The research, carried out by Vay Siu Lo, Huy Hoang Truong, and Thien Tich Truong and funded by Vietnam National University Ho Chi Minh City, focuses on a wavemaker type that is common in small laboratories but awkward in theory. Unlike the classical piston paddle, which pushes water horizontally with well-understood linear relations between stroke and wave height, a plunger oscillates vertically, plunging into and out of the water. When the box-shaped plunger moves downward it displaces a volume of water that must go somewhere, raising the free surface in front of it; when it rises, water backfills and the surface dips. That volumetric intuition is the seed of the team’s analytical predictor.

The authors formalized this intuition using mass conservation. By equating the volume swept by the submerged plunger with the volume of water contained in the wave crest over half a wavelength, and by evaluating that wave volume with linear wave theory, they arrived at a remarkably compact expression: the generated wave height equals the plunger stroke multiplied by the wave number and the plunger side length. The wave number itself follows from the standard dispersion relation, with the wave frequency locked to the plunger’s oscillation frequency. In other words, with nothing more than the stroke amplitude, the oscillation period, the water depth, and the plunger geometry, an engineer can estimate the resulting wave height with a few lines of algebra.

Crucially, the team did not simply assert that this formula works. They verified that every operating condition they tested falls within the linear wave regime, using Le Méhauté’s classical diagram for selecting wave theories, and they validated the expression against both experiment and simulation. For the smallest stroke amplitude of 0.045 meters, the discrepancy between the analytical prediction and the CFD simulation was just 1.72 percent. As the stroke grew toward 0.15 meters, the error climbed to roughly 7 percent, a trend the authors attribute to the operating point drifting toward the boundary of the linear regime, where finite-tank reflections, viscous effects, and weak nonlinearities begin to matter. Even at the largest stroke, the simple sinusoidal theoretical waveform tracked the simulated free-surface history closely.

The experimental side of the study is a lesson in doing careful science on a modest budget. The team built a narrow flume designed to promote one-directional wave propagation and suppress three-dimensional effects, with the box-shaped plunger driven by a mechanical actuation system in prescribed harmonic motion. Free-surface elevation was measured with an HC-SR04 ultrasonic sensor paired with a lightweight floating ball constrained by vertical guides, a passive tracking arrangement that stabilizes the reading against surface ripples. The system was zero-point calibrated against the still-water level before each test, and the manufacturer-specified measurement uncertainty of plus or minus 3 millimeters was carried explicitly into the validation analysis.

On the computational side, the researchers constructed a two-dimensional numerical wave tank in ANSYS Fluent that reproduces the actual finite length of the physical facility, including the downstream oscillating water column chamber geometry. The two-phase water-air flow is captured with the volume of fluid method, the incompressible Navier-Stokes equations are solved on a pressure-based transient solver, and the plunger’s motion is imposed through a user-defined function driving a dynamic mesh with smoothing, layering, and remeshing. A mesh sensitivity study comparing grids of roughly 23,000, 39,000, and 60,000 elements showed that the medium and fine meshes produced nearly identical wave patterns, while the coarse grid introduced visible discrepancies, so the medium mesh was adopted as the practical compromise between accuracy and cost.

The validation result is striking given the deliberately realistic setup: the flume has no absorbing beach, so the simulation retained a stationary no-slip wall downstream to reproduce the reflected-wave environment of the real tank. Even so, the numerical and experimental wave signals agreed well in both amplitude and period, with a root-mean-square relative amplitude error of 6.5 percent over a 20-second steady-state window, and residual crest discrepancies traced to measurement noise and finite-tank effects, mitigated with a Kalman filter. A spectral analysis added a further layer of rigor: total harmonic distortion rose monotonically from 5.5 percent at the smallest stroke to 12.6 percent at the largest, confirming that the generated waves remained predominantly monochromatic and justifying the linear-theory machinery across the tested range.

The most forward-looking element is the physics-informed neural network, or PINN, a machine learning surrogate trained not on mountains of labeled data but on the governing equations themselves. The team formulated the problem in the frequency domain, enforcing Laplace’s equation in the fluid interior, the linearized free-surface Robin condition, the impermeable bottom, no-penetration on the plunger sides, the prescribed plunger velocity on its bottom face, and a Sommerfeld-type outgoing-wave condition at the outlet. The network, a fully connected multilayer perceptron with eight hidden layers of 96 neurons and tanh activations, was trained for 12,000 epochs with weighted collocation residuals, sampling different stroke and period combinations each iteration so that a single trained model can answer queries across the entire operating domain.

The surrogate’s performance against the validated CFD simulations is the headline number: for the four largest stroke cases the difference was below 2.14 percent, and for the largest stroke of 0.15 meters it was a mere 0.66 percent. Against the analytical predictor across both stroke-driven and period-driven cases, the mean error was about 6.5 percent with a worst case near 14 percent at the smallest amplitude. A hyperparameter sensitivity study showed the results were more sensitive to learning rate than to network depth or width, and training took between roughly four and five and a half hours on a common CPU, a modest investment compared with the transient two-phase CFD runs the surrogate is designed to replace during screening.

The payoff is a dense response surface mapping wave amplitude across the entire stroke-period plane, generated in seconds rather than by weeks of simulation, letting engineers zero in on candidate wavemaker settings before committing to high-fidelity runs or laboratory time. The authors are careful about scope: the analytical formula and the PINN are valid only within the linear regime, the surrogate models the dominant propagating wave rather than the full reflected field, and the workflow is configuration-specific rather than a new general wave theory. But the framework is extensible, with numerical quadrature able to handle higher-order wave volume terms and nonlinear constraints replaceable as training methods mature. For the coastal laboratories developing the next generation of oscillating water columns, point absorbers, and integrated wind-wave systems, this study offers something quietly transformative: a wave tank that, in effect, already knows what wave it is going to make.

Subject of Research: Calibration of plunger-type wavemaker wave generation in a numerical wave tank using experiments, CFD, analytical prediction, and a physics-informed neural network surrogate

Article Title: Experiment, simulation, analytical prediction and a PINN surrogate for generated wave analysis in a numerical wave tank with box-shaped plunger wavemaker type

Article References: Lo, V. S., Truong, H. H., & Truong, T. T. (2026). Experiment, simulation, analytical prediction and a PINN surrogate for generated wave analysis in a numerical wave tank with box-shaped plunger wavemaker type. Results in Engineering, 32, Article 113245. https://doi.org/10.1016/j.rineng.2026.113245

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113245

Keywords: numerical wave tank, plunger wavemaker, physics-informed neural network, wave energy, computational fluid dynamics, linear wave theory, volume of fluid method, oscillating water column, wave tank calibration, Le Méhauté diagram, machine learning surrogate, renewable energy