Concrete is a quiet giant. It bears the weight of skylines, carries the nerve impulses of water and power across cities, and endures the daily grind of heat, cold, and load. Yet inside its gray face there is a restless underworld of waves, zigzagging through the cement like commuters in a crowded subway. These ultrasonic waves don’t simply travel in neat, straight lines; they scatter, bounce, and fade as they scrape past gravel, air pockets, microcracks, and the tiny imperfections that define a material’s inner life. For engineers who want to listen to a structure without tearing it apart, diffuse ultrasonic waves are the most sensitive reporters on the scene. A new study from Delft University of Technology shines a clearer light on how to read those reporters with less math and more practical sense.
Core idea The researchers set out to measure two diffusive properties of energy moving through concrete—how fast energy spreads (diffusivity) and how much energy is lost along the way (dissipation). Their twist is as simple as it is powerful: instead of solving a heavy diffusion equation that accounts for every possible reflection inside a bounded object, they model the reflections as if they were produced by a small set of mirror-image sources. In other words, boundaries behave like mirrors, and you can tally their effects with a handful of image sources rather than a sprawling, exact analytic solution. The result is a cleaner, more robust way to extract meaningful material properties from real-world tests.
Led by Hao Cheng at Delft University of Technology and conducted with collaborators from NTNU in Trondheim, the team worked with a geopolymer concrete beam. They embedded piezoelectric “smart aggregates” inside the beam to act as both shakers and receivers of ultrasonic energy, and they measured how the energy evolves after a short pulse. The aim wasn’t to perfect a theory in the abstract but to craft a practical toolkit for non-destructive assessment: if we know how diffusivity and dissipation respond to damage, temperature, or stress, we can use those signals to flag problems long before visible cracks appear. The paper’s core contribution is methodological as much as it is scientific: a simpler, robust way to estimate two key transport properties in a material that is anything but simple to model in the real world.
The paper’s authors are Hao Cheng, Katrin Löerb, Max A.N. Hendriks, and Yuguang Yang. Their institutions span the Netherlands and Norway, with the experimental work rooted in Delft University of Technology and a nod to NTNU for collaborative strength. The study sits at the intersection of civil engineering, acoustics, and materials science, and it addresses a practical bottleneck in structural health monitoring: how to quantify the codependent language of energy spreading and energy loss when the boundaries of a real structure are doing their best to complicate the message.
What this means in plain terms If you look at energy as a kind of ink spreading through a sponge, diffusivity is how quickly the ink blurs outward, and dissipation is how fast the ink leaks away as heat and friction erase the image. In a boundless block of material, this is already a subtle dance; in a real beam with top and bottom faces, internal interfaces, and layered construction, it can become a tangled, almost unreadable map. The Delft approach shows that you don’t need to re-derive the entire diffusion picture every time you have a new boundary layout. You can instead think in terms of a few reflected sources that recreate the boundary’s influence. That makes the curve fitting—matching the experimental energy evolution to a diffusion model—significantly more robust and accessible for engineers who want real-time or near-real-time insight into a structure’s condition.
Rethinking Diffusion in Concrete
In acoustics and wave physics, the diffusion equation is a trusty workhorse when waves are scattered enough to behave like a spreading fog rather than a single beam. In its classic form, it envisions energy moving through an infinite medium with a single slowing term for dissipation. But concrete structures are rarely infinite. They have edges, interfaces, and layers that throw energy back into the middle, altering both the timing and the intensity of what arrives at a detector. The traditional fix—solving a diffusion equation with specific boundary conditions—gets mathematically thorny quickly, especially for real-world shapes and layered materials. The Delft team sidesteps that complexity with a practical approximation: treat the boundary effects as if they originate from a small number of image sources that sit beyond the boundary as perfect mirrors.
Key idea The energy at a receiver can be thought of as the sum of energy from the actual ultrasonic source plus contributions from these image sources. The math stays simple, but the physics respects boundaries. In their experiments, distant image sources contributed so little that you could safely ignore them without losing fidelity in the results. That insight makes the method both fast and robust, which matters when you’re trying to monitor a structure in near real time or during a load test when conditions are changing and the signal is noisy.
One of the elegant threads in the paper is the balance between simplicity and accuracy. The analytic diffusion equation with Neumann boundary conditions (which theoretically accounts for all reflections) exists in the literature, but its complexity makes curve fitting a bit brittle in practice. By embracing a limited, physically meaningful set of image sources, the authors keep the model tractable enough to fit their measured energy curves across a range of frequencies, while still capturing the boundary’s essential influence. The result is a method that can be used by engineers who do not live in the pages of a math tome but who need reliable readings from a real concrete member.
The authors also lay out the intuition behind why this works. In the Rayleigh and stochastic scattering regimes that dominate diffuse ultrasound in concrete, energy transport can be described by diffusion-like behavior. Boundaries matter because they bounce energy back into the interior, effectively changing the path lengths that different parts of the energy take. If you can account for the most energetically important reflections, you preserve the integrity of the diffusivity and dissipation estimates without getting lost in a forest of mathematical terms that offer little practical advantage.
Boundaries, Reflections, and Image Sources
The experimental playground for the study is a geopolymer concrete beam that blends two materials in a single structure. The lower portion is a prestressed beam, and a cast-in-situ topping layer sits on top. The team embedded 12 smart aggregates along the midspan to act as both transmitters and receivers of ultrasonic energy, operating around 80 kHz. The geometry is deliberately chosen to keep the energy path realistic for a real building component while remaining small enough to manage in a controlled lab test. In total they performed 20 measurements by swapping who sends and who receives in a pair of sensors, with a careful layout that kept the sensors well away from the physical edges to minimize near-edge distortions.
