where $q$ is an integer determining the winding number. The intensity remains reciprocal (no net power circulation), but the phase winds around the core, creating a vortex analogue without physical rotation.
Analogue Black Hole:
- A "black hole" is constructed by defining edge lengths such that outward-directed lines become infinitely long (or effectively matched loads/leads) as the flow velocity approaches the wave speed ( $M \to 1$ ).
- Result: Waves entering the horizon ( $r < r_h$ ) can travel inward but cannot escape, as the outward path is effectively infinite. A beam aimed near the structure bends inward and spirals into the horizon. The authors note that while this captures waves like a black hole, it does not produce Hawking radiation, as that would require negative lengths (active elements) beyond the horizon.
Wormhole Analogue:
- By connecting the "leads" (absorbing boundaries) of the black hole to a "white hole" (the time-reverse of the black hole, where lengths are exchanged), the authors create a wormhole.
- Result: A beam entering the black hole is routed through the leads and emerges from the white hole. The network effectively folds space, allowing a wave to traverse from one region to another without traversing the intervening space in the laboratory frame.

Significance and Claims
The paper claims that the Doppler effect and the physics of moving media can be fully reproduced in a static system by trading velocity for a difference in path length. The significance of this work lies in:

Static Realization: It provides a method to study moving media and analogue gravity without requiring bulk motion or temporal modulation of material properties.
Design Flexibility: Because the velocity field can be assigned edge-by-edge, the network can emulate complex flows (vortices, horizons) that are difficult or impossible to set up in real physical media.
Universality: The results apply to any classical wave (acoustic, microwave, optical) guided by a network where time-reversal symmetry can be broken to separate forward and backward paths.
Accessibility: The approach opens the physics of moving media to researchers who can construct waveguide networks, offering a simple laboratory for studying phenomena in flowing and curved media.

The authors conclude that the construction relies on a single rule (Eq. 4) and that existing network design methods (Green's functions, inverse design) apply directly to these static, direction-dependent networks.

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