Boundary reality is stubborn but manageable: the study enumerates the kinds of reflections that matter for their bottom-row sensor pair and finds that the most influential reflections come from the bottom and top surfaces and a set of secondary reflections. They show that a diffusion model that includes these reflections matches the energy evolution almost as well as a model that pretends there are no boundaries at all, yet the two yield noticeably different diffusivity and dissipation values. In concrete terms, ignoring reflections leads to underestimating how fast energy spreads and how much energy is dissipated along the way. When the researchers added the most impactful reflections into the model, the diffusivity crept higher and the dissipation rose as well—substantially enough to matter for assessing damage visibility and the confidence you place in the readings.
The practical takeaway is nuanced. You don’t need to chase every possible reflection—not all image sources contribute meaningfully to the main energy at the timescales that matter for the diffusion fit. The authors propose a thresholding rule: consider enough reflections so that the total contribution to the main energy at the maximum elapsed time is above a set threshold (they used 0.3 in their tests). This keeps the fitting robust without exploding the model with negligible terms. And because the method relies on a time window where the energy evolution is either rising due to spreading or falling due to dissipation, the researchers were careful to balance the data points in both phases so that the fit does not overemphasize one part of the curve at the expense of the other.
Another practical twist sits in the concrete’s interior. The experiment used a two-layer system with different mixtures, raising the question of whether a solid–solid interface would alter the transport of energy. The results suggest the top layer’s presence does not dramatically derail diffusivity readings in this setup, which is encouraging for real-world monitoring of composite or layered concrete structures. The authors caution that their conclusions are context-specific, but the overall message is hopeful: our tools for listening to concrete can tolerate real-world complexity without turning into a maze of bespoke equations for every project.
From Diffusion to Damage Sensing
What makes this line of work nervy exciting is its path from abstract properties to concrete applications. If you know how energy diffuses and dissipates in a piece of concrete, you can watch for changes that betray microstructural damage, crack initiation, or evolving interfaces. There are two complementary ambitions here. One is regional monitoring: by characterizing the whole region’s diffusive properties, you can track global changes in material state as loads evolve. The other is localizing disturbances: by combining diffuse wave measurements with sensitivity kernels built from the same diffusive properties, you can narrow down the location of the change on a sensor grid—a Coda Wave Interferometry style approach but grounded in the measured diffusivity and dissipation rather than assumed constants.
The authors walk through a careful signal-processing pipeline to extract the two key numbers from the data. They convert time-domain signals into a time-frequency portrait using a continuous wavelet transform with a Morlet wavelet, then they filter the spectrum with a bank of Gaussian filters to isolate frequency bands from roughly 50 kHz to 500 kHz. For each band, they compute the energy as the average squared amplitude within short time windows. The evolution of the logarithm of this ensemble-averaged energy over time becomes the playground for fitting the diffusion model. The early rise of energy encodes diffusivity, while the later, exponential-like decay encodes dissipation. The trick is to choose windows and centers that balance the signal content in both phases, so you don’t bias the fit toward the part of the curve that has more data points.
To decide which image sources to include, the researchers estimate contributions by the boundaries at the timescale of interest. They offer a kind of pragmatic scoring: if a boundary reflection’s contribution is large enough at the time when the energy peaks, you should include it. If it’s small, you can leave it out. In their tests, bottom and top surface reflections plus the secondary reflections provided a robust account of the energy evolution; front and back reflections added only marginally. This is a useful, real-world rule of thumb for practitioners who want to apply the method to different geometries without drowning in a barrage of calculations.
Stepping back, the paper’s results reinforce a simple but important intuition: the diffusive description is meaningful only if you acknowledge the energy’s many paths through a real object. The same energy that helps you see a crack can also hide in plain sight if you pretend the boundaries aren’t part of the story. The authors emphasize that, in the low- to mid-frequency regime typical for diffuse ultrasound in concrete, the diffusion equation remains a robust umbrella under which you can organize your observations. But you must account for reflections that would otherwise masquerade as the material’s intrinsic properties. If you do, the diffusivity you infer becomes a more trustworthy compass for deciding whether a beam is aging, whether a joint is decoupling, or whether a region is heating up under load.
This approach is not a universal panacea. The authors are clear that at very high frequencies, where energy might localize rather than diffuse, the diffusion model can fail. They also caution that surface-bonded sensors and multi-material interfaces require careful experimental design and post-processing to keep surface waves from muddying the body-wave signal. Still, for a broad swath of concrete structures, the method offers a practical route to quantify two vital properties that echo through the structure’s health: how fast energy spreads and how much energy is lost as it travels. And it does so using a concept that feels almost poetic in its simplicity: a handful of mirrors behind the boundary can tell you a lot about what’s happening inside a living, loaded structure.
In the end, the Delft study is a reminder that good science often travels in circles. You don’t always need a new equation; you sometimes need a smarter way to count reflections. The work of Hao Cheng and colleagues contributes a valuable, accessible tool to the growing field of structural health monitoring, one that translates the complexity of real boundaries into actionable metrics. For engineers who want to keep bridges standing, floors safe, and tunnels quiet, that straight line from theory to practice can make the difference between a quick diagnostic and a costly, unseen failure. It’s a small shift in how we listen to concrete, but it could be a big leap in how we protect the built world we rely on every day.
